Random Oscillatory and Pulsatory Models in Elliptic Functions

Victor A. Miroshnikov

Department of Mathematics and Data Analytics, School of Natural and Mathematical Sciences, University of Mount Saint Vincent, New York, USA.

DOI: 10.4236/ajcm.2025.151002                

Abstract

To explore experimental quantization of stochastic chaos and exact wave turbulence in exponential oscillons, it is necessary to construct smooth random functions of time. In the current paper, we develop a new method of modeling stochastic variables described by a closed system of ordinary differential and algebraic equations. Primarily, oscillatory and pulsatory dynamic models produced by the first triplet of copolar elliptic functions are studied from the viewpoint of the Hamiltonian and Newtonian dynamics. Secondly, the Hamiltonian systems of the first triplet and the first triplet squared are meticulously investigated in the hyperbolic limit that results in oscillations and pulsations with rectangular and point pulses and a variable period. Thirdly, the relative Hamiltonian systems are used to develop two stochastic models of a random oscillatory cn-noise and a random pulsatory cn²-noise. Numerical experiments show that for the Bernoulli frequencies the random oscillatory cn-noise approaches a smooth random oscillatory variable with an unbounded period and the Gaussian probability distribution and the random pulsatory cn²-noise tends to a smooth random pulsatory variable with an unbounded period and the truncated Gaussian probability distribution as the number of elliptic modes approaches infinity.

Keywords

Stochastic Chaos, Exact Wave Turbulence, Experimental Quantization, Smooth Random Functions, Truncated Gaussian Probability Distribution

1. Introduction

Experimental implementation of theoretical quantization of stochastic chaos [1] and exact wave turbulence [2] [3] in exponential oscillons and pulsons requires construction of smooth random functions of time, which will give an opportunity to visualize and analyze experimental quantization, like it was done for deterministic chaos with the Fourier [4] and Bernoulli [5] sets of wave parameters.

Primarily, stochastic variables have been developed with the help of elliptic functions in [6] [7] to model spatiotemporal cascades of exposed, hidden, and dual perturbations of the Couette and Poiseuille-Hagen flows. The cascade approach results in an open-ended system of algebraic equations, where deterministic variables of basic flows are interconnected with random variables of fluid-dynamic perturbations modeling transition and intermittency.

In the current paper, we develop a novel approach to modeling stochastic variables described by a closed system of ordinary differential and algebraic equations, which are separated from deterministic variables of basic flows. The contents of this paper are as follows. In Section 2, oscillatory and pulsatory dynamic models produced by the first triplet of copolar elliptic functions are studied from the viewpoint of the Hamiltonian and Newtonian dynamics. We continue exploration of the first triplet squared in Section 3 using the hyperbolic limit that results in oscillations and pulsations with rectangular and point pulses and a variable period.

Appropriate Hamiltonian systems are used to construct two stochastic models of the random oscillatory cn-noise and the random pulsatory cn²-noise in Sections 4 and 5, respectively. Numerical experiments show that for the Bernoulli frequencies the random oscillatory cn-noise approaches a smooth random oscillatory variable with an unbounded period and the Gaussian probability distribution and the random pulsatory cn²-noise tends to a smooth random pulsatory variable with an unbounded period and the truncated Gaussian probability distribution as the number of elliptic modes approaches infinity. Section 6 contains a summary of results on the Hamiltonian systems and the stochastic models.

2. Oscillatory and Pulsatory Dynamic Models of the First Triplet

2.1. Definitions of Elliptic Functions of the First Triplet

Define $\left[f,g,h\right]\left(\tau ,\epsilon \right)$ as the first triplet of copolar elliptic functions $[\text{sn}\left(\tau ,\epsilon \right),\text{cn}\left(\tau ,\epsilon \right),$ $\text{dn}\left(\tau ,\epsilon \right)]$ , which are called elliptic sine, elliptic cosine, and elliptic dine [8], respectively,

$f\left(\tau ,\epsilon \right)=\text{sn}\left(\tau ,\epsilon \right),g\left(\tau ,\epsilon \right)=\text{cn}\left(\tau ,\epsilon \right),h\left(\tau ,\epsilon \right)=\text{dn}\left(\tau ,\epsilon \right).$ (1)

The triplet depends on time $\tau$ and elliptic modulus $\epsilon \in \left[0,1\right]$ , where the elliptic modulus and complementary modulus $\delta \in \left[0,1\right]$ are related by the Pythagorean identity as follows:

${\epsilon }^2+{\delta }^2=1.$ (2)

If $\epsilon \to 0_{+}$ , then members of the first triplet approach the trigonometric asymptotes:

$f\left(\tau ,0\right)=\text{sin}\left(\tau \right),g\left(\tau ,0\right)=\text{cos}\left(\tau \right),h\left(\tau ,0\right)=1.$ (3)

If $\epsilon \to 1_{-}$ , then members of the first triplet tend to the hyperbolic asymptotes:

$f\left(\tau ,1\right)=\tanh \left(\tau \right),g\left(\tau ,1\right)=\mathrm{sech}\left(\tau \right),h\left(\tau ,1\right)=\mathrm{sech}\left(\tau \right).$ (4)

For the aim of clarity, we will typically drop argument $\tau$ and parameter $\epsilon$ in further results, i.e. a reduced form of the definition becomes

$f=\text{sn},g=\text{cn},h=\text{dn}.$ (5)

From the viewpoint of the theory of dynamical systems, the first triplet is specified by the following system of Ordinary Differential Equations (ODEs) of the first order:

$\frac{\text{d}f}{\text{d}\tau }=gh,\frac{\text{d}g}{\text{d}\tau }=-fh,\frac{\text{d}h}{\text{d}\tau }=-{\epsilon }^{2}fg,$ (6)

where the first derivative of each member of the first triplet is proportional to a product of two comembers. So, the first triplet is complete with respect to differentiation of any order.

There are two independent algebraic relations between the squared members of the first triplet

${\text{cn}}^2\text{+}{\text{sn}}^2=1,{\epsilon }^2{\text{sn}}^2\text{+}{\text{dn}}^2=1.$ (7)

Using the independent algebraic relations and the Pythagorean identity for $\epsilon$ and $\delta ,$ we compute a table of all algebraic relations between the squared members of the first triplet

$\begin{array}{l}{f}^2+g^2=1,f^2=1-g^2,g^2=1-f^2,\\ {\epsilon }^2{f}^2+h^2=1,f^2=\frac{1-h^2}{{\epsilon }^2},h^2=1-{\epsilon }^2{f}^2,\\ -{\epsilon }^2{g}^2+h^2={\delta }^2,g^2=\frac{h^2-{\delta }^2}{{\epsilon }^2},h^2={\delta }^2+{\epsilon }^2{g}^2.\end{array}$ (8)

Therefore, only a single member from the first triplet is algebraically independent since two other members may be expressed via the single member by the above quadratic relations.

Taking the first derivative of the quadratic relations, we obtain that only a single member of the first triplet is differentially independent because

$h\frac{\text{d}h}{\text{d}\tau }={\epsilon }^{2}g\frac{\text{d}g}{\text{d}\tau }=-{\epsilon }^{2}f\frac{\text{d}f}{\text{d}\tau }=-{\epsilon }^{2}fgh,$ (9)

in accordance with the dynamical definition of the first triplet.

2.2. Polynomial Potentials of the Fourth Order

Calculating squares of the dynamical definition of the first triplet, separating variables by the quadratic relations, factoring, expanding, and collecting like terms yields the following Hamiltonian ODEs:

$K_{e,f}+P_{4,f}=E_{4,f}=0,K_{e,g}+P_{4,g}=E_{4,g}=0,K_{e,h}+P_{4,h}=E_{4,h}=0,$ (10)

where

$K_{e,f}=\frac{1}{2}{\left(\frac{\text{d}f}{\text{d}\tau }\right)}^2,K_{e,g}=\frac{1}{2}{\left(\frac{\text{d}g}{\text{d}\tau }\right)}^2,K_{e,h}=\frac{1}{2}{\left(\frac{\text{d}h}{\text{d}\tau }\right)}^2$ (11)

are kinetic energies of the Hamiltonian systems $f,g,h$ with a unit mass,

$P_{4,f}=-\frac{1}{2}\left(f-1\right)\left(f+1\right)\left(\epsilon f-1\right)\left(\epsilon f+1\right)=-\frac{{\epsilon }^2}{2}{f}^4+\frac{\left({\epsilon }^2+1\right)}{2}{f}^2-\frac{1}{2},$ (12)

$P_{4,g}=+\frac{1}{2}\left(g-1\right)\left(g+1\right)\left({\epsilon }^2{g}^2+{\delta }^2\right)=+\frac{{\epsilon }^2}{2}{g}^4+\frac{\left({\delta }^2-{\epsilon }^2\right)}{2}{g}^2-\frac{{\delta }^2}{2},$ (13)

$P_{4,h}=+\frac{1}{2}\left(h-1\right)\left(h+1\right)\left(h-\delta \right)\left(h+\delta \right)=+\frac{1}{2}{h}^4-\frac{\left({\delta }^2+1\right)}{2}{h}^2+\frac{{\delta }^2}{2}$ (14)

are even polynomial potentials of the fourth order in $f,g,h$ , respectively, and $E_{4,f},E_{4,g},E_{4,h}$ are vanishing total energies of $f,g,h$ , correspondingly.

2.3. Polynomial Forces of the Third Order

We then take the first derivative of the dynamical definition of the first triplet, separate variables by the quadratic relations, factor, expand, and collect like terms to obtain the following Newtonian ODEs:

$\frac{{\text{d}}^{2}f}{\text{d}{\tau }^2}=F_{3,f},\frac{{\text{d}}^{2}g}{\text{d}{\tau }^2}=F_{3,g},\frac{{\text{d}}^{2}h}{\text{d}{\tau }^2}=F_{3,h},$ (15)

where

$\frac{{\text{d}}^{2}f}{\text{d}{\tau }^2},\frac{{\text{d}}^{2}g}{\text{d}{\tau }^2},\frac{{\text{d}}^{2}h}{\text{d}{\tau }^2}$ (16)

are accelerations of $f,g,h$ ,

$F_{3,f}=+\left(\sqrt{2}\epsilon f-\sqrt{{\epsilon }^2+1}\right)f\left(\sqrt{2}\epsilon f+\sqrt{{\epsilon }^2+1}\right)=+2{\epsilon }^2{f}^3-\left({\epsilon }^2+1\right)f$ (17)

is an odd third-order polynomial force in $f$ ,

$F_{3,g}=-\left(\sqrt{2}\epsilon g-\sqrt{{\epsilon }^2-{\delta }^2}\right)g\left(\sqrt{2}\epsilon g+\sqrt{{\epsilon }^2-{\delta }^2}\right)=-2{\epsilon }^2{g}^3+\left({\epsilon }^2-{\delta }^2\right)g$ (18)

is an odd third-order polynomial force in $g$ for $\sqrt{2}/2\le \epsilon \le 1$ ,

$F_{3,g}=-g\left(2{\epsilon }^2{g}^2-{\epsilon }^2+{\delta }^2\right)=-2{\epsilon }^2{g}^3+\left({\epsilon }^2-{\delta }^2\right)g$ (19)

is an odd third-order polynomial force in $g$ for $0\le \epsilon \le \sqrt{2}/2$ ,

$F_{3,h}=-\left(\sqrt{2}h-\sqrt{{\delta }^2+1}\right)h\left(\sqrt{2}h+\sqrt{{\delta }^2+1}\right)=-2h^3+\left({\delta }^2+1\right)h$ (20)

is an odd third-order polynomial force in $h$ .

It is a straightforward procedure to verify that the polynomial forces of the third order are potential since they are connected with the polynomial potentials of the fourth order by the following relationships:

$F_{3,f}=-\frac{\text{d}{P}_{4,f}}{\text{d}f},F_{3,g}=-\frac{\text{d}{P}_{4,g}}{\text{d}g},F_{3,h}=-\frac{\text{d}{P}_{4,h}}{\text{d}h}.$ (21)

2.4. Static Visualizations

Polynomial force $F_{3,f}$ , polynomial potential $P_{4,f}$ , and the first Hamiltonian system $f$ are shown in Figure 1(a), Figure 1(b), and Figure 1(c), respectively, for various values of $\epsilon$ . Blue solid curves correspond to a hyperbolic value of $\epsilon ={\epsilon }_h$ , black dotted curves to a critical value of $\epsilon ={\epsilon }_c$ , and green dashed curves to a trigonometric value of $\epsilon ={\epsilon }_t$ , where

${\epsilon }_t=0.0000001,{\epsilon }_c=\sqrt{2}/2=0.7071070,{\epsilon }_h=\mathrm{0.9999999.}$ (22)

(a)

(b)

(c)

(d)

Figure 1. Plots of the first Hamiltonian system: (a) $-F_{3,f}$ , (b) $-P_{4,f}$ ,(c) $-f$ for various values of $\epsilon :{\epsilon }_t,{\epsilon }_c,{\epsilon }_h$ , (d) the last frame of the animated 3-d Hamiltonian map of $f\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{4,f}\left(f,\tau ,{\epsilon }_h\right)$ .

The correspondent values of the complete elliptic integral of the first kind $K=K\left(\epsilon \right)$ [8] are following:

$K_t=K\left({\epsilon }_t\right)=1.5707963,K_c=K\left({\epsilon }_c\right)=1.8540747,K_h=K\left({\epsilon }_h\right)=\mathrm{9.0987690.}$ (23)

In Figure 1(c), $f$ is shown using the green dashed curve and the black dotted curve on domains $\left[-K_t,5K_t\right]$ and $\left[-K_c,5K_c\right]$ , respectively.

Calculation of three zeros of $F_{3,f}$ gives

$f_{F,1}=-\frac{\sqrt{2\left({\epsilon }^2+1\right)}}{2\epsilon },f_{F,2}=0,f_{F,3}=+\frac{\sqrt{2\left({\epsilon }^2+1\right)}}{2\epsilon }.$ (24)

As $\epsilon$ varies from 0 to 1, $f_{F,1}$ moves from −∞ to −1, $f_{F,2}$ is stationary, and $f_{F,3}$ moves from +∞ to +1 since $F_{3,f}$ transforms from a linear polynomial $-f$ into the cubic polynomial.

