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The third order explicit autonomous differential equations named as jerk equations represent an interesting subclass of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we show that an algebraically simple system, the Genesio system can be recast into a jerky dynamics and its jerk equation can be derived from one-dimensional Newtonian equation. We also investigate the global dynamical properties of the corresponding jerk system.

The term jerk [

As jerky dynamics can be considered a subclass of three-dimensional dynamical systems an interesting question [

In [_{1} to JD_{7}) of jerky dynamics as a hierarchy of quadratic jerk equations with increasingly many terms as seen in _{1} and JD_{2} in more detail and identified the regions of parameter space over which they exhibit chaos.

In this paper, we show that the Genesio system can be recast into a jerky dynamics by an affine transformation and the resulting form belongs to class JD_{2}. Moreover it is derived from one-dimensional Newtonian equation that is, it is a Newtonian jerky dynamics. Furthermore we investigate the global dynamics of that jerk equation and also show that it shares the common route to chaos as systems in class JD_{2}.

Consider the class of systems which can be written as a

scalar ordinary differential equations:

where n is the order and denotes the nth derivative of the scalar state variable. Clearly an nth order scalar ODE can be written as a system of n first order ODEs. On the other hand, the following systems of equations can be rewritten as a scalar ODE:

For third order scalar ODEs, where is position, is the change in acceleration which is generally called the jerk, and the resulting dynamics are called jerky dynamics. In [

Theorem 2.1 Consider a three-dimensional system of the form

where, is a matrix with constant coefficients and a three-dimensional vector solely nonlinear functions in x, y, z that are twice differentiable and do not contain additive constants. If

and

then the system is topologically conjugate to a jerky dynamics via a state transformation.

The state transformation in Theorem 2.1 has the restriction that the state variable in the corresponding jerky dynamics is equal to one of the state variables in Equation (2). It was shown that 16 out of 20 simple chaotic systems considered in [

Definition 2.1. Let A be an n by n matrix and b be an n by 1 vector. The pair is controllable if the matrix

is nonsingular. The matrix K is called the controllability matrix.

Theorem 2.2. Consider the system

where A is an n by n matrix, b, c are n by 1 vectors and f is a real-valued function. If is controllable, then the system is topologically conjugate to a scalar ODE via an affine transformation.

It is possible that a dynamical system that is contained in the class specified by Equation (5) can be converted simultaneously into two or three jerky dynamics in different variables (the jerky dynamics in the certain variable is unique, if it exists).

To obtain dynamical systems of the class in Theorem 2.2 with two simultaneously existing jerky dynamics, e.g., in x and y one has to restrict the nonlinear function such that it is only function of y, i.e.,. This follows directly from Equation (2). In addition to the conditions (3) and (4)

that ensure the existence of the jerky dynamics in x, there are also corresponding constraints for the jerky dynamics in y that read explicitly

where f_{1} and f_{2} are functions of the indicated arguments. Any dynamical system of functional form (5) with that fulfills the conditions (6)-(9) can be recast into an equivalent jerky dynamics in its variables x and y. For simultaneously existing jerky dynamics in two other variables one has to take into account permutations of variables and indices, respectively.

For dynamical systems that possess simultaneously three jerky dynamics, further constraints apply. Clearly, must hold. Furthermore, in addition to Equations (6)-(9) there is a third condition reading explicitly

If a jerky dynamics can be derived from one-dimensional Newtonian equation by taking its derivative with respect to time we call the dynamics Newtonian jerky. The following theorem [

Theorem 2.3. Any jerky dynamics of the functional form

with p and q being differentiable and integrable functions of their arguments x and, is Newtonian jerky.

In the qualitative theory of dynamical systems [12,13] gradient systems play an interesting role. For these systems, one can rule out the existence of oscillatory solutions just by considering their vector fields. In particular, a dynamical system is a gradient system if its vector field results from the gradient of a scalar potential. In [

Theorem 2.4. Newtonian jerky dynamics are not gradient systems.

Looking at the functional form of a jerky dynamics, it is highly nontrivial to decide whether it can have chaotic solutions for some parameter ranges or not. For some subclasses of jerky dynamics one can derive a simple criterion under what circumstances aperiodic or chaotic solutions cannot appear. Consider the jerky dynamics (12) with

where r and s are functions of the indicated arguments and fulfills the Schwarz condition . As a consequence the jerky dynamics (12) can be rewritten as

or equivalently,

Direct integration of Equation (15) yields

This shows most clearly that the left-hand side of Equation (16) can be interpreted as an oscillator coupled to an internal driving mechanism or feedback (the righthand side of Equation (16)) that is an integral over the history of its motion. This fact has some consequences for the possible dynamics of the jerky system (15).

Theorem 2.5. [

The Genesio system, which was proposed by Genesio and Tesi [

where a, b, c are real parameters.

Theorem 3.1. The Genesio system (17) can be recast into a jerky dynamics, and the resulting form belongs to class JD_{2}.

Proof. The Equation (17) can be written as

where

The matrix is nonsingular since. So, by Definition 2.1 K is the controllability matrix and the pair is controllable. Hence the Genesio system can be recast into a jerky dynamics via an affine transformation by Theorem 2.2.

