Bifurcation Analysis in a Kind of Fourth-Order Delay Differential Equation

A kind of fourth-order delay differential equation is considered. Firstly, the linear stability is investigated by analyzing the associated characteristic equation. It is found that there are stability switches for time delay and Hopf bifurcations when time delay cross through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the analytic results.


Introduction
Sadek 1 has considered the following fourth-order delay differential equation: By constructing Lyapunov functionals, it was given a group of conditions to ensure that the zero solution of 1.1 is globally asymptotically stable when the delay τ is suitable small, but if the sufficient conditions are not satisfied, what are the behaviors of the solutions? This is a interesting question in mathematics. The purpose of the present paper is to study the dynamics of 1.1 from bifurcation. We will give a detailed analysis on the above mentioned question. By regarding the delay τ as a bifurcation parameter, we analyze the distribution of the roots of the characteristic equation of 1.1 and obtain the existence of stability switches and Hopf bifurcation when the delay varies. Then by using the center manifold theory and normal form method, we derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.

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We would like to mention that there are several articles on the stability of fourth-order delay differential equations, we refer the readers to 1-8 and the references cited therein.
The rest of this paper is organized as follows. In Section 2, we firstly focus mainly on the local stability of the zero solution. This analysis is performed through the study of a characteristic equation, which takes the form of a fourth-degree exponential polynomial. Using the approach of Ruan and Wei 9 , we show that the stability of the zero solution can be destroyed through a Hopf bifurcation. In Section 3, we investigate the stability and direction of bifurcating periodic solutions by using the normal form theory and center manifold theorem presented in Hassard et al. 10 . In Section 4, we illustrate our results by numerical simulations. Section 5 with conclusion completes the paper.

Stability and Hopf Bifurcation
In this section, we will study the stability of the zero solution and the existence of Hopf bifurcation by analyzing the distribution of the eigenvalues. For convenience, we give the following assumptions: with φ and f are both continuous functions and those three-order differential quotients at origin are existent. We rewrite 1.1 as the following form: x y, y u,

2.1
It is easy to see that 0, 0, 0, 0 is the only trivial solution to the system 2.1 and the linearization around 0, 0, 0, 0 is given byẋ y,
Proof. When τ 0, 2.3 becomes By Routh-Hurwitz criterion, all roots of 2.4 have negative real parts if and only if The conclusion follows from H 1 and H 2 .
Let iω ω > 0 be a root of 2.3 , then we have Separating the real and imaginary parts gives

Direction and Stability of the Hopf Bifurcation
In this section, we will study the direction, stability, and the period of the bifurcating periodic solution. The method we used is based on the normal form method and the center manifold theory presented by Hassard et al. 10 .
We first rescale the time by t → t/τ to normalize the delay so that system 2.1 can be written as the formẋ τy,

3.1
The linearization around 0, 0, 0, 0 is given bẏ 3.2 and the nonlinear term is The characteristic equation associated with 3.2 is Comparing 3.4 with 2.3 , one can find out that γ τλ, and hence, 3.4 has a pair of imaginary roots ±iτ j k ω k , when τ τ j k for k 1, 2, 3, 4, j 0, 1, 2, . . ., and the transversal condition holds. For By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions η θ, μ in θ ∈ −1, 0 such that In fact, we choose

3.8
Discrete Dynamics in Nature and Society 9 For ϕ ∈ C 1 −1, 0 , C 4 , define 3.9 Hence, we can rewrite 3.1 in the following form:
By direct computation, we obtain that is the eigenvector of A 0 corresponding to iτ 0 ω 0 , and 14 is the eigenvector of A * corresponding to −iτ 0 ω 0 . Moreover, 3.16 Using the same notation as in Hassard et al. 10 , we first compute the coordinates to describe the center manifold C 0 at μ 0. Let w t be the solution of 3.1 when μ 0. Define On the center manifold C 0 , we have z and z are local coordinates for center manifold C 0 in the direction of q * and q * . Note that W is real if w t is real. We consider only real solutions. For solution w t in C 0 of 3.1 , since μ 0,

3.20
We rewrite this asż t iτ 0 ω 0 z t g z, z , 3.21 Nature and Society   11 where

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3.23 Compare the coefficients of 3.20 and 3.21 , noticing 3.23 , we have By 3.10 and 3.21 , it follows thaṫ

3.29
Thus 3.30 and we have Discrete Dynamics in Nature and Society

13
Then we have

3.32
So we only need to find out W

3.33
Comparing the coefficients with 3.26 , we get

3.35
Then we can get

3.36
Notice that

14
Discrete Dynamics in Nature and Society We obtain where Discrete Dynamics in Nature and Society 15 Consequently, from 3.32 ,

3.40
Substituting g 20 , g 11 , g 02 , and g 21 into 3.41 we can obtain Re C 1 0 . Then we obtain the sign of

3.42
By the general theory due to Hassard et al. 10 , we know that the quantity of β 2 determines the stability of the bifurcating periodic solutions on the center manifold, and μ 2 determines the direction of the bifurcation; and we have the following. ii If β 2 < 0 > 0 , then the bifurcating periodic solutions of system 1.1 are asymptotically stable (unstable).

An Example and Numerical Simulations
In this section, we give an example and present some numerical simulations to illustrate the analytic results. Example 4.1. Consider the following equation: Clearly,
According to the formula given in Section 3, we can obtain that −0.217 0.098i, g 02 g 11 g 02 E * 0.

Conclusion of 4.1
The zero solution of system 4.1 is asymptotically stable when τ ∈ 0, 0.061 ∪ 0.158, 0.596 . The Hopf bifurcation at the origin when τ 0 τ 0 k is supercritical, and the bifurcating periodic solutions are asymptotically stable.
The following is the results of numerical simulations to system 4.1 .
i We choose τ 0.4 ∈ 0.158, 0.596 , then the zero solution of system 4.1 is asymptotically stable, as shown in Figure 2.
ii We choose τ 0.64 being near to τ 0 2 0.596, a periodic solution bifurcates from the origin and is asymptotically stable, as shown in Figure 3.

Conclusion
In this paper, we consider a certain fourth-order delay differential equation. The linear stability is investigated by analyzing the associated characteristic equation. It is found that there may exist the stability switches when delay varies, and the Hopf bifurcation occurs when the delay passes through a sequence of critical values. Then the direction and the stability of the Hopf bifurcation are determined using the normal form method and the center manifold theorem. Finally, an example is given and numerical simulations are carried out to illustrate the results. By using Lyapunov's second method, Sadek 1 investigated the stability of system 1.1 . The main result is as the following. i There are constants α 1 > 0, α 2 > 0, α 3 > 0, α 4 > 0, and Δ > 0 such that for all y. for all x, where ε is a positive constant such that with d 0 α 1 α 2 α 2 α 3 α −1 4 . iii φ 0 0 and φ y ≥ α 3 > 0 for all y, and 0 ≤ φ y − φ y /y ≤ δ 1 < 2Δα 4 /α 1 α 2 3 for all y / 0.
Comparing Theorem 5.1 with Theorem 2.5 obtained in Section 2, one can find out that if the sufficient conditions to ensure the globally asymptotical stability of system 1.1 given in 10 are not satisfied, we can also get the stability of system 1.1 , but here the stability means local stability, and the system undergoes a Hopf bifurcation at the origin. Otherwise, here we just need to give the condition on the origin of f x and φ x , the condition is relatively weak.