Multiple Hopf bifurcations of four coupled van der Pol oscillators with delay

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Introduction
Recently, the dynamic behavior of coupled oscillator systems has become a hot research topic.
More and more research results show that the coupled oscillator system can describe the dynamic behavior in physics, chemistry, biology and other disciplines [1,2].Van der Pol oscillator is an important nonlinear oscillator, and coupled van der Pol oscillators are widely used in various fields of natural science.Many scholars pay attention to the dynamic behavior caused by the interaction between coupled van der Pol oscillators, such as stability, instability and periodicity.
For example, in [3], van der pol oscillators were used to describe reproduce the dynamic behavior of thecardiac muscle .in[4], the circadian rhythm of eyes has been studied by using three coupled van der Pol oscillators.In [5,6], the dynamic behavior of van der Pol oscillator in circuit has been studied.In [7], the occurrence of explosion death has been studied by using coupled van der Pol oscillator.In [8], Batool et al. have studied the limit cycle behavior of the van der Pol model.
Since the delay in signal propagation and chemical reaction, time delay is inevitable in coupling system.In addition, time delay is ubiquitous in the coupled system and has an important impact on the dynamic behavior of the system.For example, in [9], the effect of synchronization of delayed van der Pol oscillator has been discussed.In [10,11], some dynamic behaviors of delayed van der Pol oscillator have been verified.
Symmetric system is also one of the research hot spots.In general, symmetry describes some kind of spatial invariance of the system.In [12,13], the Hopf bifurcation behavior of three symmetric coupled van der Pol oscillators with delay has been has been analysed.In the aspect of artificial intelligence, the movement of robots is controlled by imitating the rhythmic movement of quadruped animals when designing robots, and people often use van der Pol oscillator to design CPG(Central Pattern Generators) network [14].For example, in [14], it is shown that CPG can be controlled by a two-mode van der Pol oscillator under certain conditions.
The van der Pol oscillator is often used in biological modeling, as follow where x is the output signal from oscillator.α, p and w are variable parameters which can influence the character of oscillators.Commonly, the shape of the wave is affected by parameter α, and the amplitude of an output counts on the parameter p mostly.The output frequency is mainly relying on the parameter w when the amplitude parameter p is fixed.But the alteration of parameter p can lightly change the frequency of the signal, and α also can effect the output frequency.
Let w 2 = 1, we study following four coupled van der Pol oscillator systems with time delay.
where i = 1, 2, 3, 4, i(mod 4), and a, b, c are constants, τ ≥ 0. System (1.1) is symmetrical, and the classical Hopf bifurcation theory cannot be used to study it, since symmetrical systems usually have multiple eigenvalues.
The content of this paper is as follows.In Section 2, the conditions for the occurrence of multiple Hopf bifurcations of the system (1.1) are given.In Section 3, the spatiotemporal patterns of the multi-periodic solutions of the system are obtained by using the results of the symmetric system in [15].In Section 4, the normal form of the system on the central manifold is obtained by using the normal form theory in [16].Finally, some numerical simulations are carried out to verify the theoretical results.
A similar lemma is as follows.
Solving the above equation, we have From the above analysis, we have the following lemma.

