Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay

In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method.


Introduction
In more recent decades, the bifurcation theory in dynamical system has become a research hotspot, which is widely used in the fields of physics, chemistry, medicine, finance, biology, engineering and so on [3,12,13,15,19,20,22,24,[30][31][32][33][34][35]. In particular, the bifurcation phenomenon of the system is analyzed, and the dynamic characteristics of the nonlinear model are characterized. In real life, the mathematical models to solve practical problems have been described based on nonlinear dynamical systems, in which, how to study the dynamic characteristics of the high-dimensional nonlinear system is very important. In engineering, Toda oscillators (named after the Morikazu-Toda) refer to a dynamical system employed to model a chain of particles with exponential potential interaction between neighbors. The Toda oscillators model has been extensively applied in engineering [21,23,25]. It is noteworthy that it acts as a simple model to clarify the phenomenon of self-pulsation, namely, a quasiperiodic pulsation of the output intensity of a solid-state laser in the transient regime [16].
For the different control parameters of the chains of coupled Toda oscillators with external force, most scholars have investigated the effect of external force (F, ω) on the system from the numerical continuation and experimental simulation, they suggest that the system eventually produces the coexistence stable limit cycle in synchronous region [10,11,17]. Under the action of external force (F, ω), the coexistence stability limit cycle is generated in the synchronous region. This shows that the final trajectories of the Toda oscillators are asymptotically stable in the synchronous region, that is to say, there is a regular stable transmission between the oscillators, and the industrial productivity reaches the maximum. Thus far, the bifurcation phenomenon caused by coupled delay in nonlinear differential equations has been extensively studied, especially, the Hopf-zero bifurcation [1,2,4,5,18,[26][27][28][29], but no studies have been conducted on the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay as we know today. Therefore, it is of great significance to probe into the Hopf-zero bifurcation of system (1). Under the guidance of the bifurcation theory, our study is majorally under the premise that the external force does not exist (considering only the coupled effect between oscillators).
In this study, we investigate the ring system of three unidirectionally coupled Toda oscillators [6]ẋ 1 = y 1 , y 1 = 1 − exp(x 1 ) − αy 1 + γH(x 3 ) + F sin(ωt), x 2 = y 2 , y 2 = 1 − exp(x 2 ) − αy 2 + γH(x 1 ), where x 1 , y 1 , x 2 , y 2 , x 3 and y 3 are the dynamical variables of system (1), α is the damping coefficient, γ is the coupled coefficient (γ > 0), H(x) = exp(x) − 1 is the coupling function, F and ω represent the amplitude and the frequency of the external force, respectively. The symbol (F, ω) denotes the effect of external force on the coupled system (1). In [6], with XPPAUT software package, the author has summarized the dynamic characteristics of system (1) with numerical integration of differential equations and Runge-Kutta method, such as exponential spectrum, projections of phase portraits, bifurcation diagram and periodic multiplier on the parameter plane. The author mainly investigated the peculiar phenomena of the ring structure of the coupled oscillators with the change of coupled coefficient γ under the effect of external force (F, ω). In other words, with the threshold of self-oscillation birth, the further increase of coupled coefficient γ will be followed by the new ring resonance phenomenon, besides when the threshold of self-oscillation birth is exceeded, the interaction between external oscillations and the inner oscillatory mode of system (1) will result in the synchronization region in the parameter plane [6,7].
In the small neighborhood of the equilibrium point of system (1), the coupled time delay should be considered since the propagations between oscillators x 1 , x 2 and x 3 are no longer instantaneous. Assuming (F, ω) = 0, system (1) generated the rich dynamic characteristics with the joint change of the coupled coefficient γ, the coupled time delay τ and control parameters. Furthermore, the appropriate coupled time delay is introduced to facilitate the understanding of high-dimensional bifurcation analysis followed by the generation of a new Toda oscillators model with delay, and the coupled direction in the model is shown in Fig. 1.
In this paper, we mainly study the dynamic behaviors of the following simplified system:ẋ In engineering fields, such a time-delay feedback indicates an unidirectional interaction between oscillators x 1 , x 2 and x 3 . That is to say, the dynamic characteristics of system (2) are determined uniquely by the effect of connection when the oscillators have coupled delay in a ring structure.
The rest of the paper is organized as follows. In Section 2, the existence of Hopf-zero bifurcation in the ring of unidirectionally coupled Toda oscillators with delay is considered. In Section 3, according to the normal form of system (2), the dynamic bifurcation analysis and numerical simulation are conducted. Lastly, in Section 4, the conclusion is drawn.

