2021
263
18112019
Su Rina srnmath@163.com
Northeast Forestry University, China
Zhang Chunrui math@nefu.edu.cn
Northeast Forestry University, China
Abstract: 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.
Keywords: Toda oscillators, normal form, Hopf-zero.
Article
Hopf-zero bifurcation of the ring unidirectionallycoupled Toda oscillators with delay
Zhang Chunrui math@nefu.edu.cn
Recepción: 18 Noviembre 2019
Revisado: 11 Noviembre 2020
Publicación: 01 Mayo 2021
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 [3,12,13,15,19,20,22,24,30–35]. 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 engineer- ing [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
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
, 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, 26–29], 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]
(1)where 
and 
are the dynamical variables of system (1), 
is the damping coefficient, 
is the coupled coefficient
is the coupling func- tion, . and . represent the amplitude and the frequency of the external force, respectively. The symbol
denotes the effect of external force on the coupled system (1). In [6], with XPPAUT software package, the author has summarized the dynamic characteris- tics 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 pecu- liar phenomena of the ring structure of the coupled oscillators with the change of coupled
coefficient 
under the effect of external force
. In other words, with the threshold of self-oscillation birth, the further increase of coupled coefficient . 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 .., .. and .. are no longer instantaneous. Assuming
= 0, system (1) generated the rich dynamic characteristics with the joint change of the coupled coefficient ., the coupled time delay . 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:
(2)In engineering fields, such a time-delay feedback indicates an unidirectional interaction between oscillators
and 
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 consid- ered. 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
In this section, we will give the existence conditions of Hopf-zero bifurcation of sys- tem (2). To convenience for expression, we make the following assumptions:
Then system (2) can be written as
(3)Clearly, system (3) is equivalent to system (2), where 
is an equilibrium point of system (3). The characteristic equation according to the linear part of system (3) at the equilibrium point is given by
(4)Where
There exists three types of bifurcations as follows:
Case 1: Fixed-point bifurcation. Substituting 
into equation (4), we obtain 
= 1 u√n.er which the three roots of equation (4) with 
are 
= 
then system (3) undergoes a fixed-point bifurcation.
Case 2: Hopf-bifurcation. Substituting
into 
and separating the real and imaginary part, we obtain
(5)Then the time delay 
can be solved from system (5) as
(6)where 
Under the assumption
we have 
After a simple calculation, the transversality conditions are shown by
(7)where “Re” represents the real part.
Next, let 
be the root of
It is not difficult to verify that 
is the root of 
In order to study the distribution of the roots of
, we only need to investigate the distribution of the roots of 
. Substituting the root 
into 
and separating the real and imaginary parts, we obtain
(8)then the time delay 
can be solved from system (5) as
(9)Under the assumption 
, we have 
After a simple calculation, the transversality conditions are shown as follows
(10)Case 3: Hopf-zero bifurcation. Combining cases 1 and 2, we have the following theorem.
Assume 
and 
we have 
The critical time delays and the transversality conditions are given by systems (6), (9), (7) and (10), respectively. Moreover, when
, where 
theroots of characteristic equation (4) are 0 and -
. Then the Hopf-zero bifurcation occursassociated with eigenvalues 0 and 
of system (3). When 
the trivial equilibriumof system (3) is asymptotically unstable.
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:
(11)The detailed calculation and analysis of system (11) are made in the Appendix. Since 
through the change of variables 
and 
if we use double polar coordinates
then we get
and system (11) becomes
Since the third equation describes a rotation around the .-axis, it is irrelevant to our discussion, and we shall omit it. Hence we get the system in the 
-plane up to the third order
(12)Where
From Section 2 we have
then the coefficient
is sufficiently small. For simplicity, we only discuss the case of
. So system (12) becomes
(13)System (13) is equivalent to system (3), we can analyze the dynamic behaviors of sys- tem (13) in the neighborhood of the bifurcation critical point, which are obtained by
When
system (13) has no solution, then the spatially inhomogeneous steady states does not exist. In order to give a mor√e .lear bifurcation picture, we choose 
which satisfy the assumption
of Theorem 1. Then we have
We take 
the characteristic equation (4) has a zero root and a pair of purely imaginary eigenvalues 
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 
, 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 
near the critical parameters 
are as follows:
In D1, system (13) has only one trivial equilibrium Mo, which is a sink.
