Synchronous dynamics of a delayed two-coupled oscillator

This paper presents a detailed analysis on the dynamics of a delayed twocoupled oscillator. Linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. By means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of time delay on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. Moreover, we illustrate our results by numerical simulations.


Introduction
Synchronization phenomena are common in nature (see Nijmeijer and Rodriguez-Angeles [26] and references therein).An important avenue of study in synchronization focuses on coupled oscillators.One classical example is the Kuramoto model [22], which assumes full connectivity of the network.By using a combination of the Lyapunov functional method, matrix inequality techniques and properties of Kronecker product, Alofi et al. [1] investigated a so-called power-rate synchronization problem for the collective dynamics among genetic oscillators with unbounded time-varying delay.Wang et al. [27] investigated the synchronization of coupled Duffing-type oscillators.By means of the residue harmonic balance method, Xiao et al. [29] investigated the approximations to the periodic oscillations of the fractional order van der Pol equation.
The study of the dynamical behavior of oscillating systems is a central issue in physics and in mathematics.These systems provide basic and general results that found major applications not only in physics, but also in all the other branches of science.The harmonic oscillator is the simplest, and more fundamental theoretical model of oscillatory phenomena.Damped and forced oscillators provide, also, very fundamental results in physics and engineering.
In this paper, we study the existence and stability of periodic orbits in a delayed twocoupled harmonic oscillator modelled by the following system of delay differential equations üi (t) + u i (t) + ε ui (t) = ε f (u i+1 (t − τ)), (1.1) where f ∈ C 1 (R; R) with f (0) = 0, τ ≥ 0 and ε > 0 are constants, and as well as in all subsequent expressions, the index i is taken to modulo 2, so that, for instance, x 3 = x 1 .We also assume that each oscillator has no self-feedback and signal transmission is delayed due to the finite switching speed of oscillator.It can be seen that in system (1.1) the growth rate of one oscillator depends on the feedback from the other.Such a network has been found in a variety of neural structures and even in chemistry and electrical engineering.Despite the low number of units, two-oscillator networks with delay often display the same dynamical behaviors as large networks and, can thus be used as prototypes for us to understand the dynamics of large networks with delayed feedback (see, for example, [8, 14-16, 20, 21]).
Here, we emphasize the importance of temporal delays in the coupling between cells, since in many chemical and biological oscillators (cells coupled via membrane transport of ions), the time needed for transport of processing of chemical components or signals may be of considerable length.Since we have symmetric coupling of identical oscillators, (1.1) has the reflection symmetry of interchange of two oscillators.Although model (1.1) is a little simple, it allows us to have a depth analysis and then to gain insight into possible mechanisms behind the observed behavior.
It is easy to see that every continuous and can be characterized by the scalar delay differential equation Such solutions are said to be synchronous.Equation (1.2) has been used to model a variety of other biological and physical phenomena, and studied by many researchers (see, for example, [24,25]).More precisely, the local stability analysis has been discussed by a lot of investigators [3-6, 9, 10, 23] and complex dynamics including limit cycles and tori are also obtained by Campbell [7], Hou and Guo [19], Zhang and Guo [30,31].The existence of nonconstant periodic solutions of (1.2) has been proved in [2].Our goal in this paper is to study the existence and stability of periodic orbits of (1.1).The plan for this paper is as follows.In Section 2, we consider the linear stability of the trivial solution (1.1).Section 3 is devoted to the spatio-temporal patterns of Hopf bifurcated periodic solutions when the the trivial solution lose its stability.In Section 4, we discuss the bifurcation direction and stability of periodic solutions emerging from from the trivial solution.In Section 5, we illustrate our results with some numerical simulations.Finally, some conclusions are made in Section 6.
, then for each j and τ ≥ 0, p j (τ, •) has only zero points λ satisfying Re λ < 0 and has no purely imaginary zero point.
Proof.It follows from (ε, α) ∈ D 1 ∪ D 4 that ε|α| < 1.We first notice the fact that there exist at most a finite number of zeros of p j (τ, λ) in right half-plane for each j ∈ {0, 1}.Indeed, for any zero λ of p j (τ, λ), This implies that there is a real number η such that all zeros of p j (τ, λ) satisfy Re λ < η.
Clearly, p j (τ, λ) is an entire function.Hence, there can only be a finite number of zeros of p j (τ, λ) in any compact set.Namely, there exist only a finite number of zeros in any vertical strip in the complex plane.We can regard λ as the continuous function of τ according to the implicit function theorem.Notice that which has exactly two zero points with negative real parts.Recall the fact that all zeros of p j (τ, λ) are simple and continuously depend on τ, then there exists a critical value τ 0 such that p j (τ, λ) has only zero points with negative real parts if τ ∈ [0, τ 0 ), and that as τ increases and passes through τ 0 , the zero points with positive real parts may appear.Thus, p j (τ 0 , λ) has a pair of purely imaginary zero points ±iω, where ω > 0 is a solution to (2.5).In view of (ε, α) ∈ D 1 ∪ D 4 , we see that τ 0 = ∞.This completes the proof.
(iii) Using a similar argument as that in the proof of Lemma 2.1, we can regard λ as the continuous function of τ according to the implicit function theorem.If τ = 0 and (ε, , then p 0 (τ, λ) (respectively, p 1 (τ, λ)) has exactly one zero point with positive real parts but p 1 (τ, λ) (respectively, p 0 (τ, λ)) has only zero points with negative real parts.Recall the fact that all zeros of p j (τ, λ) are simple and continuously depend on τ, then there exists a critical value τ j,0 such that the number of zero points of p j (τ, λ) with positive real parts keeps the same if τ ∈ [0, τ 0 ).It follows from conclusions (i) and (ii) that as τ increases and passes through τ 0 , only one zero point of p j (τ, λ), denoted by λ * (τ), varies from a complex number with a negative real part to a purely imaginary number and then to a complex number with a positive real part.In fact, the proof of conclusion (i) yields that τ 0 = τ + j,0 > 0. We can repeat the same analysis to conclude that there exists next critical value τ j,1 such that the number of zero points of p j (τ, λ) with positive real parts keeps the same if τ ∈ (τ + j,0 , τ j,1 ), and that as τ increases and passes through τ j,1 , a new zero point of p j (τ, λ) varies from a complex number with a negative real part to a purely imaginary number and then to a complex number with a positive real part.Similarly, it follows from the proof of conclusion (i) that τ j,1 = τ + j,1 .By induction, we can draw the conclusion that the number of zeros of p j (τ, λ) with positive real parts increases as τ increases.This completes the proof.
Proof.Using a similar argument as that in the proof of Lemma 2.2, we can prove conclusions (i)-(iii).We now prove conclusion (iv).First we notice the fact that when 0 < α ≤ 1, and when −1 < α ≤ 0, and Thus, there exists an nonnegative integer m j such that Lemma 2.4.
It follows from the above lemma that we have the following results on the linear stability of the equilibrium x * = 0 of system (1.1).
where m 0 and m 1 are given in Lemma 2.3.