Hopf-zero Bifurcation analysis and Numerical simulations
In the present section, a clearer analysis of the dynamic bifurcation of system (3) is to be conducted. The normal form on the center manifold of system (3) is reduced to the following form: The detailed calculation and analysis of system (11) are made in the Appendix. Since Since the third equation describes a rotation around the χ-axis, it is irrelevant to our discussion, and we shall omit it. Hence we get the system in the ρ, ζ-plane up to the third orderρ where From Section 2 we have e ±iτ ω = ∓iαω−ω 2 +1, then the coefficient β 11 is sufficiently small. For simplicity, we only discuss the case of β 11 = 0. So system (12) becomeṡ System (13) is equivalent to system (3), we can analyze the dynamic behaviors of system (13) in the neighborhood of the bifurcation critical point, which are obtained by ρ =ζ = 0. Note that: M 0 = (ρ, ζ) = (0, 0) stands for the coexistence equilibrium point; When ρ = 0, system (13) has no solution, then the spatially inhomogeneous steady states does not exist.
In order to give a more clear bifurcation picture, we choose α = 1.35, which satisfy the assumption 0 < α < √ 2 of Theorem 1. Then we have We take τ 0 = τ 2 0 = 10.9919, the characteristic equation (4) has a zero root and a pair of purely imaginary eigenvalues ±0.421307i, and all the other eigenvalues have negative real part. Assume that system (3) undergoes a Hopf-zero bifurcation from the equilibrium point (0, 0, 0, 0, 0, 0). After a simple calculation, we get β 30 = Re[b 11 +c 11 ] = 0.534691, By analyzing the above contents we can obtain the bifurcation critical lines and phase portraits as shown in Fig. 2. From [14] the dynamic characteristic of system (13) in D 1 −D 4 near the critical parameters (α, τ 0 ) are as follows: In D 1 , system (13) has only one trivial equilibrium M 0 , which is a sink. In D 2 , the trivial equilibrium (corresponding to M 0 ) becomes a saddle from a sink, and an unstable periodic orbit (corresponding to M 1 ) appears.
In D 3 , the trivial equilibrium (corresponding to M 0 ) becomes a source from a saddle, the periodic orbit (corresponding to M 1 ) becomes stable, and a pair of unstable periodic solution equilibria (corresponding to M ± 2 ) appears. In D 4 , the trivial equilibrium (corresponding to M 0 ) becomes a saddle from a source, the periodic orbit (corresponding to M 1 ) remains unstable, and the unstable periodic orbit (corresponding to M ± 2 ) disappear.       The first, we choose a group of perturbation parameter value: (κ 1 (µ), κ 1 (µ)) = (0.012, −0.1), which determined by (µ 1 , µ 2 ) = (0.012, −0.1) and belong to region D 1 (see Fig. 3).

Conclusions
In this paper, we have investigated the Hopf-zero bifurcation in the ring of unidirectionally coupled Toda oscillators with delay. We mainly conclude the singularity in a small enough neighborhood of system (3) near equilibrium point, such as the phenomenon of stable coexistence periodic solution and unstable quasiperiodic solution caused by the selfpulsation without external force (F, ω). Moreover, the Hopf zero bifurcation of system (3) on the parameter plane is determined according to the change of control parameters (µ 1 , µ 2 ) and the time delay τ . It shows that there may exhibit a stable trivial equilibrium point, an unstable periodic equilibrium point orbit and a pair of unstable periodic solutions (see Fig. 4(b)). Firstly, the stability of equilibrium and existence of Hopf-zero bifurcation induced by delay were discussed through characteristic equation. It is then followed by the normal form on the center manifold so that the bifurcation direction and stability of Hopf-zero bifurcation could be determined. Finally, in two-dimensional space, we presented numerical simulations to illustrate a homogeneous periodic solution bifurcating from the positive equilibrium when α = 1.35 and τ = τ 0 .
According to the physical meaning of the Toda oscillator model, the trajectory of the oscillator in the synchronous region is asymptotically stable because the more stable the system is, the more regular the transmission between the oscillators is, and the higher the industrial productivity is. In our conclusion, Hopf-zero bifurcation of the model with (F, ω) = 0, γ = 1 occurs when the control parameters α and time delay τ vary in a small range. Then we find that the dynamic characteristics of system (3) will change from stable to unstable, that is, an unstable region will be generated near the equilibrium point. However, the neighborhood is small enough to be ignored in industrial production and can almost be regarded as a stable region. We further confirm that the effect of time delay and the mutual coupling between oscillators will eventually lead to the synchronization region in the parameter plane indicating that the external oscillations and internal oscillations of the system are equally important. Therefore, it is meaningful to analyze the Hopf-zero bifurcation of the Toda oscillators model without external intervention (F, ω).

Equation (A.3) is decomposed to the forṁ
Using the method of Faria and Magalhães [8,9], the above system can be written aṡ where f i (x, y, µ) is homogeneous polynomials of degree j about (x, y, µ) with coefficients in C 3 . Here, define M j to be the operator in V 5 j (C 3 × Ker π) with the range in the same space by represents the linear space of the second-order homogeneous polynomials in seven variables (x 1 , x 2 , x 3 , µ 1 , µ 2 ) with coefficients in C 3 , and it is easy to verify that one can choose such a decomposition 2 ) and Ker(M 1 3 ) represent the complimentary space, respectively, and the span elements of Ker(M 1 2 ) are Next, we will do a little more detailed calculation: On the center manifold, the first equation of system (A.4) can be transform as the following normal form: Step 1. Calculate g 1 2 (x, 0, µ): where a 11 =De −iτ0ω0 e 2iτ0ω0 + e 4iτ0ω0 + 1 1 + ω 2 0 − 1 e iτ0ω0 , a 13 = −Dτ 0 e −iτ0ω0 e iτ0ω0 − 1 e 2iτ0ω0 + e 4iτ0ω0 + 1 , a 21 = 3D 1 τ 0 a 22 = 0, a 23 = 0, a 24 = 0.