In agreement with the first derivative of the polynomial potentials, zeros of $F_{3,f}$ correspond to extrema of $P_{4,f}$ . Namely, there are the first local maximum, the local minimum, and the second local maximum that are given by

$P_{4,f,\max ,1}=\frac{{\left({\epsilon }^2-1\right)}^2}{8{\epsilon }^2},P_{4,f,\min }=-\frac{1}{2},P_{4,f,\max ,2}=\frac{{\left({\epsilon }^2-1\right)}^2}{8{\epsilon }^2}.$ (25)

As $\epsilon$ varies from 0 to 1, $P_{4,f,\max ,1}$ at $f=f_{F,1}$ and $P_{4,f,\max ,2}$ at $f=f_{F,3}$ change from ∞ to 0 and $P_{4,f,\min }$ at $f=f_{F,2}$ remains constant since $P_{4,f}$ transforms from a quadratic polynomial $\left(f^2-1\right)/2$ into the fourth-order polynomial.

Four zeros of $P_{4,f}$ are computed as follows:

$f_1=-\frac{1}{\epsilon },f_2=-1,f_3=+1,f_4=+\frac{1}{\epsilon }.$ (26)

As $\epsilon$ varies from 0 to 1, $f_1$ moves from -∞ to -1, $f_2=f_{\min }=-1$ and $f_3=f_{\max }=+1$ are steady, and $f_4$ moves from +∞ to +1 since $P_{4,f}$ alters the order from two to four.

Depth of the sn-potential well

$d_{4,f}=P_{4,f,\max ,2}-P_{4,f,\min }=\frac{{\left({\epsilon }^2+1\right)}^2}{8{\epsilon }^2}$ (27)

and width of the sn-potential well

$w_{4,f}=f_{F,3}-f_{F,1}=\frac{\sqrt{2\left({\epsilon }^2+1\right)}}{\epsilon }.$ (28)

Thus, the first Hamiltonian system $f\in \left[f_{\min ,}{f}_{\max }\right]=\left[-1,+1\right]$ is locked in the sn-potential well $\left[f_{F,1},f_{F,3}\right]$ . The vanishing total energy $E_{4,f}$ of $f$ is indicated in Figure 1(b) by a space dashed magenta line and the ultimate positions of $f$ are specified by magenta dots. As $\epsilon \to 1_{-}$ , $f$ approaches the nonlinear sn-oscillation with positive pulses of a rectangular shape for $4nK\left(\epsilon \right)\le \tau \le 2K\left(\epsilon \right)+4nK\left(\epsilon \right)$ and negative pulses of a rectangular shape for $2K\left(\epsilon \right)+4nK\left(\epsilon \right)\le \tau \le 4K\left(\epsilon \right)+4nK\left(\epsilon \right)$ , where $n=0,\pm 1,\pm 2,\cdots$ is an integer. For $\epsilon ={\epsilon }_h$ , the shape of rectangular pulses coincides with the graph accuracy with $\tanh \left(\tau \right)$ for $-K\left(\epsilon \right)+4nK\left(\epsilon \right)\le \tau$ $\le K\left(\epsilon \right)+4nK\left(\epsilon \right)$ and with $-\tanh \left(\tau \right)$ for $K\left(\epsilon \right)+4nK\left(\epsilon \right)\le \tau \le 3K\left(\epsilon \right)+4nK\left(\epsilon \right)$ .

The period of the sn-oscillation is $4K\left(\epsilon \right)$ since periodic minimums $f_{\min }=f\left({\tau }_{\min }\right)=-1$ and periodic maximums $f_{\max }=f\left({\tau }_{\max }\right)=+1$ are reached at

${\tau }_{\min }=-K\left(\epsilon \right)+4nK\left(\epsilon \right),{\tau }_{\max }=+K\left(\epsilon \right)+4nK\left(\epsilon \right).$ (29)

The zeros of $f$ are periodic points of inflection $f_{\inf }=f\left({\tau }_{\inf }\right)=f_{F,2}=0$ , which the first Hamiltonian system attains at

${\tau }_{\inf }=2nK\left(\epsilon \right).$ (30)

Polynomial force $F_{3,g}$ , polynomial potential $P_{4,g}$ , and the second Hamiltonian system $g$ are visualized in Figure 2(a), Figure 2(b), and Figure 2(c), correspondingly.

Calculation of three zeros of $F_{3,g}$ gives

$g_{F,1}=-\frac{\sqrt{2\left({\epsilon }^2-{\delta }^2\right)}}{2\epsilon },g_{F,2}=0,g_{F,3}=+\frac{\sqrt{2\left({\epsilon }^2-{\delta }^2\right)}}{2\epsilon }.$ (31)

As $\epsilon$ varies from 0 to 1, $g_{F,1}$ moves from $-\text{I}\infty$ to $-\sqrt{2}/2$ , $g_{F,2}$ is stationary, and $g_{F,3}$ moves from $+\text{I}\infty$ to $+\sqrt{2}/2$ since $F_{3,g}$ transforms from a linear polynomial $-g$ into the cubic polynomial, where $\text{I}$ is the imaginary unit. All imaginary roots become real for $\epsilon ={\epsilon }_c$ as $g_{F,1}=g_{F,2}=g_{F,3}=0$ .

In the view of the first derivative of the polynomial potentials, zeros of $F_{3,g}$ produce extrema of $P_{4,g}$ . Specifically, the first local minimum, the local maximum, and the second local minimum are computed by

(a)

(b)

(c)

(d)

Figure 2. Plots of the second Hamiltonian system: (a) $-F_{3,g}$ , (b) $-P_{4,g}$ , (c) $-g$ for various values of $\epsilon :{\epsilon }_t,{\epsilon }_c,{\epsilon }_h$ , (d) the last frame of the animated 3-d Hamiltonian map of $g\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{4,g}\left(g,\tau ,{\epsilon }_h\right)$ .

$P_{4,g,\min ,1}=-\frac{1}{8{\epsilon }^2},P_{4,g,\max }=\frac{{\epsilon }^2-1}{2},P_{4,g,\min ,2}=-\frac{1}{8{\epsilon }^2}.$ (32)

As $\epsilon$ varies from ${\epsilon }_c$ to 1, $P_{4,g,\min ,1}$ at $g=g_{F,1}$ and $P_{4,g,\min ,2}$ at $g=g_{F,3}$ change from -1/4 to -1/8 and $P_{4,g,\max }$ at $g=g_{F,2}$ changes from -1/4 to 0 because $P_{4,g}$ transforms from a quadratic polynomial $\left(g^2-1\right)/2$ for $\epsilon =0$ into the fourth-order polynomial for $\epsilon =1$ , while $P_{4,g,\min ,1}=P_{4,g,\max }=P_{4,g,\min ,2}=-1/4$ when $\epsilon ={\epsilon }_c$ as $P_{4,g}$ becomes $\left(g^4-1\right)/4$ .

Four zeros of $P_{4,g}$ are calculated in the following form:

$g_1=-1,g_2=-\frac{\text{I}\sqrt{1-{\epsilon }^2}}{\epsilon },g_3=+\frac{\text{I}\sqrt{1-{\epsilon }^2}}{\epsilon },g_4=+1.$ (33)

As $\epsilon$ varies from 0 to 1, $g_1$ and $g_4$ are stationary, $g_2$ moves from $-\text{I}\infty$ to 0, and $g_3$ from $+\text{I}\infty$ to 0, whereas $g_2=-\text{I}$ and $g_3=+\text{I}$ when $\epsilon ={\epsilon }_c$ because $P_{4,g}$ transforms into $\left(g^2-1\right)\left(g^2+1\right)/4$ .

Depths of the cn-potential well at the locations of the local extrema $g_{F,1}$ , $g_{F,2}$ , and $g_{F,3}$ of $P_{4,g}$ for $\epsilon >{\epsilon }_c$ are

$d_{4,g,1}=-P_{4,g,\min ,1}=\frac{1}{8{\epsilon }^2},d_{4,g,2}=-P_{4,g,\max }=\frac{1-{\epsilon }^2}{2},d_{4,g,3}=-P_{4,g,\min ,2}=\frac{1}{8{\epsilon }^2},$ (34)

respectively. A width of the cn-potential well does not depend on $\epsilon$ since

$w_{4,g}=g_4-g_1=2.$ (35)

The second Hamiltonian system $g\in \left[g_{\min ,}{g}_{\max }\right]=\left[-1,+1\right]$ oscillates in the cn-potential well $\left[g_1,g_4\right]$ . The vanishing total energy $E_{4,g}$ of $g$ is visualized in Figure 2(b) by a space dashed magenta line and the ultimate positions of $g$ are shown by magenta dots. As $\epsilon \to 1_{-}$ , $g$ approaches the nonlinear cn-oscillation with positive point pulses for $-K\left(\epsilon \right)+4nK\left(\epsilon \right)\le \tau \le K\left(\epsilon \right)+4nK\left(\epsilon \right)$ and negative point pulses for $K\left(\epsilon \right)+4nK\left(\epsilon \right)\le \tau \le 3K\left(\epsilon \right)+4nK\left(\epsilon \right)$ . For $\epsilon ={\epsilon }_h$ , the shape of point pulses coincides with the graph accuracy with $\mathrm{sech}\left(\tau \right)$ for $-K\left(\epsilon \right)+4nK\left(\epsilon \right)\le \tau \le K\left(\epsilon \right)+4nK\left(\epsilon \right)$ and with $-\mathrm{sech}\left(\tau \right)$ for $K\left(\epsilon \right)+4nK\left(\epsilon \right)\le \tau \le 3K\left(\epsilon \right)+4nK\left(\epsilon \right)$ , which model the Dirac delta function.

The period of the cn-oscillation is $4K\left(\epsilon \right)$ because periodic minimums $g_{\min }=g\left({\tau }_{\min }\right)=-1$ and periodic maximums $g_{\max }=g\left({\tau }_{\max }\right)=+1$ are attained at

${\tau }_{\min }=-2K\left(\epsilon \right)+4nK\left(\epsilon \right),{\tau }_{\max }=4nK\left(\epsilon \right).$ (36)

The zeros of $g$ are again periodic points of inflection $g_{\inf }=g\left({\tau }_{\inf }\right)=g_{F,2}=0$ , which $g$ reaches at

${\tau }_{\inf }=K\left(\epsilon \right)+2nK\left(\epsilon \right).$ (37)

Polynomial force $F_{3,h}$ , polynomial potential $P_{4,h}$ , and the third Hamiltonian system $h$ are displayed in Figure 3(a), Figure 3(b), and Figure 3(c), sequentially.

We then calculate three zeros of $F_{3,h}$ and obtain

$h_{F,1}=-\frac{\sqrt{2\left(2-{\epsilon }^2\right)}}{2},h_{F,2}=0,h_{F,3}=+\frac{\sqrt{2\left(2-{\epsilon }^2\right)}}{2}.$ (38)

As $\epsilon$ varies from 0 to 1, $h_{F,1}$ moves from −1 to $-\sqrt{2}/2$ , $h_{F,2}$ is stationary, and $h_{F,3}$ moves from +1 to $+\sqrt{2}/2$ because $F_{3,h}$ remains the cubic polynomial for all $\epsilon$ .

In the accordance with the first derivative of the polynomial potentials, zeros of $F_{3,h}$ correspond to extrema of $P_{4,h}$ . Precisely, the first local minimum, the local maximum, and the second local minimum are given by

$P_{4,h,\min ,1}=-\frac{{\epsilon }^4}{8},P_{4,h,\max }=\frac{1-{\epsilon }^2}{2},P_{4,h,\min ,2}=-\frac{{\epsilon }^4}{8}.$ (39)

As $\epsilon$ varies from 0 to 1, $P_{4,h,\min ,1}$ at $h=h_{F,1}$ and $P_{4,h,\min ,2}$ at $h=h_{F,3}$ change from 0 to −1/8 and $P_{4,h,\max }$ at $h=h_{F,2}$ changes from 1/2 to 0 since $P_{4,h}$ is the fourth-order polynomial for all $\epsilon .$

Four zeros of $P_{4,h}$ are computed in the following form:

(a)

(b)

(c)

(d)

Figure 3. Plots of the third Hamiltonian system: (a) $-F_{3,h}$ , (b) $-P_{4,h}$ , (c) $-h$ for various values of $\epsilon :{\epsilon }_t,{\epsilon }_c,{\epsilon }_h$ , (d) the last frame of the animated 3-d Hamiltonian map of $h\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{4,h}\left(h,\tau ,{\epsilon }_h\right)$ .

$h_1=-1,h_2=-\sqrt{1-{\epsilon }^2},h_3=+\sqrt{1-{\epsilon }^2},h_4=+1.$ (40)

As $\epsilon$ varies from 0 to 1, $h_1$ and $h_4$ are stationary, $h_2$ moves from −1 to 0, and $h_3$ from +1 to 0 as $P_{4,h}$ has the same order for all $\epsilon$ .

Depth of the dn-potential well

$d_{4,h}=P_{4,h,\max }-P_{4,h,\min ,2}=\frac{{\left({\epsilon }^2-2\right)}^2}{8}.$ (41)

Width of the dn-potential well

$w_{4,h}=h_w=\sqrt{2-{\epsilon }^2}$ (42)

is a solution of the following algebraic equation:

$P_{4,h}-P_{4,h,\max }-\frac{h^2\left(h^2-h_w^2\right)}{2}=0.$ (43)

Contrary to the cn-potential well in Figure 2(b), potential sub-wells in Figure 3(b) are separated since $P_{4,h,\max }=+\left(1-{\epsilon }^2\right)/2>0$ and $P_{4,g,\max }=-\left(1-{\epsilon }^2\right)/2<0$ at the origin. Therefore, a periodic motion of the third Hamiltonian system $h\in \left[h_{\min },h_{\max }\right]=\left[\sqrt{1-{\epsilon }^2},+1\right]$ is bounded in the dn-potential well $\left[0,h_w\right]$ . The total vanishing energy $E_{4,h}$ of $h$ is displayed in Figure 3(b) using a space dashed magenta line and the ultimate positions of $h$ are visualized by magenta dots. As $\epsilon \to 1_{-}$ , $h$ tends to the nonlinear dn-pulsation with only positive point pulses for $-K\left(\epsilon \right)+2nK\left(\epsilon \right)\le \tau \le +K\left(\epsilon \right)+2nK\left(\epsilon \right)$ . For $\epsilon ={\epsilon }_h$ , the shape of point pulses co-incides with the graph accuracy with $\mathrm{sech}\left(\tau \right)$ for $-K\left(\epsilon \right)+2nK\left(\epsilon \right)\le \tau \le +K\left(\epsilon \right)+2nK\left(\epsilon \right)$ .

