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.
Sadek  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 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 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.
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 , 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. . In Section 4, we illustrate our results by numerical simulations. Section 5 with conclusion completes the paper.
2. 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 and are both continuous functions and those three-order differential quotients at origin are existent. We rewrite (1.1) as the following form:It is easy to see that is the only trivial solution to the system (2.1) and the linearization around is given byIts characteristic equation is
Proof. When , (2.3) becomesBy Routh-Hurwitz criterion, all
roots of (2.4) have negative real parts if and only ifThe conclusion follows from () and ().
Let be a root of (2.3), then we haveSeparating the real and imaginary parts givesAdding up the squares of both equations yieldsLet , and denoteThen (2.8) becomesSetThen we haveConsiderLet . Then (2.13) becomeswhereDefineThen by Lemma 2.2 in Li and Wei , we have the following results on the distribution of the roots of (2.10).
Without loss of generality, we assume that equation has four positive roots denoted by , and , respectively. Then (2.8) also has four positive roots, say .
Lemma 2.3. Suppose . If , then is a pair of simple purely imaginary roots of (2.3); and when ; and when .
Proof. Substituting into (2.3) and differentiating with respet to givesThenwhereand for , and , we can know that when ; and when . This completes the proof.
From and , it is easy to know that: if satisfies , if the equation has positive roots, then the number of the roots must be even; and from Lemma 2.3, we have that the sign of changes as varies, and then the stability switches may happen.
Lemma 2.4. Suppose that (), () and are
(i)If conditions (i) and (ii) in Lemma 2.2 are not satisfied, then all the roots of (2.3) have negative real parts for all .(ii)If one of conditions (i) and (ii) in Lemma 2.2 is satisfied, let then all roots of (2.3) have negative real parts when ; and there may exist an integer such that , and all the roots of (2.3) have negative real parts when , and (2.3) has at least a pair of roots with positive real parts when , where .
Theorem 2.5. Suppose (), (), and are
(i)If conditions (i) and (ii) in Lemma 2.2 are not satisfied, then the trivial solution of system (2.1) is asymptotically stable when .(ii)If one of conditions (i) and (ii) in Lemma 2.2 is satisfied, let , then the trivial solution of system (2.1) is asymptotically stable when ; and there may exist an integer such that , and the trivial solution of system (2.1) is asymptotically stable when , and is unstable when , where .(iii)The system (2.1) undergoes a Hopf bifurcation at the origin when , with
3. 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. .
We first rescale the time by to normalize the delay so that system (2.1) can be written as the formThe linearization around is given byand the nonlinear term isThe characteristic equation associated with (3.2) isComparing (3.4) with (2.3), one can find out that , and hence, (3.4) has a pair of imaginary roots , when for , , and the transversal condition holds.
For , letBy the Riesz representation theorem, there exists a matrix whose components are bounded variation functions in such thatIn fact, we choosewhere
For , defineHence, we can rewrite (3.1) in the following form:where , for .
For , defineFor and , define the bilinear formwhere . Then and are adjoint operators, and are eigenvalues of . Thus, they are also eigenvalues of .
By direct computation, we obtain thatis the eigenvector of corresponding to , andis the eigenvector of corresponding to . Moreover,where
DefineOn the center manifold , we havewhere and are local coordinates for center manifold in the direction of and . Note that is real if is real. We consider only real solutions.
For solution in of (3.1), since ,We rewrite this aswhere Compare the coefficients of (3.20) and (3.21), noticing (3.23), we haveBy (3.10) and (3.21), it follows thatwhereExpanding the above series and comparing the coefficients, we obtainNotice thatthat is,Thusand we haveThen we haveSo we only need to find out , , , and to obtain .
When , we haveComparing the coefficients with (3.26), we getFrom (3.27), (3.32), (3.33), and (3.34), we deriveThen we can getNotice thatWe obtainwhere HenceConsequently, from (3.32),Substituting , , , and intowe can obtain . Then we obtain the sign of
By the general theory due to Hassard et al. , we know that the quantity of determines the stability of the bifurcating periodic solutions on the center manifold, and determines the direction of the bifurcation; and we have the following.
Theorem 3.1. (i) If , then the Hopf
bifurcation at the origin of system (1.1) is supercritical (subcritical).
(ii) If , then the bifurcating periodic solutions of system (1.1) are asymptotically stable (unstable).
4. 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,
By direct computation, we know () and () are satisfied. That is, the data satisfy the conditions of Lemma 2.1. The characteristic equation isand we can obtain As shown in Figure 1, the equation has four roots asand HenceFor , we obtain that the zero solution of system (4.1) is asymptotically stable when .
According to the formula given in Section 3, we can obtain thatThen we haveHence, when , we have
Conclusion of (4.1)
The zero solution of system (4.1) is asymptotically stable when . The Hopf bifurcation at the origin when 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 , then the zero solution of system (4.1) is asymptotically stable, as shown in Figure 2.(ii) We choose being near to , a periodic solution bifurcates from the origin and is asymptotically stable, as shown in Figure 3.
(a) y-x phase plots
(b) y waveform
(c) x waveform
(d) v-u phase plots
(e) v waveform
(f) u waveform
(a) y-x phase plots
(b) y waveform
(c) x waveform
(d) v-u phase plots
(e) v waveform
(f) u waveform
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  investigated the stability of system (1.1). The main result is as the following.
Theorem 5.1. Suppose that the following hold.
(i)There are constants and such thatfor all .(ii) as , andfor all , where is a positive constant such thatwith .(iii) and for all , and for all . Then the zero solution of (1.1) is asymptotically stable, provided thatwith .
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  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 and , the condition is relatively weak.
This work was supported by the National Natural Science Foundation of China (No10771045), and Program of Excellent Team in Harbin Institute of Technology.
C. Tunç, “Some stability results for the solutions of certain fourth order delay differential equations,” Differential Equations and Applications, vol. 4, pp. 133–140, 2005.View at: Google Scholar