Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay ∗

Abstract. In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis.


Introduction or the first section
The coupling of nonlinear systems comes naturally from physics and engineering, for example, in electronics, nonlinear systems have been long used as an efficient system to generate higher harmonics from a given signal [26].Studying of coupled nonlinear systems is significant in a number of areas of fundamental and applied mathematics, such as bifurcation in the presence of symmetries, chaos theory, nonlinear electronics.
In the research of nonlinear dynamical system, van der Pol equation is one of the most intensely studied equation (see [14,15]).This celebrated equation has a nonlinear damping ẍ + ε x 2 − 1 ẋ + x = f (x), which originally was a model for an electrical circuit with a triode valve and was extensively studied as a host of a rich class of dynamical behavior, including relaxation oscillations, quasi-periodicity, elementary bifurcations, and chaos [1].Noting that most practical implementations of feedback have inherent delays, some researchers have considered the effect of time delay in van der Pol's oscillator [8,18,19,25,32,35,36].It is shown that the presence of time delay can change the amplitude of limit cycle oscillations.Thus, time delay is inevitable in coupled systems, and effects of time delay are also very popular in the study of dynamical systems with many delay factors that appear in state variables, and some of them appear in parameters [42].When time delay becomes a parameter, structural properties of dynamical systems (such as the number of equilibrium, stability, ect.) will change, then the question belongs to the bifurcation question.Now, there are some articles on Hopf bifurcation in delay differential equations (see [2,29,34]).However, there are other articles on Hopf-pitchfork bifurcation in delay differential equations (see [3-6, 10, 17, 22, 24, 27, 30, 31, 38, 41]).Particularly, by our existing knowledge, there is no study in Hopf-pitchfork bifurcation of coupled van der Pol's equation with time delay.In recent years, various aspects of the van der Pol have been studied [28,40].Wen et al. [33] investigate the dynamics of Mathieu equation with two kinds of van der Pol fractional-order terms.Euzebio and Llibre [9] discuss some aspects on the periodic solutions of the extended Duffing-van der Pol oscillator.They show that it can bifurcate one or three periodic solutions from a two-dimensional manifold filled by periodic solutions of the referred system.Kumar et al. [20] carry out investigations on the bifurcation characteristics of a Duffing-van der Pol oscillator subjected to white noise excitations.Fu et al. [13] discuss noise-induced and delay-induced bifurcations in a bistable Duffing-van der Pol oscillator under time delay and join noises theoretically and numerically.Dubkov and Litovsky [7] investigate that the exact Fokker-Planck equation for the joint probability distribution of amplitude and phase of a van der Pol oscillator perturbed by both additive and multiplicative noise sources with arbitrary nonlinear damping is first derived by the method of functional splitting of averages.Yonkeu et al. [39] propose to compute the effective activation energy, usually referred to a pseudopotential or quasipotential, of a birhythmic system -a van der Pol-like oscillator -in the presence of correlated noise.Ji and Zhang [16] use the method of multiple time scales to investigate the following system with both external force and feedback control: Njah [23] studied the synchronization and antisynchronization of the following van der Pol systems based on the theory of Lyapunov stability and Routh-Hurwitz criteria: Yamapi and Filatrella [37] studied the strange attractors of the following coupled van der Pol systems: where H(t) is the Heaviside function.
In [37], they have obtained the stability of the equilibrium and the existence of Hopf bifurcation.Using the center manifold reduction technique and normal form theory, they give the direction of the Hopf bifurcation.Therefore, I want to know if this model can produce Hopf-pitchfork bifurcation, and whether we can apply these theories to analyse of the Hopf-pitchfork bifurcation.
Because there are only some articles to study Hopf-pitchfork bifurcation of coupled van der Pol with delay, in order to get more dynamic behaviors, we have the reason to believe that investigating Hopf-pitchfork bifurcation of coupled van der Pol with delay is interesting and worthwhile.Consider the following coupled van der Pol systems: Let x = x 1 , ẋ = x 2 , y = x 3 , ẏ = x 4 , then equation ( 1) can be written as follows: Because the activate function g(t) belongs to sigmoidal function (see [12, p. 356]), we assume g(0) = g (0) = 0, g (0) = 1, and g (0) = 0 throughout this paper.Clearly, we probe dynamical behaviors of system (1) equaling to investigate that of system (2).The rest of the article is organized as follows.In Section 2, we will give the existence condition of the Hopf-pitchfork bifurcation by taking interaction coefficient and delay as two parameters.In Section 3, we use center manifold theory and normal form method [11,30] to investigate Hopf-pitchfork bifurcation with original parameters.In Section 4, we given some numerical simulations to support the analytic results.Finally, we draw the conclution in Section 5.