The period of the dn-pulsation is $2K\left(\epsilon \right)$ since periodic minimums $h_{\min }=h\left({\tau }_{\min }\right)=\sqrt{1-{\epsilon }^2}$ and periodic maximums $h_{\max }=h\left({\tau }_{\max }\right)=1$ are attained at

${\tau }_{\min }=-K\left(\epsilon \right)+2nK\left(\epsilon \right),{\tau }_{\max }=2nK\left(\epsilon \right).$ (44)

The third Hamiltonian system does not have zeros since $h$ is strictly positive.

2.5. Dynamic Visualizations

From the viewpoint of the Hamiltonian dynamics, a Hamiltonian system of the first triplet

$\varphi =\left[f,g,h\right]$ (45)

is a solution of the Hamiltonian ODE

$K_{e,\varphi }+P_{4,\varphi }=E_{4,\varphi }=\frac{1}{2}{\left(\frac{\text{d}\varphi }{\text{d}\tau }\right)}^2+{\sigma }_4{\varphi }^{{}^4}+{\sigma }_2{\varphi }^{{}^2}+{\sigma }_0=0,$ (46)

where $K_{e,\varphi }$ is the kinetic energy, $P_{4,\varphi }$ is the even polynomial potential in $\varphi$ of the fourth order, total energy $E_{4,\varphi }=0$ , and polynomial coefficients of $f,g,h$ are

$\left[{\sigma }_4,{\sigma }_2,{\sigma }_0\right]=\left[-{\epsilon }^2/2,\left({\epsilon }^2+1\right)/2,-1/2\right],$ (47)

$\left[{\sigma }_4,{\sigma }_2,{\sigma }_0\right]=\left[+{\epsilon }^2/2,\left({\delta }^2-{\epsilon }^2\right)/2,-{\delta }^2/2\right],$ (48)

$\left[{\sigma }_4,{\sigma }_2,{\sigma }_0\right]=\left[+1/2,-\left({\delta }^2+1\right)/2,+{\delta }^2/2\right],$ (49)

respectively.

From the perspective of the Newtonian dynamics, $\varphi$ represents a solution of the Newtonian ODE

$\frac{{\text{d}}^2\varphi }{\text{d}{\tau }^2}=F_{3,\varphi }={\omega }_3{\varphi }^3+{\omega }_1\varphi ,$ (50)

where the second derivative of $\varphi$ is the acceleration, $F_{3,\varphi }$ is the odd polynomial force in $\varphi$ of the third order, and polynomial coefficients of $f,g,h$ are

$\left[{\omega }_3,{\omega }_1\right]=\left[+2{\epsilon }^2,-\left({\epsilon }^2+1\right)\right],$ (51)

$\left[{\omega }_3,{\omega }_1\right]=\left[-2{\epsilon }^2,+\left({\epsilon }^2-{\delta }^2\right)\right],$ (52)

$\left[{\omega }_3,{\omega }_1\right]=\left[-2,+\left({\delta }^2+1\right)\right],$ (53)

correspondingly.

The polynomial force is connected with the polynomial potential by the following relations:

$F_{3,\varphi }=-\frac{\text{d}{P}_{4,\varphi }}{\text{d}\varphi },{\omega }_3=-4{\sigma }_4,{\omega }_1=-2{\sigma }_2.$ (54)

The dynamical problems for the Hamiltonian or Newtonian ODEs are subjected to the following initial conditions for $f,g,h$ , respectively:

${f|}_{\tau =0}=0,{\frac{\text{d}f}{\text{d}\tau }|}_{\tau =0}=1,$ (55)

${g|}_{\tau =0}=1,{\frac{\text{d}g}{\text{d}\tau }|}_{\tau =0}=0,$ (56)

${h|}_{\tau =0}=1,{\frac{\text{d}h}{\text{d}\tau }|}_{\tau =0}=0.$ (57)

Solutions of the Hamiltonian or Newtonian ODEs with these initial conditions provide unique solutions for the first triplet.

Figure 1(b) and Figure 1(c) are integrated for $\epsilon ={\epsilon }_h$ in a three-dimensional (3-d) Hamiltonian map animated on $\tau \in \left[-K_h,3K_h\right]$ , the last frame of which is shown in Figure 1(d). A 3-d trajectory of the first Hamiltonian system $f\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{4,f}\left(f,\tau ,{\epsilon }_h\right)$ is indicated by a black curve. A two-dimensional (2-d) projection of the 3-d trajectory on plane $\left[f,\tau \right]$ is visualized in Figure 1(d) by a blue dashed curve, which coincides with the 2-d trajectory of $f$ displayed by the blue solid curve in Figure 1(c).

The periodic 3-d trajectory of $f$ starts on the first potential barrier at $f=f_{\min }$ when $\tau ={\tau }_{\min }=-K_h+4n{K}_h$ and $K_{e,f}=-P_{4,f}=0$ . As $P_{4,f}$ decreases towards the bottom of the sn-potential well, kinetic energy, $K_{e,f}=-P_{4,f}>0$ , which determines the magnitude of system’s velocity, initially increases and then decreases when the system transits in time interval $\tau \approx \left[-0.25K_h+4n{K}_h,+0.25K_h+4n{K}_h\right]$ from the first potential barrier to the second one at $f=f_{\max }$ . The kinetic energy reaches its maximal value $K_{e,f}=-P_{4,f}=1/2$ at the bottom of the sn-potential well at $f=f_{\inf }$ when $\tau ={\tau }_{\inf }=2n{K}_h$ . The first Hamiltonian system decelerates while moving along the potential barrier since $K_{e,f}\approx 0$ in time interval $\tau \approx [0.25K_h+4n{K}_h,1.75K_h+4n{K}_h]$ , while $K_{e,f}=-P_{4,f}=0$ at $f=f_{\max }$ when $\tau ={\tau }_{\max }=+K_h+4n{K}_h$ . Analogously, $f$ primarily accelerates and sequentially decelerates in the sn-potential well and, finally, returns to the first potential barrier when $\tau \approx \left[1.75K_h+4n{K}_h,2.25K_h+4n{K}_h\right]$ . The periodic 3-d trajectory finishes at $f=f_{\min }$ when $\tau ={\tau }_{\min }=3K_h+4n{K}_h$ . The described details of the 3-d Hamiltonian map are clearly visualized in the animated Figure 1(d).

In Figure 2(d), the last frame of the animated 3-d Hamiltonian map on $\tau \in \left[-K_h,3K_h\right]$ combines for $\epsilon ={\epsilon }_h$ the 2-d polynomial potential in Figure 2(b) and the 2-d trajectory in Figure 2(c). A black curve shows a 3-d trajectory of the second Hamiltonian system $g\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{4,g}\left(g,\tau ,{\epsilon }_h\right)$ and a blue dashed curve in Figure 2(d), which coincides with the 2-d trajectory of $g$ displayed by the blue solid curve in Figure 2(c), presents a 2-d projection of the 3-d trajectory on plane $\left[g,\tau \right]$ .

The periodic 3-d trajectory of $g$ begins on the crest of the potential barrier at $g=g_{\inf }$ when $\tau ={\tau }_{\inf }=-K_h+2n{K}_h$ and $K_{e,g}=-P_{4,g}=\left(1-{\epsilon }^2\right)/2$ . As $P_{4,g}$ decreases in the first sub-well, where $g\in \left[0,1\right]$ , kinetic energy, $K_{e,g}=-P_{4,g}>0$ , initially increases towards the bottom of the first sub-well at $g=g_{F,3}$ and then decreases in time interval $\tau \approx \left[-0.50K_h+4n{K}_h,4n{K}_h\right]$ . After reflection by the potential wall at $g=g_{\max }$ when $\tau ={\tau }_{\max }=4n{K}_h$ and $K_{e,g}=-P_{4,g}=0$ , $g$ again primarily accelerates and sequentially decelerates towards the potential barrier for $\tau \approx \left[4n{K}_h,0.50K_h+4n{K}_h\right]$ , finishing this positive pulsation on the crest of the potential barrier at $g=g_{\inf }$ when $\tau ={\tau }_{\inf }=K_h+2n{K}_h$ . The second Hamiltonian system then decelerates while moving along the potential barrier since $K_{e,g}\approx 0$ for $\tau \approx \left[0.50K_h+2n{K}_h,1.50K_h+2n{K}_h\right]$ . Similarly to the positive pulsation, $g$ consequently makes the negative pulsation in the second sub-well, where $g\in \left[-1,0\right]$ , when $\tau \approx \left[1.50K_h+4n{K}_h,2.50K_h+4n{K}_h\right]$ due to the reflection from the potential wall at $g=g_{\min }$ when $\tau ={\tau }_{\min }=2K_h+4n{K}_h$ and $K_{e,g}=-P_{4,g}=0$ . The periodic 3-d trajectory terminates at $g=g_{\inf }$ when $\tau ={\tau }_{\inf }=3K_h+2n{K}_h$ . The specified features of the 3-d Hamiltonian map are unambiguously displayed in the animated Figure 2(d).

In Figure 3(d), the last frame of the 3-d Hamiltonian map animated on $\tau \in \left[-K_h,+K_h\right]$ synthesizes for $\epsilon ={\epsilon }_h$ Figure 3(b), which contains the 2-d polynomial potential, with Figure 3(c), which displays the 2-d trajectory. A black curve in Figure 3(d) shows a 3-d trajectory of the third Hamiltonian system $h\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{4,h}\left(h,\tau ,{\epsilon }_h\right)$ . A 2-d projection of the 3-d trajectory on plane $\left[h,\tau \right]$ is visualized in Figure 3(d) by a blue dashed curve, which coincides with the 2-d trajectory of $h$ visualized with the help of the blue solid curve in Figure 3(c).

Similar to the positive pulsation in Figure 2(d), the periodic 3-d trajectory of $h$ starts on the potential barrier at $h=h_{\min }$ when $\tau ={\tau }_{\min }=-K_h+2n{K}_h$ and $K_{e,h}=-P_{4,h}=0$ . As $P_{4,h}$ decreases in the dn-potential well, kinetic energy, $K_{e,h}=-P_{4,h}>0$ , initially increases towards the bottom of the dn-potential well and then decreases in time interval $\tau \approx \left[-0.50K_h+2n{K}_h,2n{K}_h\right]$ . The kinetic energy reaches its maximal value $K_{e,h}=-P_{4,h}={\epsilon }^4/8$ at the bottom of the dn-potential well at $h=h_{F,3}$ . The third Hamiltonian system is then reflected by the potential wall at $h=h_{\max }$ when $\tau ={\tau }_{\max }=2n{K}_h$ and $K_{e,h}=-P_{4,h}=0$ . After the reflection, $h$ once more primarily accelerates and sequentially decelerates towards the potential barrier for $\tau \approx \left[2n{K}_h,+0.50K_h+2n{K}_h\right]$ , terminating this positive pulsation at the potential barrier. After the positive pulsation, $h$ decelerates along the potential barrier when $\tau \approx \left[+0.50K_h+2n{K}_h,+1.50K_h+2n{K}_h\right]$ because then $K_{e,h}\approx 0$ , whereas $K_{e,h}=-P_{4,h}=0$ on the potential barrier at the end of the periodic 3-d trajectory for $h=h_{\min }$ and $\tau ={\tau }_{\min }=K_h+2n{K}_h$ . The mentioned aspects of the 3-d Hamiltonian map are obviously represented in the animated Figure 3(d).

3. Pulsatory Dynamic Models of the First Triplet Squared

3.1. Definitions of Elliptic Functions of the First Triplet Squared

We specify $\left[p,q,r\right]\left(\tau ,\epsilon \right)$ as the first triplet of copolar elliptic functions squared $[{\text{sn}}^2\left(\tau ,\epsilon \right),$ ${\text{cn}}^2\left(\tau ,\epsilon \right),$ ${\text{dn}}^2\left(\tau ,\epsilon \right)]$ , correspondingly,

$p\left(\tau ,\epsilon \right)={\text{sn}}^2\left(\tau ,\epsilon \right),q\left(\tau ,\epsilon \right)={\text{cn}}^2\left(\tau ,\epsilon \right),r\left(\tau ,\epsilon \right)={\text{dn}}^2\left(\tau ,\epsilon \right).$ (58)

In agreement with the definition of the first triplet,

$p\left(\tau ,\epsilon \right)=f^2\left(\tau ,\epsilon \right),q\left(\tau ,\epsilon \right)=g^2\left(\tau ,\epsilon \right),r\left(\tau ,\epsilon \right)=h^2\left(\tau ,\epsilon \right).$ (59)

If $\epsilon \to 0_{+}$ , then trigonometric asymptotes of members of the first triplet squared become:

$p\left(\tau ,0\right)={\text{sin}}^2\left(\tau \right),q\left(\tau ,0\right)={\text{cos}}^2\left(\tau \right),r\left(\tau ,0\right)=1.$ (60)

If $\epsilon \to 1_{-}$ , then hyperbolic asymptotes of members of the first triplet squared are

$p\left(\tau ,1\right)={\tanh }^2\left(\tau \right),q\left(\tau ,1\right)={\mathrm{sech}}^2\left(\tau \right),r\left(\tau ,1\right)={\mathrm{sech}}^2\left(\tau \right).$ (61)

For the aim of conciseness, we will further omit argument $\tau$ and parameter $\epsilon$ in computed results, viz. a simplified form of the definition of the first triplet squared is

$p=f^2,q=g^2,r=h^2.$ (62)

In agreement with the theory of dynamical systems, the triplet squared is determined by the following system of ODEs of the first order:

$\frac{\text{d}p}{\text{d}\tau }=2fgh,\frac{\text{d}q}{\text{d}\tau }=-2fgh,\frac{\text{d}r}{\text{d}\tau }=-2{\epsilon }^{2}fgh,$ (63)

where the first derivative of each member of the triplet squared is proportional to a product of all members of the triplet. Contrary to the first triplet, the first triplet squared is opened with respect to differentiation of the first order.

With the help of the identities for the squared members of the first triplet and the squares of $\epsilon$ and $\delta$ , a table of all algebraic relations between members of the first triplet squared takes the following form:

$\begin{array}{l}p+q=1,p=1-q,q=1-p,\\ {\epsilon }^{2}p+r=1,p=\frac{1-r}{{\epsilon }^2},r=1-{\epsilon }^{2}p,\\ -{\epsilon }^{2}q+r={\delta }^2,q=\frac{r-{\delta }^2}{{\epsilon }^2},r={\delta }^2+{\epsilon }^{2}q.\end{array}$ (64)

Consequently, only a single member from the first triplet squared is algebraically independent since two other members may be computed in terms of the single member using the above linear relations.