In D2, the trivial equilibrium (corresponding to Mo) becomes a saddle from a sink, and an unstable periodic orbit (corresponding to M1) appears.
In D3, the trivial equilibrium (corresponding to Mo) becomes a source from a saddle, the periodic orbit (corresponding to M1) becomes stable, and a pair of unstable periodic solution equilibria (corresponding to
.) appears.
In D4, the trivial equilibrium (corresponding to Mo) becomes a saddle from a source, the periodic orbit (corresponding to M1) remains unstable, and the unstable periodic orbit (corresponding to
) disappear
The first, we choose a group of perturbation parameter value:
, which determined by 
and belong to region D1 (see Fig. 3).
The second, we choose a group of perturbation parameter value:
, which determined by
and belong to the region D2 (see Fig. 4).
The third, we choose a group of perturbation parameter value:
, which determined by
and belong to the region .. (see Fig. 5).
The last, we choose a group of perturbation parameter value:
, which determined by 
and belong to the region .. (see Fig. 6).
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 self- pulsation without external force
. Moreover, the Hopf zero bifurcation of system (3) on the parameter plane is determined according to the change of control parameters
and the time delay 
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 
and . = 
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 
occurs when the control parameters . and time delay . 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 
.
In this section, we will derive the normal form of system (3) by using the Faria and M√a- galhães method [8, 9] regrading γ and τ as bifurcation parameters. Suppose
, then system (3) undergoes a Hopf-zero bifurcation from the trivial equilibrium at the critical point 
. After scaling 
, system (3) can be written as
(A.1)Consider the linearization of system (A.1)
Choose the space 
and, for any 
define that 
, then the phase space C can be chosen as the Banach space of con- tinuous function from 
with the supremum norm. Since L0 is a bounded v linear operator, then we define
where 
is a matrix-valued function whose components are bounded variation, and its definition is given by
And
Take the Taylor expansion of system (A.1) at the equilibrium point, and let 
, then µ is the bifurcation parameter, and system (A.1) becomes
(A.2)Let
Where
Then system (A.2) can be transformed into
and the bilinear form on C and C* (* stand for adjoin) is
where
Because L0 has a simple 0 and a pair of purely imaginary eigenvalues 
all other eigenvalues have negative real part. 
can be the generalized eigenspace associated with Λ and PΛ∗ the space adjoint with PΛ. Then C can be decomposed as 
Choose the bases 
and 
Suppose that the bases for 
are given by
Where
Thus system (A.2) becomes an abstract ODE in the space BC, which can be identified as 
(A.3)The elements of BC can be written as 
, and
where u ∈ C and A is defined by
Then the enlarged phase space BC can be decomposed as 
is the restriction of A as an operator from Q1 to the Banach space Ker π, namely
Equation (A.3) is decomposed to the form (
Using the method of Faria and Magalhães [8, 9], the above system can be written as
(A.4)where fi(x, y, µ) is homogeneous polynomials of degree j about (x, y, µ) with coeffi- cients in C3. Here, define Mj to be the operator in 
with the range in the same space by
where 
represents the linear space of the second-order homogeneous polynomials in seven variables 
with coefficients in C3, and it is easy to verify that one can choose such a decomposition
where Ker(M 1/2) and Ker(M 1/3) represent the complimentary space, respectively, and thespan elements of Ker(M1/2 ) are
Similarly, we can get that Ker(M 1/3) is spanned by
Let
where
Next, we will do a little more detailed calculation:
then we have
And
On the center manifold, the first equation of system (A.4) can be transform as the Following normal form:
with 
.
Step 1. Calculate g1 
where
Step 2. Calculate 
according to the expression from [27] as follows
We need to calculate step by step. First, calculate
.
where
then we have
Where
Second, calculate 
since
Where
with the elements of Im(M 1/2) (µ = 0)
Then we have
Where
Combining to the above, we obtain
Where
Step 3. We compute 
. Define 
Then we have
And
where 
and
. Comparing the coefficients of 
from the equation
, we can get
. According to the definition of AQ1 and , we obtain that h200, h101 satisfy the followingdifferential equations, respectively:
Since
We have
Continue to find the derivative of y and get the value of 
Then we have
(A.5)By expanding the original equation 
we have
thus 
So we know that 
According to system (A.5), we have 
From Steps 1–3 the normal form (11) has been obtained
The authors wish to express their gratitude to the editors and the reviewers for the helpful comments.
