Abstract

We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis.

1. Introduction

Since its introduction in 1927, the van der Pol equation [1] has served as a basic model of self-excited oscillations in physics, electronics, biology, neurology and other disciplines [2–15]. The intensively studied van der Pol equation is governed by the following second-order nonlinear damped oscillatory system: where , and are real constants.

In 1999, Murakami [16] introduced a discrete time delay into system (1.1), and obtained the following pair of delay differential equations By using the center manifold theorem, he [16] found that periodic solutions existed in system (1.2). The stability of bifurcating periodic solutions was discussed in detail by Yu and Cao [17].

It is well known that dynamical systems with distributed delay are more general than those with discrete delay. So Liao et al. [18] proposed the following van der Pol equation with distributed delay: The existence of Hopf bifurcation and the stability of the bifurcating periodic solutions of system (1.3) were analyzed in [18–25] for the weak and strong kernels, respectively.

In [26], Liao considered the following system with two discrete time delays: By choosing one of the delays as a bifurcation parameter, system (1.4) was found to undergo a sequence of Hopf bifurcations. The author had also found that resonant codimension two bifurcation occurred in this system.

In this paper, we consider the following van der Pol equation with discrete and distributed time delays: with initial conditions , , , , where and are bounded and are continuous functions. The weight function is a nonnegative bounded function, which describes the influence of the past states on the current dynamics. It is assumed that the presence of the distributed time delay does not affect the system equilibrium. Hence, is the unique equilibrium of system (1.5).

We normalize the kernel in the following way: Usually, the following form: is taken as the kernel. The kernel is called “weak” when and “strong” when , respectively. The analysis of weak and strong kernels is similar, so we only consider the weak kernel in this paper, that is, where reflects the mean delay of the weak kernel.

The purpose of this paper is to discuss the stability and bifurcation of system (1.5), which is an extension of the aforementioned systems. By taking the discrete delay as the bifurcation parameter, we will show that the equilibrium of system (1.5) loses its stability and Hopf bifurcation occurs when passes through a certain critical value.

The remainder of this paper is organized as follows. In Section 2, the linear stability of system (1.5) is discussed and some sufficient conditions for the existence of Hopf bifurcations are derived. The properties of Hopf bifurcation are analyzed in detail by using the center manifold theorem and the normal form theory in Section 3. In Section 4, some numerical simulations are performed to illustrate and verify the theoretical analysis. Finally, conclusions are drawn in Section 5.

2. Linear Stability and Existence of Hopf Bifurcation

In this section, we discuss the linear stability of the equilibrium of system (1.5) and the existence of Hopf bifurcations. For analysis convenience, we define the following variable: Then by the linear chain trick technique, system (1.5) can be transformed into the following system with only discrete time delay: It is obvious that system (2.2) has a unique equilibrium .

The linearization of system (2.2) at the equilibrium is

The associated characteristic equation of (2.3) is which is equivalent to

In the following, we investigate the distribution of roots of (2.5) and obtain the conditions under which system (2.2) undergoes Hopf bifurcation.

We know that is a root of (2.5) if and only if satisfies Separating the real and imaginary parts, yields Taking square on both sides of the equations in system (2.7) and adding them up yield Let , , , and . Then (2.8) becomes

Denote Since , then , and , we can conclude that (2.9) has at least one positive root.

Without loss of generality, we assume that (2.9) has three positive roots, defined by , , and , respectively. Then (2.8) has three positive roots From (2.7), we have that Thus, if we denote where , and , then is a pair of purely imaginary roots of (2.5) with . Define

In order to further investigate (2.5), we need to introduce a result proposed by Ruan and Wei [27], which is stated as follows.

Lemma 2.1 (see [27]). Consider the exponential polynomial where and are constants. As vary, the sum of the order of the zeros of on the right half plane can change only if a zero appears on or crosses the imaginary axis.

By applying Lemma 2.1, one can easily obtain the following result on the distribution of roots of (2.5).

Lemma 2.2. For the third-degree transcendental (2.5), if , then all roots with positive real parts of (2.5) have the same sum to those of the polynomial (2.5) for .

Let be the root of (2.5) near , satisfying Then the following transversality condition holds.

Lemma 2.3. Suppose that and , where is defined by (2.10); then and the sign of is consistent with that of .

Proof. The proof is similar to those in [28–32], so we omit it here.
When , (2.5) becomes According to the Routh-Hurwitz criterion, if the following conditions: hold, then all roots of (2.19) have negative real parts, which means that the equilibrium of system (2.19) is stable.

By applying Lemmas 2.2 and 2.3 to (2.5), we have the following theorem.

Theorem 2.4. Let , , and be defined by (2.13) and (2.14), respectively. Suppose that and . Then one has the following. (i)If , then the equilibrium of system (2.2) is asymptotically stable for .(ii)If and , then system (2.2) undergoes a Hopf bifurcation at its equilibrium when .

3. The Properties of Hopf Bifurcation

In the previous section, we obtain some conditions for Hopf bifurcations to occur at the critical value . In this section, we analyze the properties of the Hopf bifurcation by virtue of the method proposed by Hassard et al. [33], namely, to determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions bifurcating from the equilibrium for system (2.2) by applying the normal form theory and the center manifold theorem.

For analysis convenience, let , , , , and , . Denote ; then system (2.2) is transformed into the following form: where .

Its linear part is and the nonlinear part is as follows: Denote , each component of has -order continuous derivative}. For , we define where is a one-parameter family of bounded linear operators in . By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions in such that In fact, we can choose where is the Dirac delta function, which is defined by

For , define Then, we can transform system (2.2) into an operator equation as the following form: where for .

For , define and a bilinear form where . Then and are adjoint operators. By the discussion in Section 2, we know that are eigenvalues of , and they are also eigenvalues of .

Suppose that is the eigenvector of corresponding to ; then . It follows from the definition of and (3.5) and (3.6) that Thus, we can easily obtain On the other hand, suppose that is the eigenvector of corresponding to . By the definition of and (3.5) and (3.6), we have Therefore, we have In order to assure , we need to determine the value of . From (3.10), we have that Thus, we can choose as Similarly, we can get .

Using the same notation as in Hassard et al. [33], we compute the coordinates to describe the center manifold at . Let be the solution of (3.8) when .

Define On the center manifold we have that where in which and are local coordinates for center manifold in the direction of and , respectively. Note that is real if is real. We consider only real solutions.

For the solution of (3.8), since , we have that We rewrite this equation as where Noting that and , we have that