We take the first derivative of the linear relations to show that only a single member of the first triplet squared is differentially independent since

$\frac{\text{d}r}{\text{d}\tau }={\epsilon }^2\frac{\text{d}q}{\text{d}\tau }=-{\epsilon }^2\frac{\text{d}p}{\text{d}\tau }=-2{\epsilon }^{2}fgh,$ (65)

in agreement with the dynamical definition of the first triplet squared.

3.2. Polynomial Potentials of the Third Order

We then compute squares of the dynamical definition of the first triplet squared, separate variables by the linear relations, expand, and collect like terms to find the following Hamiltonian ODEs:

$K_{e,p}+P_{3,p}=E_{3,p}=0,K_{e,q}+P_{3,q}=E_{3,q}=0,K_{e,r}+P_{3,r}=E_{3,r}=0,$ (66)

where

$K_{e,p}=\frac{1}{2}{\left(\frac{\text{d}p}{\text{d}\tau }\right)}^2,K_{e,q}=\frac{1}{2}{\left(\frac{\text{d}q}{\text{d}\tau }\right)}^2,K_{e,r}=\frac{1}{2}{\left(\frac{\text{d}r}{\text{d}\tau }\right)}^2$ (67)

are kinetic energies of the Hamiltonian systems $p,q,r$ with a unit mass,

$P_{3,p}=-2\left(p-1\right)\left({\epsilon }^{2}p-1\right)p=-2{\epsilon }^2{p}^3+2\left({\epsilon }^2+1\right)p^2-2p,$ (68)

$P_{3,q}=+2\left(q-1\right)q\left({\epsilon }^{2}q+{\delta }^2\right)=+2{\epsilon }^2{q}^3+2\left({\delta }^2-{\epsilon }^2\right)q^2-2{\delta }^{2}q,$ (69)

$P_{3,r}=+2\left(r-1\right)\left(r-{\delta }^2\right)r=+2r^3-2\left({\delta }^2+1\right)r^2+2{\delta }^{2}r$ (70)

are polynomial potentials of the third order in $p,q,r$ , correspondingly, and $E_{3,p},E_{3,q},E_{3,r}$ are vanishing total energies of $p,q,r$ , respectively. Therefore, the first triplet squared becomes closed with respect to differentiation starting from the first-order derivatives squared.

3.3. Polynomial Forces of the Second Order

Taking the first derivative of the dynamical definition of the first triplet squared, substituting the first derivatives of the first triple, separating variables by the quadratic relations, collecting like terms, using the definition of the first triple squared, and factoring yield the following Newtonian ODEs:

$\frac{{\text{d}}^{2}p}{\text{d}{\tau }^2}=F_{2,p},\frac{{\text{d}}^{2}q}{\text{d}{\tau }^2}=F_{2,q},\frac{{\text{d}}^{2}r}{\text{d}{\tau }^2}=F_{2,r},$ (71)

where

$\frac{{\text{d}}^{2}p}{\text{d}{\tau }^2},\frac{{\text{d}}^{2}q}{\text{d}{\tau }^2},\frac{{\text{d}}^{2}r}{\text{d}{\tau }^2}$ (72)

are accelerations of $p,q,r$ ,

$\begin{array}{l}{F}_{2,p}=+\frac{2}{3{\epsilon }^2}\left(3{\epsilon }^{2}p-{\epsilon }^2-1-\sqrt{{\epsilon }^4-{\epsilon }^2+1}\right)\left(3{\epsilon }^{2}p-{\epsilon }^2-1+\sqrt{{\epsilon }^4-{\epsilon }^2+1}\right)\\ \text{}\text{}\text{}\text{}\text{}\text{}=+6{\epsilon }^2{p}^2-4\left({\epsilon }^2+1\right)p+2\end{array}$ (73)

is a second-order polynomial force in $p$ ,

$\begin{array}{l}{F}_{2,q}=-\frac{2}{3{\epsilon }^2}\left(3{\epsilon }^{2}q-{\epsilon }^2+{\delta }^2-\sqrt{{\epsilon }^4+{\epsilon }^2{\delta }^2+{\delta }^4}\right)\left(3{\epsilon }^{2}q-{\epsilon }^2+{\delta }^2+\sqrt{{\epsilon }^4+{\epsilon }^2{\delta }^2+{\delta }^4}\right)\\ =-6{\epsilon }^2{q}^2+4\left({\epsilon }^2-{\delta }^2\right)q+2{\delta }^2\end{array}$ (74)

is a second-order polynomial force in $q$ ,

$\begin{array}{l}{F}_{2,r}=-\frac{2}{3}\left(3r-{\delta }^2-1-\sqrt{{\delta }^4-{\delta }^2+1}\right)\left(3r-{\delta }^2-1+\sqrt{{\delta }^4-{\delta }^2+1}\right)\\ =-6r^2+4\left({\delta }^2+1\right)r-2{\delta }^2\end{array}$ (75)

is a second-order polynomial force in $r$ .

Similarly to the first triple, the polynomial forces are related with the polynomial potentials of the first triple squared by the following relations:

$F_{2,p}=-\frac{\text{d}{P}_{3,p}}{\text{d}p},F_{2,q}=-\frac{\text{d}{P}_{3,q}}{\text{d}q},F_{3,r}=-\frac{\text{d}{P}_{3,r}}{\text{d}r}.$ (76)

3.4. Static Visualizations

Polynomial force $F_{2,p}$ , polynomial potential $P_{3,p}$ , and the first Hamiltonian system squared $p$ are shown in Figure 4(a), Figure 4(b), and Figure 4(c), respectively, for various values of $\epsilon$ .

Computation of two zeros of $F_{2,p}$ yields

$p_{F,1}=\frac{{\epsilon }^2+1-\sqrt{{\epsilon }^4-{\epsilon }^2+1}}{3{\epsilon }^2},p_{F,2}=\frac{{\epsilon }^2+1+\sqrt{{\epsilon }^4-{\epsilon }^2+1}}{3{\epsilon }^2}.$ (77)

As $\epsilon$ varies from 0 to 1, $p_{F,1}$ moves from 1/2 to 1/3 and $p_{F,2}$ moves from +∞ to +1 since $F_{2,p}$ transforms from a linear polynomial $-4p+2$ into the quadratic polynomial.

In accordance with the first derivative of the polynomial potentials, zeros of $F_{2,p}$ correlate with extrema of $P_{3,p}$ . Specifically, there are the local minimum and the local maximum that are computed by

$\begin{array}{l}{P}_{3,p,\min }=+\frac{2}{27{\epsilon }^4}\left(R_{\epsilon }-{\epsilon }^2-1\right)\left(R_{\epsilon }-{\epsilon }^2+2\right)\left(R_{\epsilon }+2{\epsilon }^2-1\right),\\ P_{3,p,\max }=-\frac{2}{27{\epsilon }^4}\left(R_{\epsilon }+{\epsilon }^2+1\right)\left(R_{\epsilon }+{\epsilon }^2-2\right)\left(R_{\epsilon }-2{\epsilon }^2+1\right),\end{array}$ (78)

where

$R_{\epsilon }=\sqrt{{\epsilon }^4-{\epsilon }^2+1}.$ (79)

(a)

(b)

(c)

(d)

Figure 4. Plots of the first Hamiltonian system squared: (a) $-F_{2,p}$ , (b) $-P_{3,p}$ , (c) $-p$ for various values of $\epsilon :{\epsilon }_t,{\epsilon }_c,{\epsilon }_h$ , (d) the last frame of the animated 3-d Hamiltonian map of $p\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{3,p}\left(p,\tau ,{\epsilon }_h\right)$ .

As $\epsilon$ varies from 0 to 1, $P_{3,p,\min }$ at $p=p_{F,1}$ changes from −1/2 to −8/27 and $P_{3,p,\max }$ at $p=p_{F,2}$ from +∞ to 0 since $P_{3,p}$ transforms from a quadratic polynomial $2p^2-2p$ into the third-order polynomial.

Three zeros of $P_{3,p}$ are calculated as follows:

$p_1=0,p_2=1,p_3=\frac{1}{{\epsilon }^2}.$ (80)

As $\epsilon$ varies from 0 to 1, $p_1=p_{\min }=0$ and $p_2=p_{\max }=1$ are steady, and $p_3$ moves from +∞ to +1 since $P_{3,p}$ alters the order from two to three.

Depth of the sn²-potential well

$d_{3,p}=P_{3,p,\max }-P_{3,p,\min }=\frac{8R_{\epsilon }^3}{27{\epsilon }^4}.$ (81)

Width of the sn²-potential well

$w_{4,p}=p_{F,2}-p_w=\frac{R_{\epsilon }}{{\epsilon }^2},$ (82)

where

$p_w=\frac{{\epsilon }^2+1-2R_{\epsilon }}{3{\epsilon }^2}$ (83)

is a solution of the following algebraic equation:

$P_{3,p}-P_{3,p,\max }+2{\epsilon }^2\left(p-p_w\right){\left(p-p_{F,2}\right)}^2=0.$ (84)

Thus, the first Hamiltonian system squared $p\in \left[p_{\min ,}{p}_{\max }\right]=\left[0,1\right]$ is confined in the sn²-potential well $\left[p_w,p_{F,2}\right]$ . The vanishing total energy $E_{3,p}$ of $p$ is displayed in Figure 4(b) by a space dashed magenta line and the ultimate positions of $p$ are shown by magenta dots. As $\epsilon \to 1_{-}$ , $p$ approaches the nonlinear sn²-pulsation with positive pulses of a rectangular shape for $2nK\left(\epsilon \right)\le \tau \le 2K\left(\epsilon \right)+2nK\left(\epsilon \right)$ . For $\epsilon ={\epsilon }_h$ , the shape of rectangular pulses coincides with the graph accuracy with ${\tanh }^2\left(\tau \right)$ for $-K\left(\epsilon \right)+2nK\left(\epsilon \right)\le \tau \le +K\left(\epsilon \right)+2nK\left(\epsilon \right)$ .

The period of the sn²-pulsation is $2K\left(\epsilon \right)$ since periodic minimums $p_{\min }=p\left({\tau }_{\min }\right)=0$ and periodic maximums $p_{\max }=p\left({\tau }_{\max }\right)=1$ are reached at

${\tau }_{\min }=2nK\left(\epsilon \right),{\tau }_{\max }=K\left(\epsilon \right)+2nK\left(\epsilon \right).$ (85)

Polynomial force $F_{2,q}$ , polynomial potential $P_{3,q}$ , and the second Hamiltonian system squared $q$ are represented in Figure 5(a), Figure 5(b), and Figure 5(c), correspondingly.

Calculating two zeros of $F_{2,q}$ , we have

$q_{F,1}=\frac{2{\epsilon }^2-1-R_{\epsilon }}{3{\epsilon }^2},q_{F,2}=\frac{2{\epsilon }^2-1+R_{\epsilon }}{3{\epsilon }^2}.$ (86)

As $\epsilon$ varies from 0 to 1, $q_{F,1}$ moves from −∞ to 0 and $q_{F,2}$ from 1/2 to 2/3 since $F_{2,q}$ transforms from a linear polynomial $-4q+2$ into the quadratic polynomial.

(a)

(b)

(c)

(d)

Figure 5. Plots of the second Hamiltonian system squared: (a) $-F_{2,q}$ , (b) $-P_{3,q}$ , (c) $-q$ for various values of $\epsilon :{\epsilon }_t,{\epsilon }_c,{\epsilon }_h$ , (d) the last frame of the animated 3-d Hamiltonian map of $q\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{3,q}\left(q,\tau ,{\epsilon }_h\right)$ .

In accordance with the first derivative of the polynomial potentials, zeros of $F_{2,q}$ correlate with extrema of $P_{3,q}$ . Namely, the local maximum and the local minimum are obtained by

$\begin{array}{l}{P}_{3,q,\max }=-\frac{2}{27{\epsilon }^4}\left(R_{\epsilon }-2{\epsilon }^2+1\right)\left(R_{\epsilon }+{\epsilon }^2-2\right)\left(R_{\epsilon }+{\epsilon }^2+1\right),\\ P_{3,q,\min }=+\frac{2}{27{\epsilon }^4}\left(R_{\epsilon }+2{\epsilon }^2-1\right)\left(R_{\epsilon }-{\epsilon }^2+2\right)\left(R_{\epsilon }-{\epsilon }^2-1\right).\end{array}$ (87)

As $\epsilon$ varies from ${\epsilon }_c$ to 1, $P_{3,q,\max }$ at $q=q_{F,1}$ changes from +∞ to 0 and $P_{3,q,\min }$ at $q=q_{F,2}$ from −1/2 to −8/27 since $P_{3,q}$ transforms from a quadratic polynomial $2q^2-2q$ to the third-order polynomial.

We then find three zeros of $P_{3,q}$ as follows:

$q_1=\frac{1-{\epsilon }^2}{{\epsilon }^2},q_2=0,q_3=1.$ (88)

As $\epsilon$ varies from 0 to 1, $q_1$ changes from −∞ to 0, while $q_2=q_{\min }=0$ and $q_3=q_{\max }=1$ are stationary because $P_{3,q}$ alters the order from two to three.

Depth of the cn²-potential well

$d_{3,q}=P_{3,q,\max }-P_{3,q,\min }=\frac{8R_{\epsilon }^3}{27{\epsilon }^4}=d_{3,p}.$ (89)

Width of the cn²-potential well

$w_{3,q}=q_w-q_{F,1}=\frac{R_{\epsilon }}{{\epsilon }^2}=w_{3,p},$ (90)

where

$q_w=\frac{2{\epsilon }^2-1+2R_{\epsilon }}{3{\epsilon }^2}$ (91)

is a solution of the following algebraic equation:

$P_{3,q}-P_{3,q,\max }-2{\epsilon }^2{\left(q-q_{F,1}\right)}^2\left(q-q_w\right)=0.$ (92)

So, the second Hamiltonian system squared $q\in \left[q_{\min ,}{q}_{\max }\right]=\left[0,1\right]$ pulsates in the cn²-potential well $\left[q_{F,1},q_w\right].$ The vanishing total energy $E_{3,q}$ of $q$ is visualized in Figure 5(b) by a space dashed magenta line and the ultimate positions of $q$ are shown by magenta dots. As $\epsilon \to 1_{-}$ , $q$ approaches the nonlinear cn²-pulsation with positive point pulses for $-K\left(\epsilon \right)+2nK\left(\epsilon \right)\le \tau \le +K\left(\epsilon \right)+2nK\left(\epsilon \right)$ . For $\epsilon ={\epsilon }_h$ , the shape of point pulses coincides with the graph accuracy with ${\mathrm{sech}}^2\left(\tau \right)$ for $-K\left(\epsilon \right)+2nK\left(\epsilon \right)\le \tau \le +K\left(\epsilon \right)+2nK\left(\epsilon \right)$ , which also models the Dirac delta function.

