Love Dynamical Models with Delay

A sufficient condition for the asymptotic stability of the equilibrium point of a system, which appears as a model for couple of the love affair with time delay, is obtained by applying the technique of linearized method and Hopf-bifurcation.


Introduction
In a pioneering paper [1] and a famous book [2], Strogatz considered a simple pedagogical model describing a love affair. He treated harmonic oscillation phenomena using a topic that is already on the minds of many college students, which is the time evolution of a love affair between a couple. Later, Sprott [3] proposed more realistic nonlinear triangle models for love dynamics (cf. [4] [5]). Moreover, Rinaldi who is an authority in this area, has studied several types of models describing love affairs and published many papers (cf. [6] [7] [8] [9] [10]). They treated the technique of standard linearized method. On the other hand, we study the effect of time delay on the nonlinear dynamical model describing a love affair with feedback between two individuals.
In this paper, we consider the following delay differential equation with feed-  Equation (1) is an extending model of the without delay differential equation which has been proposed by Rinaldi [7] and Rinaldi et al. [11] as a model for the linear system of love dynamics, where R r describes the direct effect to her love on the partner ( ) R t . Next, we introduce another differential model of love with delay proposed by Liao and Ran [12] and Son and Park [13], where are the little bit strong restricted functions with same delay τ . To consider more reality love regime than ordinary differential system (2), they investigate that the stable equilibrium point is destabilized for a delay larger than a threshold value and then bifurcates to a limit cycle via a Hopf bifurcation when Romeo is secure and Juliet is non-secure.
We investigate the first problem how is the condition of the asymptotically stable of the equilibrium point of Equation (1

Stability Criteria of Equilibrium Points
In this section we study the stability of equilibrium points of Equation (1). We have the equilibrium point We investigate the stability of the equilibrium point ( ) where ( ) ( ) Remark 1. We consider ( ) f J are particular forms by taken as two cases: for some odd integer 1 l ≥ , where 0 K > is a real number and 0 J is the concentration parameters related to the switching of the love individual by a Juliet's love function ( ) J t . For 1 l = , the function f in (H 1 ) is considered by [6] and the function f of (H 2 ) is treated in [14]. It seems that (H 2 ) is a more adjust condition than (H 1 ) as situation of love affairs. So, in this paper, we mainly employ the condition (H 2 ).
In the case where (H 1 ) and (H 2 ), respectively, * J is given by the solution of the equation In Equation (5), the case where each assumption (H 1 ) and (H 2 ), we have respectively.
We can show the next theorem by using Routh-Hurwitz theorem (cf. [2] [11] and [15]) for the second-order differential equation.
Then, the equilibrium point * E of Equation (1)  D iz τ = then for such τ , the characteristic Equation (7) has a pair of pure imaginary roots and hence the trivial solution of (5) is not asymptotically stable. Setting i λ µ ν = + in (7) and separating the real and imaginary parts, we get a system of transcendental equations: Here, in formula (9), the variables of the trigonometric functions are covered with (7) and (8). One can write (7) (6) and (8). For implying that zτ will vary in [ ] 0, 2π . This means that e izτ will vary over unit circle. Thus we can let for 0 z ≠ , zτ to be another independent variable σ (where Here, eliminating σ from (11) and (12), we get A necessary and sufficient condition for (11) and (12), we have Then, we obtain the real values of σ which satisfy (11) and (12). Thus, a set of necessary and sufficient condition for the asymptotic stability of the interior equilibrium is c b < . This completes the proof of Theorem 2. Remark 2. The above Theorem 1 and 2 hold for the both functions (H 1 ) and (H 2 ). This talk is motivated by Das et al. [17] [18] and Hamaya et al. [19], that is "Study the stability and the existence of almost periodic solutions of the Equation (5)", and we also regard Theorem 1, 2 and next Theorem 3, 4 as a partial answer in the affirmative for their research.
For the more complicated equation of (3), [4] [10] [11] and [20] have shown the asymptotic stability of the equilibrium point * E under the more complicated conditions using a bifurcation technique and others.

Estimation for the Length of Delay to Preserve Stability and Bifurcation Results
In this section, we suppose that in the absence of delay is asymptotically stable. This is guaranteed if (6) holds. By continuity of solutions and for sufficiently small 1 2 0 τ τ τ = + > , all eigenvalues of (7) have negative real parts provided that no eigenvalue bifurcates from +∞ , which could happen since this is a retarded delay system. It is then possible to use a criterion of Nyquist which we describe below to estimate the range of τ for which * E remains asymptotically stable. Here we follow the approach by [16] [17] [18] for such estimation of τ . We consider the system (5) (4).
then there exists a τ + given by ( ) such that for all τ τ + < , the equilibrium point * E of (5) is asymptotically stable.
Proof. Let ( ) ( ) if W is arc length of a curve encircling the right half plane, the curve ( ) x W will encircle the origin a number of times equal to the difference between the number of poles and the number of zeros of ( ) x W in the right half plane. We see that the conditions for the local asymptotically stability of * E is given by Then, for τ τ + < , the Nyquist criteria holds and τ + is the estimate for the length of the delay τ for which stability is preserved. Thus, the proof of this theorem completes.  [13]. The existence of unique τ is given by Our required τ is given by 0 n = in (18)   To analyze the change in the behavior of the stability of * E with respect to τ , we examine the sign of d d µ τ as µ crosses zero, that is we analyze the sign of ( ) If this derivative is positive (negative) then clearly a stabilization (destabilization) can not take place at that value of τ . We differentiate Equation (9) and Equation (10) with respect to τ . Then setting ˆ, 0 Substituting the values of ŝinτν and ĉosτν from (19) and (20), we get ( ) is the solution of (21) with 2 z ν = . Then, we have ( ) ( ) (C 2 ) If 2 0 P > , then (C 1 ) is satisfied, that is there can be no change of stability.

Oscillatory Criteria
We study the oscillatory behavior of the linearized system (1) involving two distinct delays which are different. But, so far as the author's knowledge goes, there are very few studies on the analysis of oscillation of model with unequal delays.
To make the study mathematically tractable, all the delays are assumed to be equal and equal to the 1/2 of the sum of all the delays. From physiological date, it's not psychology, today delay is nearly 28 -30 hours in [18], from the numerical simulation of the linearized system, it is seen that the pulsated or oscillatory behavior is present, if the individual unequal delay exceed from 5 hours to two days. For simplicity, without much loss of generality, we assume that  So, there cannot exist a bounded non oscillatory solution of (22) when the conditions (i) and (ii) of (23) hold, and therefore, the proof is complete.

Examples
We consider concrete examples of the following linearized equation of Equation  others. Moreover, we have given the simple example for Theorem 2 that the equilibrium point * E of Equation (5), that is Equation (1), is the asymptotically stable by assumptions (6), (H 2 ) and all positive delay 0 τ > .       In the case of (ii) of Examples 5, the solutions of (26) approach the equilibrium point ( ) ( ) ( ) ( ) , 0, 0 x t y t = . Figure 4 is the phase space of (x, y)-plane in Figure 3.