Application of the invertible transformation to the Equation (17) yields

Using the linear and invertible transformation

and then replacing by x we write Equation (19) as

Comparing with _{2} with

Theorem 3.2. Genesio system has no equivalent jerky dynamics in the variables y and z.

Proof. For a simultaneous existence of jerky dynamics in y and/or z first, the following conditions must be satisfied:

For the Equation (16) we have

Since the second equation in (22) does not hold we can conclude that the Equation (17) cannot have a jerky dynamics in z.

From the condition (8) we get which is absurd. This follows that the Genesio system cannot have a jerky dynamics in y also.

Theorem 3.3. The Genesio system is a Newtonian jerky dynamics.

Proof. The Equation (21) can be put in the form (12)

where and. Since both p and q are differentiable and integrable functions of their arguments x and the jerky dynamics (21) is Newtonian by Theorem 2.3.

Corollary 3.1. The Genesio system is not a gradient system.

Proof. From Theorem 3.3 we know that the Genesio system is a Newtonian jerky. Since Newtonian jerky dynamics are not gradient systems by Theorem 2.4, the Genesio system is not a gradient system.

Theorem 3.4. The Genesio system exhibits chaotic solutions for some parameter ranges.

Proof. We can write Equation (21) as

Integration of Equation (23) yields

The memory term changes sign as x variesthat is, it is neither positive semi-definite nor negative semi-definite for all x. Therefore the Genesio system can have chaotic solutions for some parameter ranges by Theorem 2.5.

Given the jerky dynamics

the equilibria can be found by assuming that it has a fixed point, , , which leads to

, or. So there are two equilibria: and. Linearizing Equation (24) about the equilibrium provides one real and a pair of complex conjugate eigenvalues along with the following characteristic equation

and linearizing the Equation (24) about the other equilibrium yields the following characteristic equation

According to Routh-Hurwitz criteria, the equilibrium is stable (i.e., the real part of all roots of Equation (25) are negative) only if the conditions

are fulfilled. For, the fixed point becomes unstable and the two complex roots of (25) cross the imaginary axes, while the third root remains real and negative. Therefore at a stable limit cycle arises via a Hopf bifurcation.

has the same stability characterization. If

then Equation (26) satisfies the Routh-Hurwitz criteria and at a stable limit cycle arises via a Hopf bifurcation.

The volume contraction rate of the Equation (24) is

i.e., , which can be solved to yield

. When c is positive, the jerky dynamics Equation (24) is dissipative with solutions for that contract at an exponential rate onto an attractor of zero volume that may be an equilibrium point, a limit cycle, or a strange attractor. When c = 0, is zero and the phase space volume conserved and the dynamical system is conservative. When c is negative, is positive and the volume expands exponentially fast and there are only unstable fixed points. Therefore the dynamics diverges for if the initial value does not lie exactly at such unstable set.

The Equation (24) has three free parameters a, b and c and the position of equilibria

depends on the parameter a. To get the parameter independent equilibria we use the transformation

for x and t yielding the new quantities and t. With the substitution of Equation (29), Equation (24) becomes

introducing as new parameters and dropping the overbars we write

Equation (30) possesses two stationary points, ,. Analyzing their stability leads to the characteristic equation

It follows that is stable only for, and and it becomes unstable at the line via a Hopf bifurcation. Similarly, is stable only for, and and it becomes unstable at the line via a Hopf bifurcation. These stability properties of fixed points also reflect the symmetry of the Genesio system.

Equation (30) is invariant under and. Therefore, knowing the solution of of Equation (30) for a certain value of parameter and certain initial values, the dynamics of the corresponding sign inverted and initial values is given by.

Summarizing the results, we conclude that possibly interesting dynamics of the Genesio system (17) is described by Equation (30). Due to the discussed symmetry of this equation, we need only to consider initial values close to one of the two stationary points. Then the most interesting region of the -parameter plane is the one for and positive. From the studies in [11,15,16], we also know that this parameter region contains homoclinic orbits of the other stationary point, , , which here is also a saddlefocus.

Besides the local stability and Hopf bifurcation analysis of Equation (30), we also computed the set of all Lyapunov exponents for different values of parameters and and use to determine and classify the long-time dynamics of the Genesio system. Numerical calculations are performed using Mathematica and iDMC softwares, and RKF45 and RK2Imp are used as numerical algorithms with step size 0.001. The initial values are chosen as, , which are close to the fixed point, ,.

For the parameter regions or, no bounded solutions have been found. This suggests that the Genesio system does probably not possess at all a stable attractor in these regions. For the region and the resulting Lyapunov spectra are shown in

its boundary is formed by two tongues that reach into the limit cycle domain. For smaller parameter values one only finds chaotic points at and. Moreover there are islands with bounded dynamics (limit cycles and strange attractors) located within the diverging region.

In

This region consists of an infinite series of perioddoubling bifurcations. It also contains many narrow windows, which are called limit cycle windows. As is further increased the limit cycle windows break down and eventually disappear. In

where the bifurcation diagram shows the limit cycle solutions, the largest Lyapunov exponent is negative as well as the dimension of the attractor is two. However, for the values of the parameter where the bifurcation plot shows the existence of aperiodic behavior (chaotic), the largest Lyapunov exponent is positive as well as the dimension of the attractor is a non-integer 2.15119 between two and three for the parameter values and.

For a computed value of, we record the successive local maxima of for a trajectory on the strange attractor.