The period of the cn²-pulsation is also $2K\left(\epsilon \right)$ since periodic minimums $q_{\min }=q\left({\tau }_{\min }\right)=0$ and periodic maximums $q_{\max }=q\left({\tau }_{\max }\right)=1$ are reached at

${\tau }_{\min }=-K\left(\epsilon \right)+2nK\left(\epsilon \right),{\tau }_{\max }=2nK\left(\epsilon \right).$ (93)

Polynomial force $F_{2,r}$ , polynomial potential $P_{3,r}$ , and the third Hamiltonian system squared $r$ are given in Figure 6(a), Figure 6(b), and Figure 6(c), sequentially.

We then compute two zeros of $F_{2,r}$ and get

$r_{F,1}=\frac{1}{3}\left(2-{\epsilon }^2-R_{\epsilon }\right),r_{F,2}=r_{F,1}=\frac{1}{3}\left(2-{\epsilon }^2+R_{\epsilon }\right).$ (94)

As $\epsilon$ varies from 0 to 1, $r_{F,1}$ moves from 1/3 to 0 and $r_{F,2}$ from 1 to 2/3 because $F_{2,r}$ remains the quadratic polynomial for all $\epsilon$ .

In agreement with the first derivative of the polynomial potentials, zeros of $F_{2,r}$ correspond to extrema of $P_{3,r}$ . Precisely, the local maximum and the local minimum are returned by

(a)

(b)

(c)

(d)

Figure 6. Plots of the third Hamiltonian system squared: (a) $-F_{2,r}$ , (b) $-P_{3,r}$ , (c) $-r$ for various values of $\epsilon :{\epsilon }_t,{\epsilon }_c,{\epsilon }_h$ , (d) the last frame of the animated 3-d Hamiltonian map of $r\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{3,r}\left(r,\tau ,{\epsilon }_h\right)$ .

$\begin{array}{l}{P}_{3,r,\max }=-\frac{2}{27}\left(R_{\epsilon }-2{\epsilon }^2+1\right)\left(R_{\epsilon }+{\epsilon }^2-2\right)\left(R_{\epsilon }+{\epsilon }^2+1\right),\\ P_{3,r,\min }=+\frac{2}{27}\left(R_{\epsilon }+2{\epsilon }^2-1\right)\left(R_{\epsilon }-{\epsilon }^2+2\right)\left(R_{\epsilon }-{\epsilon }^2-1\right).\end{array}$ (95)

As $\epsilon$ varies from 0 to 1, $P_{3,r,\max }$ at $r=r_{F,1}$ changes from +8/27 to 0 and $P_{3,r,\min }$ at $r=r_{F,2}$ from 0 to -8/27 since $P_{3,r}$ is the third-order polynomial for all $\epsilon$ .

We then find three zeroes of $P_{3,r}$ as

$r_1=0,r_2=1-{\epsilon }^2,r_3=1.$ (96)

As $\epsilon$ varies from 0 to 1, $r_1$ and $r_3=r_{\max }=1$ are stationary and $r_2=r_{\min }=1-{\epsilon }^2$ moves from 1 to 0 as $P_{3,r}$ has the same order for all $\epsilon$ .

Depth of the dn²-potential well

$d_{3,r}=P_{3,r,\max }-P_{3,r,\min }=\frac{8R_{\epsilon }^3}{27}={\epsilon }^4{d}_{3,p}={\epsilon }^4{d}_{3,q}.$ (97)

Width of the dn²-potential well

$w_{3,r}=r_w-r_{F,1}=R_{\epsilon }={\epsilon }^2{w}_{3,p}={\epsilon }^2{w}_{3,q},$ (98)

where

$r_w=\frac{1}{3}\left(2-{\epsilon }^2+2R_{\epsilon }\right)$ (99)

is a solution of the following algebraic equation:

$P_{3,r}-P_{3,r,\max }-2{\left(r-r_{F,1}\right)}^2\left(r-r_w\right)=0.$ (100)

Consequently, a periodic motion of the third Hamiltonian system squared $r\in \left[r_{\min ,}{r}_{\max }\right]=\left[1-{\epsilon }^2,1\right]$ with $E_{4,r}=0$ is bounded in the dn²-potential well $\left[r_{F,1},r_w\right]$ . The total energy $E_{3,r}$ of $r$ is shown in Figure 6(b) with the help of a space dashed magenta line and the ultimate positions of $r$ are displayed by magenta dots. As $\epsilon \to 1_{-}$ , $r$ approaches the nonlinear dn²-pulsation with positive point pulses for $-K\left(\epsilon \right)+2nK\left(\epsilon \right)\le \tau \le +K\left(\epsilon \right)+2nK\left(\epsilon \right)$ . For $\epsilon ={\epsilon }_h,$ the shape of point pulses coincides with the graph accuracy with ${\mathrm{sech}}^2\left(\tau \right)$ for $-K\left(\epsilon \right)+2nK\left(\epsilon \right)\le \tau \le +K\left(\epsilon \right)+2nK\left(\epsilon \right)$ .

The period of the dn²-pulsation is $2K\left(\epsilon \right)$ , as well, since periodic minimums $r_{\min }=r\left({\tau }_{\min }\right)=1-{\epsilon }^2$ and periodic maximums $r_{\max }=r\left({\tau }_{\max }\right)=1$ are attained at

${\tau }_{\min }=-K\left(\epsilon \right)+2nK\left(\epsilon \right),{\tau }_{\max }=2nK\left(\epsilon \right).$ (101)

3.5. Dynamic Visualizations

In terms of the Hamiltonian dynamics, a Hamiltonian system of the first triple squared

$\varphi =\left[p,q,r\right]$ (102)

is a solution of the Hamiltonian ODE

$K_{e,\varphi }+P_{3,\varphi }=E_{3,\varphi }=\frac{1}{2}{\left(\frac{\text{d}\varphi }{\text{d}\tau }\right)}^2+{\sigma }_3{\varphi }^3+{\sigma }_2{\varphi }^2+{\sigma }_1\varphi =0,$ (103)

where $K_{e,\varphi }$ is the kinetic energy, $P_{3,\varphi }$ is the polynomial potential in $\varphi$ of the third order, total energy $E_{3,\varphi }=0$ , and polynomial coefficients of $p,q,r$ are

$\left[{\sigma }_3,{\sigma }_2,{\sigma }_1\right]=\left[-2{\epsilon }^2,2\left({\epsilon }^2+1\right),-2\right],$ (104)

$\left[{\sigma }_3,{\sigma }_2,{\sigma }_1\right]=\left[+2{\epsilon }^2,2\left({\delta }^2-{\epsilon }^2\right),-2{\delta }^2\right],$ (105)

$\left[{\sigma }_3,{\sigma }_2,{\sigma }_1\right]=\left[+2,-2\left({\delta }^2+1\right),+2{\delta }^2\right],$ (106)

respectively.

From the viewpoint of the Newtonian dynamics, $\varphi$ is a solution of the Newtonian ODE

$\frac{{\text{d}}^2\varphi }{\text{d}{\tau }^2}=F_{2,\varphi }={\omega }_2{\varphi }^3+{\omega }_1\varphi +{\omega }_0,$ (107)

where the second derivative of $\varphi$ is the acceleration, $F_{2,\varphi }$ is the polynomial force in $\varphi$ of the second order, and polynomial coefficients of $p,q,r$ are

$\left[{\omega }_2,{\omega }_1,{\omega }_0\right]=\left[+6{\epsilon }^2,-4\left({\epsilon }^2+1\right),+2\right],$ (108)

$\left[{\omega }_2,{\omega }_1,{\omega }_0\right]=\left[-6{\epsilon }^2,-4\left({\delta }^2-{\epsilon }^2\right),+2{\delta }^2\right],$ (109)

$\left[{\omega }_2,{\omega }_1,{\omega }_0\right]=\left[-6,+4\left({\delta }^2+1\right),-2{\delta }^2\right],$ (110)

sequentially.

Analogous to the first triplet, the polynomial forces are related with the polynomial potentials of the first triplet squared by the following connections:

$F_{2,\varphi }=-\frac{\text{d}{P}_{3,\varphi }}{\text{d}\varphi },{\omega }_2=-3{\sigma }_3,{\omega }_1=-2{\sigma }_2,{\omega }_0=-{\sigma }_1.$ (111)

The dynamical problems for the Hamiltonian or Newtonian ODEs are complemented by the following initial conditions for $p,q,r$ , respectively:

${p|}_{\tau =0}=0,{\frac{\text{d}p}{\text{d}\tau }|}_{\tau =0}=0,$ (112)

${q|}_{\tau =0}=1,{\frac{\text{d}q}{\text{d}\tau }|}_{\tau =0}=0,$ (113)

${r|}_{\tau =0}=1,{\frac{\text{d}r}{\text{d}\tau }|}_{\tau =0}=0.$ (114)

Solutions of the Hamiltonian or Newtonian ODEs with the above initial conditions return unique solutions for the first triplet squared.

Figure 4(b) and Figure 4(c) are combined for $\epsilon ={\epsilon }_h$ in a 3-d Hamiltonian map animated on $\tau \in \left[-K_h,+K_h\right]$ , the last frame of which is displayed in Figure 4(d). A 3-d trajectory of the first Hamiltonian system squared $p\left(\tau ,{\epsilon }_h\right)$ on the surface of polynomial potential $P_{3,p}\left(p,\tau ,{\epsilon }_h\right)$ is visualized by a black curve. A 2-d projection of the 3-d trajectory on plane $\left[p,\tau \right]$ is shown in Figure 4(d) by a blue dashed curve, which coincides with the 2-d trajectory of $p$ indicated by the blue solid curve in Figure 4(c).

The periodic 3-d trajectory of $p$ begins on the potential barrier at $p=p_{\max }$ when $\tau ={\tau }_{\max }=-K_h+2n{K}_h$ and $K_{e,p}=-P_{3,p}=0$ . As $P_{3,p}$ decreases in the sn²-potential well, kinetic energy, $K_{e,p}=-P_{3,p}>0$ , initially increases towards the bottom of the sn²-potential well at $p=p_{F,1}$ and then decreases in time interval $\tau \approx \left[-0.33K_h+2n{K}_h,2n{K}_h\right]$ . The kinetic energy reaches its maximal value $K_{e,p,\max }=-P_{3,p,\min }$ at the bottom of the sn²-potential well. After reflection by the potential wall at $p=p_{\min }$ when $\tau ={\tau }_{\min }=2n{K}_h$ and $K_{e,p}=-P_{3,p}=0$ , $p$ again accelerates and then decelerates towards the potential barrier for $\tau \approx \left[2n{K}_h,+0.33K_h+2n{K}_h\right]$ . Further $p$ decelerates moving along the potential barrier since $K_{e,p}\approx 0$ for $\tau \approx \left[0.33K_h+2n{K}_h,1.66K_h+2n{K}_h\right]$ . The periodic 3-d trajectory finishes at $p=p_{\max }$ when $\tau ={\tau }_{\max }=+K_h+2n{K}_h$ . The described properties of the 3-d Hamiltonian map are evidently displayed in the animated Figure 4(d).

In Figure 5(d), the last frame of the animated 3-d Hamiltonian map on $\tau \in \left[-K_h,+K_h\right]$ integrates for $\epsilon ={\epsilon }_h$ Figure 5(b), which contains the 2-d polynomial potential, with Figure 5(c), which displays the 2-d trajectory. A black curve in Figure 5(d) visualizes a 3-d trajectory of the second Hamiltonian system squared $q\left(\tau ,{\epsilon }_h\right)$ on the surface of the polynomial potential $P_{3,q}\left(q,\tau ,{\epsilon }_h\right)$ . A blue dashed curve in Figure 5(d), which fits the 2-d trajectory of $q$ shown by the blue solid curve in Figure 5(c), represents a 2-d projection of the 3-d trajectory on plane $\left[q,\tau \right].$

The periodic 3-d trajectory of $q$ starts on the potential barrier at $q=q_{\min }$ when $\tau ={\tau }_{\min }=-K_h+2n{K}_h$ and $K_{e,q}=-P_{3,q}=0$ . As $P_{3,q}$ decreases in the cn²-potential well, kinetic energy, $K_{e,q}=-P_{3,q}>0$ , initially increases towards the bottom of the cn²-potential well and then decreases in time interval $\tau \approx \left[-0.33K_h+2n{K}_h,2n{K}_h\right]$ . The kinetic energy reaches its maximal value $K_{e,q,\max }=-P_{3,q,\min }$ at the bottom of the cn²-potential well at $q=q_{F,2}$ . After reflection by the potential wall at $q=q_{\max }$ when $\tau ={\tau }_{\max }=2n{K}_h$ and $K_{e,q}=-P_{3,q}=0$ , $q$ once more primarily accelerates and sequentially decelerates towards the potential barrier for $\tau \approx \left[2n{K}_h,+0.33K_h+2n{K}_h\right]$ . Later $q$ decelerates while moving along the potential barrier as $K_{e,q}\approx 0$ for $\tau \approx \left[0.33K_h+2n{K}_h,1.66K_h+2n{K}_h\right]$ . The periodic 3-d trajectory terminates at $q=q_{\min }$ when $\tau ={\tau }_{\min }=+K_h+2n{K}_h$ . The described details of the 3-d Hamiltonian map are clearly visualized by the animated Figure 5(d).

In Figure 6(d), the last frame of the 3-d Hamiltonian map animated on $\tau \in \left[-K_h,+K_h\right]$ synthesizes for $\epsilon ={\epsilon }_h$ the 2-d polynomial potential in Figure 6(b) and the 2-d trajectory in Figure 6(c). A black curve shows a 3-d trajectory of the third Hamiltonian system squared $r\left(\tau ,{\epsilon }_h\right)$ on the surface of the polynomial potential $P_{3,r}\left(r,\tau ,{\epsilon }_h\right)$ and a blue dashed curve in Figure 6(d), which coincides with the 2-d trajectory of $r$ displayed by the blue solid curve in Figure 6(c), presents a 2-d projection of the 3-d trajectory on plane $\left[r,\tau \right]$ .