Spatio-temporal patterns of periodic solutions
Throughout this section, we always assume that (ε, Lemmas 2.2 and 2.3, together with the Hopf theorem (see, pp. 332 in [18]), imply that a Hopf bifurcation for (1.1) occurs at each τ = τ ± j,k > 0. Namely, in every neighborhood of (x * = 0, τ * = τ ± j,k ) there is a unique branch of periodic solutions x j,k (t, τ) with x j,k (t, τ) → 0 as τ → τ ± j,k .The period P j,k (γ, τ) of x j,k (t, τ) satisfies that P j,k (γ, τ) → 2π/β ± as τ → τ ± j,k .In what follows, we aim to analyze the spatio-temporal patterns of these bifurcated periodic solutions.It is well-known that the symmetry of a system is important in determining the patterns of oscillation that it can support.To explore the possible (spatial) symmetry of the system (1.1), we need to introduce two compact Lie groups.One is the cycle group S 1 , the other is Z 2 , the cyclic group of order 2 (the order of a finite group is the number of the elements it contains).Clearly, we have Lemma 3.1.Denote by ρ the generator of the cyclic subgroup Z 2 .Define the action of 2 ) and i (mod 2).Namely, F is Z 2 -equivariant.This completes the proof.
Here, we consider the following subgroup of Z 3 × S 1 to describe the symmetry of periodic solution of system (1.1) (see [13] for more details): The two equations in (3.1) imply that the Σ-fixed-point set of SP ω is itself, i.e., Fix(Σ, SP ω ) = SP ω .Thus, the general symmetric local Hopf bifurcation theorem (Theorem 2.1 in [28]) enables us we obtain the following result on the existence of smooth local Hopf bifurcations of wave solutions.

Conclusion
The goal of this paper is to study the existence and stability of periodic orbits of delay differential equations.To achieve this, a novel model based on a delayed two-coupled harmonic oscillator is proposed.The local Hopf bifurcations and the spatio-temporal patterns of Hopf bifurcating periodic orbits are also investigated.Numerical simulations are adopted to validate the theoretical results.By using different suitable parameters and coefficient numbers, the simulation results reveal that the bifurcating periodic solutions are orbitally asymptotically stable.