The existence of Hopf-pitchfork bifurcation
In the following, if the characteristic equation ( 2) has a simple root 0 and a simple pair of purely imaginary roots ±iω 0 , and all other roots of the characteristic equation have negative real parts, then the Hopf-zero bifurcation will occur.The linearization equation of system (2) at the origin is The characteristic equation of system (3) is https://www.mii.vu.lt/NAwhere a 4), we obtain a = ±1.By above analysis, we know a > 0, so we get a = 1.If τ = 0, then We obtain that if τ = 0, then α < 0. Except a single zero eigenvalue, all the roots of equation ( 4) have negative real parts.

Normal form for Hopf-zero bifurcation
In this section, center manifold theory and normal form method [11,30] are used to study Hopf-pitchfork bifurcation.After scaling t → t/τ , system (2) can be written as Let the Taylor expansion of g be where g(0) = g (0) = 0, g (0) = 1.
By calculating, we choose To consider system (8), we need to enlarge the space C to the following: BC = ϕ is continuous functions on [−1, 0), and lim θ→0 − ϕ(θ) exists .
(Im(M 1 3 )) c is spanned by Then we get where S 1 and S 2 are spanned, respectively, by and On the center manifold, (8) can be transform as the following normal form: According to [11, Thm.2.1], we obtain the dynamical behavior of (8) near X t = 0, which is governed by the general normal form of the third order.Then the equation becomes Nonlinear Anal.Model.Control, 22(5):598-613 where Through the change of variables  12) can be written as where where σ = Re(c 21 )/c 11 , δ = c 12 / Re(c 22 ).
Since system (12) rotates around the Z-axis, the correspondences between two-dimensional flows for ( 14) and three-dimensional flows (12) can be established.Thus, for (12), the equilibria on the Z-axis in ( 14) remain equilibria, while the equilibria outside the Z-axis in ( 14) become periodic orbits.
Therefore, if case I occurs, then the detailed dynamics of system (2) in D 1 -D 6 near the original parameters (1/k 2 , τ 0 ) are as follows: • In D 1 , (2) 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 a stable periodic orbit (corresponding to M 1 ) appears.• In D 3 , the trivial equilibrium (corresponding to M 0 ) becomes a source from a saddle, a pair of unstable semitrivial equilibria (corresponding to M ± 2 ) appear, and the periodic orbit (corresponding to M 1 ) remains stable.
• In D 4 , the semitrivial equilibria (corresponding to M ± 2 ) become stable from its unstable state, a pair of unstable periodic orbits (corresponding to M ± 3 ) appear, and the periodic orbit (corresponding to M 1 ) remains stable.
• In D 5 , the unstable periodic orbit (corresponding to M ± 3 ) disappear, the periodic orbit (corresponding to M 1 ) becomes unstable, and the semitrivial equilibria (corresponding to M ± 2 ) remains stable.• In D 6 , the periodic orbit (corresponding to M 1 ) disappears, the trivial equilibrium (corresponding to M 0 ) becomes a saddle from a source, and the semitrivial equilibria (corresponding to M ± 2 ) remains stable.
Noticing that if case II arises, then the detailed dynamics of system (2) in D1, D2, D3, D5, and D6 are the same as that in case I, except in D7.In D7, system (2) has a pair of stable periodic orbits (corresponding to M ± 3 ), a pair of unstable semitrivial equilibria (corresponding to M ± 2 ), an unstable periodic orbit (corresponding to M 1 ), and an unstable trivial equilibrium (corresponding to M 0 ).
By analysing above, we can obtain the bifurcation critical lines as follows:

Conclusions
In this paper, we have investigated the Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay.Our contributions include the following: 1.By analyzing the distribution of the eigenvalues of the corresponding characteristic equation of its linearized equation, we find the conditions for the occurrence of Hopf-pitchfork bifurcation.2. By using the normal form method and the center manifold theorem, we have derived the normal form of the reduced system on the center manifold, discussed the Hopf-pitchfork bifurcation with the parameter in system (2), and analyzed the stability.Furthermore, we can obtain the coexistence of periodic orbits.3.By comparing system (2) in this paper with non-coupled van der Pol system in [20], we obtain the coexistence of a pair of stable nontrivial equilibria and the coexistence of a stable periodic orbit and a pair of stable nontrivial equilibria.We know that the periodic orbit is stable, corresponding to the periodic spiking behavior.
Our work is a further study of van der Pol oscillator, which is helpful in the study of the complex phenomenon caused by high co-dimensional bifurcation of delay differential equation.