The periodic 3-d trajectory of $r$ begins on the potential barrier at $r=r_{\min }$ when $\tau ={\tau }_{\min }=-K_h+2n{K}_h$ and $K_{e,r}=-P_{3,r}=0$ . As $P_{3,r}$ decreases in the dn²-potential well, kinetic energy, $K_{e,r}=-P_{3,r}>0$ , initially increases towards the bottom of the dn²-potential well and then decreases in time interval $\tau \approx \left[-0.33K_h+2n{K}_h,2n{K}_h\right]$ . The kinetic energy attains its maximal value $K_{e,r,\max }=-P_{3,r,\min }$ at the bottom of the dn²-potential well at $r=r_{F,2}$ . After reflection by the potential wall at $r=r_{\max }$ when $\tau ={\tau }_{\max }=2n{K}_h$ and $K_{e,r}=-P_{3,r}=0$ , $r$ again primarily accelerates and sequentially decelerates towards the potential barrier for $\tau \approx \left[2n{K}_h,+0.33K_h+2n{K}_h\right]$ . While moving along the potential barrier, $r$ decelerates since $K_{e,r}\approx 0$ for $\tau \approx \left[0.33K_h+2n{K}_h,1.66K_h+2n{K}_h\right]$ . The periodic 3-d trajectory ends at $r=r_{\min }$ when $\tau ={\tau }_{\min }=+K_h+2n{K}_h$ . The 3-d Hamiltonian map in Figure 6(d) qualitatively coincides with that in Figure 5(d) since ${\text{dn}}^2\left(\tau ,{\epsilon }_h\right)={\text{cn}}^2\left(\tau ,{\epsilon }_h\right)$ with a graph accuracy. The considered features of the 3-d Hamiltonian map are obviously manifested in the animated Figure 6(d).

4. An Oscillatory Random Model

4.1. The lth Mode of the Random Oscillatory cn-Noise

Similarly to the second Hamiltonian system $g$ , we construct the lth mode $g_l$ of the random oscillatory cn-noise in the following form:

$g_l=A_l\text{cn}\left({\upsilon }_l\tau -K\left(\epsilon \right),\epsilon \right),$ (115)

where $A_l$ is an amplitude, ${\upsilon }_l$ is a frequency of the lth mode, and $l=1,2,\cdots ,L$ .

Periodic zeroes of $g_l$

$g_{l,z}=g_l\left({\tau }_{g,z},\epsilon \right)=0$ (116)

are located at

${\tau }_{g,z}=\frac{2nK\left(\epsilon \right)}{{\upsilon }_l},$ (117)

where $n=0,\pm 1,\pm 2,\cdots$ is an integer.

The lth mode $g_l$ reaches its maximal value

$g_{l,\max }=g_l\left({\tau }_{g,\max },\epsilon \right)=+A_l$ (118)

periodically at

${\tau }_{g,\max }=\frac{\left(4n+1\right)K\left(\epsilon \right)}{{\upsilon }_l}$ (119)

and attains its minimal value

$g_{l,\min }=g_l\left({\tau }_{g,\min },\epsilon \right)=-A_l$ (120)

also periodically at

${\tau }_{g,\min }=\frac{\left(4n+3\right)K\left(\epsilon \right)}{{\upsilon }_l}$ (121)

since the period of oscillation $g_l$

$P_{o,l}=\frac{4K\left(\epsilon \right)}{{\upsilon }_l}.$ (122)

Therefore, range of $g_l\in \left[g_{l,\min },g_{l,\max }\right]=\left[-A_l,+A_l\right]$ .

Calculating the half of the first derivative of $g_l$ squared and using the independent algebraic relations, we derive the Hamiltonian ODE

$K_{e,g,l}+P_{4,g,l}=E_{4,g,l}=0,$ (123)

where $K_{e,g,l}$ is the kinetic energy of $g_l$ ,

$P_{4,g,l}=\frac{{\upsilon }_l^2}{2A_l^2}\left(g_l-A_l\right)\left(g_l+A_l\right)\left({\epsilon }^2{g}_l^2+{\delta }^2{A}_l^2\right)={\sigma }_{4,g,l}{g}_l^4+{\sigma }_{2,g,l}{g}_l^2+{\sigma }_{0,g,l}$ (124)

is an even polynomial potential of the fourth order in $g_l$ with polynomial coefficients

$\left[{\sigma }_{4,g,l},{\sigma }_{2,g,l},{\sigma }_{0,g,l}\right]=\left[\frac{{\epsilon }^2{\upsilon }_l^2}{2A_l^2},\frac{\left({\delta }^2-{\epsilon }^2\right){\upsilon }_l^2}{2},-\frac{A_l^2{\delta }^2{\upsilon }_l^2}{2}\right]$ (125)

and total energy $E_{4,g,l}=0$ . Compared with the polynomial coefficients of $g$ , ${\sigma }_{4,g,l}$ is inversely proportional to $A_l^2$ , ${\sigma }_{2,g,l}$ does not depend on $A_l^2$ , ${\sigma }_{0,g,l}$ is directly proportional to $A_l^2$ , and all coefficients are proportional to ${\upsilon }_l^2$ .

The lth mode $g_l$ of the random oscillatory cn-noise is a unique solution of the Hamiltonian ODE subjected to the initial conditions

${g_l|}_{\tau =0}=0,{\frac{\text{d}{g}_l}{\text{d}\tau }|}_{\tau =0}=0.$ (126)

In Figure 7(a), $g_l$ is shown for ${\upsilon }_l=1$ , $\epsilon =0.9999$ , $K\left(\epsilon \right)=5.6451$ , and various values of $A_l$ : a green dashed curve corresponds to $A_l=1/2$ , a black dotted curve to $A_l=1$ , and a blue solid curve to $A_l=2.$ The value of $g_l$ is directly proportional to amplitude $A_l$ and the initial value of $g_l$ vanishes in accordance with the initial conditions. All curves in Figure 7 are displayed on time domain $\tau \in \left[0,4K\left(\epsilon \right)/{\upsilon }_l\right]$ .

The Hamiltonian system $g_l$ is displayed in Figure 7(b) for $A_l=1$ , $\epsilon =0.9999$ , and different values of ${\upsilon }_l$ : a green dashed curve for ${\upsilon }_l=1/2$ , a black dotted curve for ${\upsilon }_l=1$ , and a blue solid curve for ${\upsilon }_l=2$ . Indeed, period $P_{o,l}$ is inversely proportional to ${\upsilon }_l$ and $P_{o,l}\to +\infty$ as ${\upsilon }_l\to 0_{+}$ .

The effect of $\epsilon$ on the period and the shape of $g_l$ is visualized in Figure 7(c) for $A_l=1$ , ${\upsilon }_l=1$ , and various values of $\epsilon$ : a green dashed curve for $\epsilon ={\epsilon }_c$ with $K\left(\epsilon \right)=K_c$ , a black dotted curve for $\epsilon =0.9999$ , and a blue solid curve for $\epsilon ={\epsilon }_h$ with $K\left(\epsilon \right)=K_h$ . As $\epsilon \to 1_{-}$ , $P_{o,l}\to +\infty$ and the shape of $g_l$ approaches a sequence of positive and negative pulses $A_l\mathrm{sech}\left({\upsilon }_l\tau \right)$ with truncated tails because of the vertical asymptote of $K\left(\epsilon \right)$ at $\epsilon =1$ [8].

(a)

(b)

(c)

Figure 7. Plots of the lth mode $g_l$ of the random oscillatory cn-noise for various parameters given in the text.

4.2. The Random Oscillatory cn-Noise with L modes

The random oscillatory cn-noise

$g_o={\displaystyle \sum _{l=1}^L{g}_l}={\displaystyle \sum _{l=1}^L{A}_l\text{cn}\left({\upsilon }_l\tau -K\left(\epsilon \right),\epsilon \right)}$ (127)

is a superposition of $L$ modes $g_l$ , where $A_l$ is a random amplitude on interval $\left[0,1\right]$ .

We then explore an effect of $L$ on the rate of stochastization of the random oscillatory cn-noise for two sequences of frequencies: the Fourier sequence [4] and the Bernoulli sequence [5]. Random oscillatory cn-noise $g_{o,F}$ with the Fourier frequencies and period of the random oscillatory cn-noise $P_{o,F}=4K\left(\epsilon \right)=22.5806$ with $\epsilon =0.9999$ is shown on interval $\tau \in \left[0,P_{o,F}\right]$ in Figure 8(a), Figure 8(b), Figure 8(c) for the following five random amplitudes:

$A_l=\left[0.388841,0.854563,0.425320,0.573515,0.777952\right]$ (128)

and $L=3,4,5$ , respectively.

(a)

(b)

(c)

Figure 8. Plots of the random oscillatory cn-noise for the Fourier frequencies and various $L$ : (a)—$L=3$ , (b)—$L=4$ , (c)—$L=5$ .

Period $P_{o,l,F}$ of the lth mode $g_{l,F}$ has the following values:

$P_{o,l,F}=\left[22.5806,11.2903,7.52686,5.64515,4.51612\right].$ (129)

So, numbers $n_{o,l,F}$ of periods $P_{o,l,F}$ in $P_{o,F}$ grow as the Fourier frequencies ${\upsilon }_{l,F}$ , i.e. ${\upsilon }_{l,F}=n_{o,l,F}=P_{o,F}/P_{o,l,F}=\left[1,2,3,4,5\right]$ . The rate of stochastization of $g_{o,F}$ is a lowest one since $P_{o,F}$ does not depend on $L$ and stochastization of the random oscillatory cn-noise is caused only by growth of number of amplitudes and frequencies with $L.$

However, simplicity of curves in Figures 8(a)-(c) clearly demonstrates an odd symmetry of $g_o$ for all sequences of frequencies. Interval $\left[0,P_o\right]$ contains integer number $n_{o,l}$ of subintervals $\left[0,P_{o,l}\right]$ for $g_l$ , i.e.

$P_o=n_{o,l}{P}_{o,l}.$ (130)

All $g_l$ vanish at the end points of subintervals $\left[0,P_{o,l}\right]$ . Therefore, $g_o$ vanishes at the end points of interval $\left[0,P_o\right]$ , namely,

$g_o\left(0,\epsilon \right)=0,g_o\left(P_o,\epsilon \right)=0.$ (131)

If $n_{o,l}$ is even, then the midpoint of $\left[0,P_o\right]$ becomes an endpoint of $\left[0,P_{o,l}\right]$ , where $g_l$ vanishes, viz.

$P_o=2k{P}_{o,l},\frac{P_o}{2}=k{P}_{o,l},g_{o,l}\left(\frac{P_o}{2},\epsilon \right)=0.$ (132)

If $n_{o,l}$ is odd, then the midpoint of $\left[0,P_o\right]$ becomes a midpoint of $\left[0,P_{o,l}\right]$ . Then $g_l$ vanishes at the midpoint of subinterval $\left[0,P_{o,l}\right]$ , as well, i.e.

$P_o=\left(2k+1\right)P_{o,l},\frac{P_o}{2}=\left(k+\frac{1}{2}\right)P_{o,l},g_{o,l}\left(\frac{P_o}{2},\epsilon \right)=0.$ (133)

So, $g_o$ vanishes at the midpoint of $\left[0,P_o\right]$ as the sum of $g_l$ vanishing both for even and odd $n_{o,l}$ , namely,

$g_o\left(\frac{P_o}{2},\epsilon \right)=0.$ (134)

Since an odd symmetry

$g_l\left(\tau -{\tau }_{g,z},\epsilon \right)=-g_l\left(-\left(\tau -{\tau }_{g,z}\right),\epsilon \right)$ (135)

holds for all periodic zeroes of $g_l$ (see Figures 7(a)-(c)), the odd symmetry is valid at the midpoint and endpoints for all $g_l$ and, therefore, the odd symmetry is also valid at the midpoint of $\left[0,P_o\right]$ for $g_o$ :

$g_o\left(\tau -\frac{P_o}{2},\epsilon \right)=-g_o\left(-\left(\tau -\frac{P_o}{2}\right),\epsilon \right).$ (136)

Vanishing of $g_o$ at the end points and the midpoint of $\left[0,P_o\right]$ and the odd symmetry of $g_o$ with respect to the midpoint of $\left[0,P_o\right]$ is illustrated for various $L$ and ${\upsilon }_l$ in Figures 8(a)-(c) and Figures 9(a)-(b).

Random oscillatory cn-noise $g_{o,B}$ with $\epsilon =0.9999$ , the Bernoulli frequencies ${\upsilon }_{l,B}=\left[1,1/2,1/3,1/5,1/7\right]$ , the same random amplitudes $A_l$ , and periods of the random oscillatory cn-noise $P_{o,B}=135.484,677.418,4741.92$ is visualized on interval $\tau \in \left[0,P_{o,B}\right]$ in Figure 9(a), Figure 9(b), Figure 9(c) for $L=3,4,5$ , respectively.

Periods $P_{o,l,B}=4K\left(\epsilon \right)/{\upsilon }_{l,B}$ of the lth mode $g_{l,B}$ increase, viz.

$P_{o,l,B}=\left[22.5806,45.1612,67.7418,112.903,158.064\right],$ (137)

where

$P_{o,B}=LCM\left(P_{o,1,B},\cdots ,P_{o,l,B},\cdots ,P_{o,L,B}\right)$ (138)

and LCM stands for the least common multiple.

For $L=5$ , numbers $n_{o,l,B}$ of periods $P_{o,l,B}$ in $P_{o,B}$ decrease as follows: $n_{o,l,B}=P_{o,B}/P_{o,l,B}=\left[210,105,70,42,30\right]$ . The rate of stochastization of $g_{o,B}$ has a highest value since $P_{o,B}$ grows as a product of prime numbers. For $L=3$ , $P_{o,B}$ is greater six times than the period of the first mode $P_{o,1,B}$ . For $L=4$ , $P_{o,B}$ is larger 30 times, and, for $L=5$ , $P_{o,B}$ exceeds $P_{o,1,B}$ 210 times. To summarize, $g_{o,B}$ emulates a smooth random oscillatory variable with an unbounded period at high values of $L$ since $P_{o,B}\to \infty$ as $L\to \infty$ .

(a)

(b)

(c)

Figure 9. Plots of the random oscillatory cn-noise for the Bernoulli frequencies and various $L$ : (a)—$L=3$ , (b)—$L=4$ , (c)—$L=5$ .