Spatio-temporal patterns of bifurcating periodic solutions
In this section, we study the periodic solution when β = β + .The case of β = β − can be considered similarly.The infinitesimal generator A(τ ) of the C 0 -semigroup generated by the linear system (2.2) has a pair of purely imaginary eigenvalues ±iβ + at the critical value τ + k (k = 0, 1, 2 . . .), the corresponding eigenvectors can be chosen as Lemma 3.1.R 2 is an absolutely irreducible representation of Γ , and Ker∆(τ Proof.It can be easily proved that R 2 is an absolutely irreducible representation of Γ.
Let Ker∆(τ Clearly, J is a linear isomorphism from Ker∆(τ Note that Then This completes the proof.
Lemma 3.2.The generalized eigenspace where It is easy to get k(ReV Thus we have the following lemma.
Lemma 3.3. and Let T = 2π β + and denote P T as the Banach space of all continuous T − periodic functions x.
The role of where S 1 =< e iθ > is the temporal.
Denoting SP T as the subspace of P T consisted of all T -periodic solutions of system (2.1) For the following subgroups of We have the following lemma.
Lemma 3.4.Fix( ∑ j , SP T ) (j = 1, 2, 3, 4.) are all the two-dimensional subspaces of SP T . Where (i) It is necessary for us to prove x ∈ Fix( ∑ 1 , SP T ) if and only if kx = x.By the lemma 3.3, we have Hence , kx = x if and only if x 1 = x 3 and x 2 = x 4 .This implies that Fix( (ii) It is enough to prove that x ∈ Fix( ∑ 2 , SP T ) if and only if kx(t) = x(t + T 2 ).It is clearly that where By the same token Then we have kx(t) = x(t + T 2 ) if and only if −x 1 = x 3 , x 2 = −x 4 and Fix( (iii) It is sufficient to prove that x ∈ Fix( ∑ 3 , SP T ) if and only if ρx(t) = x(t − T 4 ).By lemma 3.3 , we can get Hence It can be attained that ρx(t) = x(t − π 4 ) if and only if x 1 = x 2 = 0, such that Fix( the span of ε 3 and ε 4 . (iv) It is adequate to prove that x ∈ Fix( ∑ 4 , SP T ) if and only if ρx(t − T 4 ) = x(t).Similarly, we have Hence ρx(t − T 4 ) = x(t) if and only if x 3 = x 4 = 0, and Fix( ∑ 4 , SP T ) is the span of ε 1 and By lemma 3.1, 3.2, 3.4 and theorem 4.1 of [15] , we can get the following theorem.

Normal form on center manifolds and bifurcation direction of bifurcated periodic solutions
In this section, we employ the algorithm and notations of [16] to derive the normal forms of system (2.1) on center manifolds and direction of bifurcation periodic solution.
The base of the center space X at τ = τ + k can be taken as Φ(θ) = (φ 1 , φ 2 , φ 3 , φ 4 ), where It is easy to check that the base for the adjoint space X * is Ψ(s) = (ψ 1 (s), ψ 2 (s), ψ 3 (s), ψ 4 (s)) T , where and ⟨Ψ, Φ⟩ = I 4 for the adjoint bilinear form on C * × C as where Define a 4 × 4 matrix Using the decomposition of z t = Φx + y , system (4.2) can be decomposed as where x ∈ C 4 , y ∈ Q 1 .We have the Taylor expansion as following where f 1 j (x, y, µ) are homogeneous polynomials of degree j in (x, y, µ) with coefficients in C 4 .
Then the normal form of (4.2) on the center manifold at the origin as for µ = 0 is given by where g 1 2 , g 1 3 will be calculated in the following. where Since M 1 j p(x, µ) = D x p(x, µ) Bx − Bp(x, µ), j ≥ 2, and where e 1 , e 2 , e 3 , e 4 is the canonical basis for C 4 .We have where n = D(m 1 + n 1 ).
Then we need to compute where U 1 2 is the change of variables associated with the transformation from f 1 2 to g 1 2 and h is It's easy to see from (4.4) and (4.5) that f 1 2 (x, 0, 0) = g 1 2 (x, 0, 0) = 0 for µ = 0. Therefore, we have .
The normal form on the center manifold becomes for x ∈ C 4 .By the following coordinates transformation Let where σ is a periodic-scaling parameter, then it follows that Let's denote g(z, µ) as follows, then equation (4.8) can be written as (4.9) According to [17] (pp.296-297,Theorem 6.3 and 6.5 ), the bifurcations of small-amplitude periodic solution of (4.9) are comletely determined by the zero point of equation and (4.10) can be written as where From the expressions of D and those above, we get the following expressions We have the following theorem by the result of [17] (pp.383,Theorem 3.1).
Theorem 4.1.The following statements are true.

Conclusions
In this paper, four coupled van der Pol oscillators system (1.1) is discussed.The system is symmetrical and D 4 equivariant.By analyzing the characteristic equation of the system, the conditions and critical value τ ± k (k = 0, 1, 2 . . . ) of losing stability of the zero equilibrium are obtained.At the critical value τ ± k (k = 0, 1, 2 . . .), the system exhibits asynchronous periodic solutions: discrete waves, mirror-emitted waves and standing waves.The normal form of the system on the central manifold is obtained, and the bifurcation direction is studied.Finally, the theoretical results are supported by using numerical simulation.In addition, standing waves and mirror-reflecting waves are unstable by numerical simulation.Unfortunately, the discrete wave of the system has not been simulated.