The rates of stochastization of the random oscillatory cn-noise with the Fourier and Bernoulli frequencies are compared in Figure 10, where $\ln \left(P_{o,F}\right)$ versus $L$ is shown by green circles and a dotted black line and $\ln \left(P_{o,B}\right)$ by blue diamonds and a dashed black line. For $L\ge 4$ , $P_{o,B}$ grows exponentially as follows:

$\ln \left(P_{o,B}\right)=-4.2408+2.6552L,P_{o,B}=0.0014397\exp \left(2.6552L\right),$ (139)

where coefficients are computed by minimizing the least-squares error.

We then compare histogram of $g_o$ for the Fourier and Bernoulli frequencies with the density of the Gaussian (normal) probability distribution

$p_o=\frac{1}{\sqrt{2\pi }{\sigma }_o}\exp \left(-\frac{{\left(g_o-{\mu }_o\right)}^2}{2{\sigma }_o^2}\right),$ (140)

where ${\mu }_o$ is the mean and ${\sigma }_o^2$ is the variance of discrete data sample $g_{o,k}$ of size $K_o$ , which are computed by the formulas of the descriptive statistics [9]:

${\mu }_o=\frac{{\displaystyle \sum _{k=1}^{K_o}{g}_{o,k}}}{K_o},{\sigma }_o^2=\frac{{\displaystyle \sum _{k=1}^{K_o}{\left(g_{o,k}-{\mu }_o\right)}^2}}{K_o-1},g_{o,k}=g_o\left({\tau }_k\right).$ (141)

Figure 10. Plots of $\ln \left(P_o\right)$ versus $L$ of the random oscillatory cn-noise.

Here, moments of time ${\tau }_k$ as selected as equidistant points on interval $\tau \in \left[0,P_o\right]$ , where $P_o$ is the period of the random oscillatory cn-noise.

Histograms in Figure 11(a) and Figure 11(b) are computed with random amplitudes $A_l$ , $\epsilon =0.9999$ , and $L=5$ for the Fourier and Bernoulli frequencies, respectively. The data sample sizes $K_{o,F}=416$ and $K_{o,B}=3840$ are equal to the number of computational points of plots $g_{o,F}$ and $g_{o,B}$ in Figure 8(c) and Figure 9(c), correspondingly.

In Figure 11, ${\mu }_{o,F}=0,{\mu }_{o,B}=0$ and ${\sigma }_{o,F}=0.580233,{\sigma }_{o,F}=0.537277$ , whereas the Gaussian probability distribution is displayed by a solid orange line. As $L$ increases, $p_{o,F}$ develops into the normal distribution. However, even for $L=5$ , the tails of the normal distribution are not reproduced and noticeable deviations from the Gaussian distribution still remain. As $L$ grows, $p_{o,B}$ approaches the Gaussian distribution much faster. For $L\ge 5$ , the normal distribution is reproduced with graph accuracy.

5. A Pulsatory Random Model

5.1. The lth Mode of the Random Pulsatory cn²-Noise

Analogously to the second Hamiltonian system squared $q$ , we define the lth mode $q_l$ of the random pulsatory cn²-noise as follows:

$q_l=A_l{\text{cn}}^2\left({\upsilon }_l\tau -K\left(\epsilon \right),\epsilon \right),$ (142)

where $A_l$ is an amplitude and ${\upsilon }_l$ is a frequency of the lth mode, and $l=1,2,\cdots ,L$ .

Periodic minimums of $q_l$ , which are periodic zeroes of $q_l$ ,

$q_{l,\min }=q_l\left({\tau }_{q,\min },\epsilon \right)=0$ (143)

are positioned at

${\tau }_{q,\min }=\frac{2nK\left(\epsilon \right)}{{\upsilon }_l},$ (144)

where $n=0,\pm 1,\pm 2,\cdots$ is an integer.

The lth mode $q_l$ attains its maximal value

$q_{l,\max }=q_l\left({\tau }_{q,\max },\epsilon \right)=A_l$ (145)

periodically at

(a)

(b)

Figure 11. Histogram of the random oscillatory cn-noise: (a)—for the Fourier frequencies, (b)—for the Bernoulli frequencies.

${\tau }_{q,\max }=\frac{\left(2n+1\right)K\left(\epsilon \right)}{{\upsilon }_l},$ (146)

because the period of pulsation $q_l$

$P_{p,l}=\frac{2K\left(\epsilon \right)}{{\upsilon }_l}.$ (147)

Consequently, range of $q_l\in \left[q_{l,\min },q_{l,\max }\right]=\left[0,A_l\right]$ .

Computation of the half of the first derivative of $q_l$ squared and application of the independent algebraic relations yield the Hamiltonian ODE

$K_{e,q,l}+P_{3,q,l}=E_{3,q,l}=0,$ (148)

where $K_{e,q,l}$ is the kinetic energy of the Hamiltonian system $q_l$ ,

$P_{3,q,l}=\frac{2{\upsilon }_l^2}{A_l}\left(q_l-A_l\right)q_l\left({\epsilon }^2{q}_l+{\delta }^2{A}_l\right)={\sigma }_{3,q,l}{q}_l^3+{\sigma }_{2,q,l}{q}_l^2+{\sigma }_{1,q,l}{q}_l$ (149)

is a polynomial potential of the third order in $q_l$ with polynomial coefficients

$\left[{\sigma }_{3,q,l},{\sigma }_{2,q,l},{\sigma }_{1,q,l}\right]=\left[\frac{2{\epsilon }^2{\upsilon }_l^2}{A_l},2\left({\delta }^2-{\epsilon }^2\right){\upsilon }_l^2,-2{\delta }^2{A}_l{\upsilon }_l^2\right]$ (150)

and total energy $E_{3,q,l}=0$ . Compared with the polynomial coefficients of $q$ , ${\sigma }_{3,q,l}$ is inversely proportional to $A_l$ , ${\sigma }_{2,q,l}$ does not depend on $A_l$ , ${\sigma }_{1,q,l}$ is directly proportional to $A_l$ , and all coefficients are proportional to ${\upsilon }_l^2$ .

The lth mode $q_l$ of the pulsatory cn²-noise represents a unique solution of the Hamiltonian ODE with the following initial conditions:

${q_l|}_{\tau =0}=0,{\frac{\text{d}{q}_l}{\text{d}\tau }|}_{\tau =0}=0.$ (151)

In Figure 12(a), $q_l$ is displayed for ${\upsilon }_l=1$ , $\epsilon =0.9999$ , $K\left(\epsilon \right)=5.6451$ , and different values of $A_l$ : a green dashed curve corresponds to $A_l=1/2$ , a black dotted curve to $A_l=1$ , and a blue solid curve to $A_l=2$ . The value of $q_l$ is directly proportional to amplitude $A_l$ and the initial value of $q_l$ vanishes in agreement with the initial conditions. Pulsations $q_l$ are sharper than pulsations $g_l$ in the same manner as pulsations $A_l{\mathrm{sech}}^2\left({\upsilon }_l\tau \right)$ are sharper than pulsations $A_l\mathrm{sech}\left({\upsilon }_l\tau \right)$ . All curves in Figure 12 are visualized on time domain $\tau \in \left[0,4K\left(\epsilon \right)/{\upsilon }_l\right]$ .

The Hamiltonian system $q_l$ is visualized in Figure 12(b) for $A_l=1$ , $\epsilon =0.9999$ , and various values of ${\upsilon }_l$ : a green dashed curve for ${\upsilon }_l=1/2$ , a black dotted curve for ${\upsilon }_l=1$ , and a blue solid curve for ${\upsilon }_l=2$ . In the view of the definition, period $P_{p,l}$ is inversely proportional to ${\upsilon }_l$ and $P_{p,l}\to +\infty$ as ${\upsilon }_l\to 0_{+}$ .

The effect of $\epsilon$ on the period and the shape of $q_l$ is represented in Figure 12(c) for $A_l=1$ , ${\upsilon }_l=1$ , and various values of $\epsilon$ : a green dashed curve for $\epsilon ={\epsilon }_c$ , a black dotted curve for $\epsilon =0.9999$ , and a blue solid curve for $\epsilon ={\epsilon }_h$ . As $\epsilon \to 1_{-}$ , $P_{p,l}\to +\infty$ and the shape of $q_l$ tends to a sequence of pulses $A_l{\mathrm{sech}}^2\left({\upsilon }_l\tau \right)$ with truncated tails.

5.2. The Random Pulsatory cn²-Noise with L Modes

The random pulsatory cn²-noise

$q_p={\displaystyle \sum _{l=1}^L{q}_l}={\displaystyle \sum _{l=1}^L{A}_l{\text{cn}}^2\left({\upsilon }_l\tau -K\left(\epsilon \right),\epsilon \right)}$ (152)

is a superposition of $L$ modes $q_l$ , where $A_l$ is the random amplitude on interval $\left[0,1\right]$ .

The random pulsatory cn²-noise $q_{p,F}$ with the Fourier frequencies ${\upsilon }_{l,F}=\left[1,2,3,4,5\right]$ , $\epsilon =0.999866$ , a period of the pulsatory noise $P_{p,F}=2K\left(\epsilon \right)=10.9978$ , and random amplitudes $A_l$ is displayed on interval $\tau \in \left[0,P_{p,F}\right]$ in Figure 13(a), Figure 13(b), Figure 13(c) for $L=3,4,5$ , correspondingly.

Period $P_{p,l,F}$ of the lth mode $q_{l,F}$ contains the following values:

$P_{p,l,F}=\left[10.9978,5.49889,3.66593,2.74945,2.19956\right].$ (153)

Therefore, numbers $n_{p,l,F}$ of periods $P_{p,l,F}$ in $P_{p,F}$ also increase as the Fourier frequencies ${\upsilon }_{l,F}$ , viz. $n_{p,l,F}=P_{p,F}/P_{p,l,F}=\left[1,2,3,4,5\right]$ . The rate of stochastization of $q_{p,F}$ has a lowest value because $P_{p,F}$ does not depend on $L$ and stochastization of the random pulsatory cn²-noise is induced by increase in number of amplitudes and frequencies with $L$ .

Curves in Figures 13(a)-(c) obviously display an even symmetry of $q_p$ for all sequences of frequencies. Interval $\left[0,P_p\right]$ includes integer number $n_{p,l}$ of subintervals $\left[0,P_{p,l}\right]$ for $q_l$ , namely,

$P_p=n_{p,l}{P}_{p,l}.$ (154)

All $q_l$ vanish at the end points of subintervals $\left[0,P_{p,l}\right]$ . So, $q_p$ also vanishes at the end points of interval $\left[0,P_p\right]$ , viz.

(a)

(b)

(c)

Figure 12. Plots of the lth mode $q_l$ of the random oscillatory cn²-noise for various parameters given in the text.

$q_p\left(0,\epsilon \right)=0,q_p\left(P_p,\epsilon \right)=0.$ (155)

If $n_{p,l}$ is even, then the midpoint of $\left[0,P_p\right]$ becomes an endpoint of $\left[0,P_{p,l}\right]$ , where $q_l$ vanishes, i.e.

$P_p=2k{P}_{p,l},\frac{P_p}{2}=k{P}_{p,l},q_{p,l}\left(\frac{P_p}{2},\epsilon \right)=0.$ (156)

If $n_{p,l}$ is odd, then the midpoint of $\left[0,P_p\right]$ becomes a midpoint of $\left[0,P_{p,l}\right]$ . Then $q_l$ reaches local maximum $A_l$ at the midpoint of subinterval $\left[0,P_{p,l}\right]$ , viz.

$P_p=\left(2k+1\right)P_{p,l},\frac{P_p}{2}=\left(k+\frac{1}{2}\right)P_{p,l},q_{p,l}\left(\frac{P_p}{2},\epsilon \right)=A_l.$ (157)

Thus, $q_p$ attains a local maximum at the midpoint of $\left[0,P_p\right]$ as the sum of $q_l=0$ for even $n_{p,l}$ and $q_l=A_l$ for odd $n_{p,l}$ , explicitly,

$q_p\left(\frac{P_p}{2},\epsilon \right)=q_{p,\max }.$ (158)

Because an even symmetry

(a)

(b)

(c)

Figure 13. Plots of the random pulsatory cn²-noise for the Fourier frequencies and various $L$ : (a)—$L=3$ , (b)—$L=4$ , (c)—$L=5$ .

$\begin{array}{l}{q}_l\left(\tau -{\tau }_{q,\min },\epsilon \right)=q_l\left(-\left(\tau -{\tau }_{q,\min }\right),\epsilon \right),\\ q_l\left(\tau -{\tau }_{q,\max },\epsilon \right)=q_l\left(-\left(\tau -{\tau }_{q,\max }\right),\epsilon \right)\end{array}$ (159)

is valid for all periodic zeroes (minimums) and periodic maximums of $q_l$ (see Figures 12(a)-(c)), the even symmetry also holds for the midpoint of $\left[0,P_p\right]$ for all $q_l$ and, consequently, for $q_p$ :

$q_p\left(\tau -\frac{P_p}{2},\epsilon \right)=q_p\left(-\left(\tau -\frac{P_p}{2}\right),\epsilon \right).$ (160)

Vanishing of $q_p$ at the end points of interval $\left[0,P_p\right]$ and the even symmetry of $q_p$ with respect to the midpoint of $\left[0,P_p\right]$ are visualized for various $L$ and ${\upsilon }_l$ in Figures 13(a)-(c), Figures 14(a)-(b). We also observe that periodic zeroes of $q_l$ frequently do not become zeroes of $q_p$ since zeroes of $q_l$ happen at the various times for different $l$ .

Random pulsatory cn²-noise $q_{p,B}$ with the Bernoulli frequencies ${\upsilon }_{l,B}=\left[1,1/2,1/3,1/5,1/7\right]$ , the same random amplitudes $A_l$ , $\epsilon =0.999866$ , and periods of the random pulsatory cn²-noise $P_{p,B}=65.9867,329.933,2309.53$ is shown on interval $\tau \in \left[0,P_{p,B}\right]$ in Figure 14(a), Figure 14(b), and Figure 14(c) for $L=3,4,5$ , respectively.

(a)

(b)

(c)

Figure 14. Plots of the random pulsatory cn²-noise for the Bernoulli frequencies and various $L$ : (a)—$L=3$ , (b)—$L=4$ , (c)—$L=5$ .

Periods $P_{p,l,B}=2K\left(\epsilon \right)/{\upsilon }_{l,B}$ of the lth mode $q_{l,B}$ represent the following increasing sequence:

$P_{p,l,B}=\left[10.9978,21.9956,32.9933,54.9889,76.9845\right],$ (161)

where

$P_{p,B}=LCM\left(P_{p,1,B},\cdots ,P_{p,l,B},\cdots ,P_{p,L,B}\right).$ (162)

For $L=5$ , numbers $n_{p,l,B}$ of periods $P_{p,l,B}$ in $P_{p,B}$ decrease the same way as for the random oscillatory cn-noise: $n_{p,l,B}=P_{p,B}/P_{p,l,B}=\left[210,105,70,42,30\right]$ . The rate of stochastization of $q_{p,B}$ has a maximal value since $P_{p,B}$ increases as a product of prime numbers. Therefore, for $L=3$ , $P_{p,B}$ is six times larger than the period of the first mode $P_{p,1,B}$ . For $L=4$ , $P_{p,B}$ is 30 times greater, and, for $L=5$ , $P_{p,B}$ is 210 times larger than $P_{p,1,B}$ . To summarize, $q_{p,B}$ models a smooth random pulsatory variable with an unbounded period at large values of $L$ because $P_{o,B}\to \infty$ as $L\to \infty$ .

We compare the rates of stochastization of the random pulsatory cn²-noise for the Fourier and Bernoulli frequencies in Figure 15, where $\ln \left(P_{p,F}\right)$ versus $L$ is displayed by green circles and a dotted black line and $\ln \left(P_{p,B}\right)$ by blue diamonds and a dashed black line. For $L\ge 4$ , the period of the pulsatory cn²-noise $P_{p,B}$ with the Bernoulli frequencies increases exponentially since

$\ln \left(P_{p,B}\right)=-4.9339+2.6552L,P_{o,B}=0.0071983\exp \left(2.6552L\right),$ (163)

where coefficients are obtained by the least-squares fit.

Density $p_o\left(g_o\right)$ of the Gaussian probability distribution with mean ${\mu }_o$ , variance ${\sigma }_o^2$ , and normalization coefficient $c_o$ for random oscillatory variable $g_o$ , which may take both positive and negative values as $-\infty <g_o<+\infty$ ,

$p_o\left(g_o\right)=\frac{c_o}{{\sigma }_o}\exp \left(-\frac{{\left(g_o-{\mu }_o\right)}^2}{2{\sigma }_o^2}\right),c_o=\frac{1}{\sqrt{2\pi }}$ (164)

satisfies the following normalization condition:

${\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{p}_o\left(g_o\right)}\text{d}{g}_o=1.$ (165)

Consider density $p_p\left(q_p\right)$ of the truncated Gaussian probability distribution [9] with mean ${\mu }_p$ , variance ${\sigma }_p^2$ , and normalization coefficient $c_p$ for random pulsatory variable $q_p$ , which may take only positive values since $0\le q_p<+\infty$ ,

$p_p\left(q_p\right)=\frac{c_p}{{\sigma }_p}\exp \left(-\frac{{\left(q_p-{\mu }_p\right)}^2}{2{\sigma }_p^2}\right).$ (166)

This density satisfies the correspondent normalization condition

${\displaystyle \underset{0}{\overset{+\infty }{\int }}{p}_p\left(q_p\right)}\text{d}{q}_p=1.$ (167)

To find the normalization coefficient of $p_p\left(q_p\right)$ , we decompose the normalization integral into two parts

Figure 15. Plots of $\ln \left(P_p\right)$ versus $L$ of the random pulsatory cn²-noise.

${\displaystyle \underset{0}{\overset{+\infty }{\int }}{p}_p\left(q_p\right)}\text{d}{q}_p={\displaystyle \underset{0}{\overset{{\mu }_p}{\int }}{p}_p\left(q_p\right)}\text{d}{q}_p+{\displaystyle \underset{{\mu }_p}{\overset{+\infty }{\int }}{p}_p\left(q_p\right)}\text{d}{q}_p$ (168)

and take the integrals separately

${\displaystyle \underset{0}{\overset{{\mu }_p}{\int }}{p}_p\left(q_p\right)}\text{d}{q}_p=c_p\sqrt{\frac{\pi }{2}}\text{erf}\left(\frac{{\mu }_p}{\sqrt{2}{\sigma }_p}\right),{\displaystyle \underset{{\mu }_p}{\overset{+\infty }{\int }}{p}_p\left(q_p\right)}\text{d}{q}_p=c_p\sqrt{\frac{\pi }{2}}.$ (169)

Thus, an algebraic form of the normalization condition becomes

$c_p\sqrt{\frac{\pi }{2}}\left[1+\text{erf}\left(\frac{{\mu }_p}{\sqrt{2}{\sigma }_p}\right)\right]=1.$ (170)

Solving it with respect to $c_p$ and back-substituting yield

$p_p\left(q_p\right)=\frac{\sqrt{\frac{2}{\pi }}\exp \left(-\frac{{\left(q_p-{\mu }_p\right)}^2}{2{\sigma }_p^2}\right)}{{\sigma }_p\left[1+\text{erf}\left(\frac{{\mu }_p}{\sqrt{2}{\sigma }_p}\right)\right]},$ (171)

where the error function

$\text{erf}\left(w\right)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{0}{\overset{w}{\int }}\exp \left(-u^2\right)}\text{d}u.$ (172)

Density $p_p\left(q_p\right)$ of the truncated Gaussian probability distribution is shown by a solid blue curve and density $p_o\left(q_p\right)$ of the Gaussian probability distribution is displayed by a dotted blue curve in Figure 16(a), Figure 16(a), Figure 16(c) for ${\sigma }_p^2=1$ and${\mu }_p=-1/2,0,+1/2$ , respectively. A common area beneath $p_p\left(q_p\right)$ and $p_o\left(q_p\right)$ is shadowed by a blue color. A truncated part under $p_o\left(q_p\right)$ is shadowed by a yellow color. Due to the normalization condition, this area is equal to an area between the solid and dotted curves for $q_p\ge 0$ , which is also visualized by yellow.

Figure 16(a) demonstrates that $p_p\left(q_p\right)$ looks like the tail of a rescaled normal probability distribution for negative ${\mu }_p$ . For ${\mu }_p=0$ in Figure 16(b), the area between $p_p\left(q_p\right)$ and the $x$ -axis exceeds twice the blue shadowed area between $p_o\left(q_p\right)$ and the $x$ -axis since $\text{erf}\left(0\right)=0$ and

$\underset{{\mu }_p\to 0}{\lim }{p}_p\left(q_p\right)=2p_o\left(q_p\right).$ (173)

As ${\mu }_p$ becomes positive in Figure 16(c), the yellow shadowed area between $p_p\left(q_p\right)$ and $p_o\left(q_p\right)$ gradually diminishes because $\text{erf}\left(\infty \right)=1$ and

$\underset{{\mu }_p\to \infty }{\lim }{p}_p\left(q_p\right)=p_o\left(q_p\right).$ (174)

Histograms in Figure 17(a) and Figure 17(b) are obtained with the same random amplitudes $A_l$ , $\epsilon =0.999866$ , and $L=5$ for the Fourier frequencies and Bernoulli frequencies, correspondingly. The data sample sizes $K_{p,F}=388$ and $K_{p,B}=4649$ are the same as the number of computational points of plots $q_{p,F}$ and $q_{p,B}$ in Figure 13(c) and Figure 14(c), respectively.

(a)

(b)

(c)

Figure 16. Plots of the truncated Gaussian probability distribution for ${\sigma }_p^2=1$ and various ${\mu }_p$ : (a)—${\mu }_p=-1/2$ , (b)—${\mu }_p=0$ , (c)—${\mu }_p=+1/2$ .

In Figure 17, ${\mu }_{p,F}=0.429347$ , ${\mu }_{p,B}=0.0007767$ together with ${\sigma }_{p,F}=0.461963$ , ${\sigma }_{p,F}=0.717548$ are computed by the numeric least-squares fit, while the truncated Gaussian probability distribution is shown by a solid orange line. As $L$ grows, $p_{p,F}$ evolves into the truncated normal distribution. Even for $L=5$ , there are obvious deviations from the truncated Gaussian distribution. As $L$ increases, $p_{p,B}$ transforms into the truncated Gaussian distribution much faster. For $L\ge 5$ , the truncated normal distribution is emulated with graph accuracy.

(a)

(b)

Figure 17. Histogram of the random oscillatory cn²-noise: (a)—for the Fourier frequencies, (b)—for the Bernoulli frequencies.

6. Conclusions

We have meticulously studied six oscillatory and pulsatory dynamic models of a conservative perturbation with vanishing total energy with the help of the Hamiltonian and Newtonian dynamics of the first triplet of copolar elliptic functions and the first triplet of copolar elliptic functions squared.

As $\epsilon \to 1_{-}$ , the first Hamiltonian system $f=f\left(\tau ,\epsilon \right)$ represents the nonlinear sn-oscillation in range $f\in \left[-1,+1\right]$ with period $4K\left(\epsilon \right)$ that displays a sequence of positive and negative rectangular pulses of the tanh-shape in $f\in \left[0,+1\right]$ ,$f\in \left[-1,0\right]$ for $\tau \in \left[4nK\left(\epsilon \right),2K\left(\epsilon \right)+4nK\left(\epsilon \right)\right]$ , $\tau \in \left[2K\left(\epsilon \right)+4nK\left(\epsilon \right),4K\left(\epsilon \right)+4nK\left(\epsilon \right)\right]$ , respectively, where $n=0,\pm 1,\pm 2,\cdots$ . The second Hamiltonian system $g=g\left(\tau ,\epsilon \right)$ describes the nonlinear cn-oscillation in range $g\in \left[-1,+1\right]$ with period $4K\left(\epsilon \right)$ that manifests a sequence of positive and negative point pulses of the sech-shape in $g\in \left[0,+1\right]$ , $g\in \left[-1,0\right]$ for $\tau \in \left[-K\left(\epsilon \right)+4nK\left(\epsilon \right),+K\left(\epsilon \right)+4nK\left(\epsilon \right)\right]$ , $\tau \in [K\left(\epsilon \right)+4nK\left(\epsilon \right),3K\left(\epsilon \right)+4nK\left(\epsilon \right)]$ , correspondingly. The third Hamiltonian system $h=h\left(\tau ,\epsilon \right)$ specifies the nonlinear dn-pulsation with period $2K\left(\epsilon \right)$ that displays a sequence of positive point pulses of the sech-shape in range $h\in \left[\delta ,1\right]$ for $\tau \in [-K\left(\epsilon \right)+2nK\left(\epsilon \right),+K\left(\epsilon \right)+2nK\left(\epsilon \right)]$ .

As $\epsilon \to 1_{-}$ , the fourth Hamiltonian system $p=p\left(\tau ,\epsilon \right)$ presents the nonlinear sn²-pulsation with period $2K\left(\epsilon \right)$ that exhibits a sequence of positive rectangular pulses of the tanh²-shape in range $p\in \left[0,1\right]$ for $\tau \in \left[2nK\left(\epsilon \right),2K\left(\epsilon \right)+2nK\left(\epsilon \right)\right]$ . The fifth Hamiltonian system $q=q\left(\tau ,\epsilon \right)$ expresses the nonlinear cn²-pulsation with period $2K\left(\epsilon \right)$ that is visualized by a sequence of positive point pulses of the sech²-shape in range $q\in \left[0,1\right]$ for $\tau \in \left[-K\left(\epsilon \right)+2nK\left(\epsilon \right),+K\left(\epsilon \right)+2nK\left(\epsilon \right)\right]$ . The sixth Hamiltonian system $r=r\left(\tau ,\epsilon \right)$ defines the nonlinear dn²-pulsation with period $2K\left(\epsilon \right)$ that is established by a sequence of positive point pulses of the sech²-shape in range $r\in \left[{\delta }^2,1\right]$ for $\tau \in \left[-K\left(\epsilon \right)+2nK\left(\epsilon \right),+K\left(\epsilon \right)+2nK\left(\epsilon \right)\right]$ .

Using the Hamiltonian systems $g$ and $q$ of conservative perturbations with the vanishing total energy, two stochastic models have been developed. First, the smooth stochastic model $g_o$ of the random oscillatory cn-noise, which depends on random amplitudes $A_l$ , elliptic modulus $\epsilon$ , frequency sequence ${\upsilon }_l$ , and number $L$ of modes $g_l$ . Dependence of $g_l$ on parameters $A_l,\epsilon ,{\upsilon }_l$ have been studied and the odd symmetry of $g_o$ for all sequences of frequencies have been shown. Numerical experiments show that for the Bernoulli frequencies $g_o$ approaches a smooth random oscillatory variable with an unbounded period and the Gaussian probability distribution as an exponential function of $L$ .

Second, the smooth stochastic model $q_p$ of the random pulsatory cn²-noise, which is determined by $A_l,\epsilon ,{\upsilon }_l$ , and number $L$ of modes $q_l$ . We then explore dependence of $q_l$ on $A_l,\epsilon ,{\upsilon }_l$ and prove the even symmetry of $q_p$ for all sequences of frequencies. Numerical experiments demonstrate that for the Bernoulli frequencies $q_p$ approaches a smooth random pulsatory variable with an unbounded period and the truncated Gaussian probability distribution as an exponential function of $L$ .

A point-by-point comparison of the stochastic models of the current paper with experimental data is impossible since smooth stochastic functions with an unbounded period are never repeated. Each implementation of the stochastic models produces another version of stochastic data. However, a qualitative comparison with experimental data is presented in the current paper, especially in Figure 9(c) and Figure 14(c), where a typical realization of white noise is represented for oscillations and pulsations, respectively. A single quantitative issue, which is used to compare theoretical and experimental stochastic data, is a probability distribution of random variables. This comparison is provided in Figure 11(b) and Figure 17(b), where the stochastic data generated by the random models fit with graph accuracy the Gaussian probability distribution and the truncated Gaussian probability distribution for the oscillatory white noise and the pulsatory white noise, correspondingly.

Although the paper was originally aimed at development of exact wave turbulence, its results are applicable in a much broader realm of applications since the stochastic models are constructed independently from the deterministic models. The developed random models describe conservative perturbations with vanishing total energies, which should be initiated once and then need not an external source of energy to maintain them. Another attractive property of elliptic functions is a variable period, which allows the models of the oscillatory white noise and the pulsatory white noise to be constructed with an unbounded period. The conventional generators of random numbers produce discrete sets of random variables. However, the mathematical analysis of stochastic processes requires smooth random variables, which have been developed in the current paper. Therefore, the presented results are of interest in various applications dealing with smooth stochastic functions.

Acknowledgements

The support of CAAM and the University of Mount Saint Vincent is gratefully acknowledged. The author thanks a reviewer for helpful comments, which have improved the paper.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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