Stability Switches and Hopf Bifurcations in a Second-Order Complex Delay Equation

Abstract

The existence of stability switches and Hopf bifurcations for the second-order delay differential equation with complex coefficients, is studied in this paper.

1. Introduction

The delayed friction equationwhere , , and and are nonnegative such that , was considered by Minorsky [1, 2] for problems of ship stability and modeling of small vibrations of a pendulum. In [3, 4], the stability of the zero solution of more general forms of the delayed friction equation with real coefficients was characterized.

Delay differential equations (DDE) with complex coefficients have attracted increasing attention in the last years (e.g., [5–7]). In [8], Wei and Zhang characterized the stability of the zero solution of the retarded equation with complex coefficientsby studying the distribution of the roots of the characteristic equation for the associated real differential system with delay and analyzed the existence of stability switches [3, 4, 9].

In [10], Li et al. presented a method for directly analyzing the stability of complex DDEs on the basis of stability switches. Their results generalize those for real DDEs, thus greatly reducing the complexity of the analysis. In [11], Roales and Rodríguez studied the stability switches of the zero solution of the neutral equation with complex coefficientsusing the results developed in [10].

The aim of this paper is to characterize the stability of the zero solution of the equationwhere is a constant delay and are complex parameters, with .

Using the results given by [10], the existence of stability switches and Hopf bifurcations for certain conditions on the parameters of (4) will be shown, discussing the conditions that may allow for delay dependent stabilization of the system.

2. Methods

To carry out our analysis, we will use some previous results that are recalled in this section (see, [4, 10, 12, 13]).

Following [10], and similar to the analysis carried out [11] for a first-order equation, we write the characteristic equation of a time-delay system with a single delay in the formwhere and are complex polynomial. To be able to apply the main result in [10], we will require the order of to be either higher than that of or, if they have the same order, that , with being, respectively, the highest order coefficients of and . Also, it is necessary that and have no roots on the imaginary axis simultaneously and that is not a root of (5): that is,In the next section, it will be shown that all these conditions hold in our problem.

As shown in [10], introducing the functionif is a zero of , then there are an infinite number of delays corresponding to satisfying

Based on a previous work of Lee and Hsu [14], Li et al. established the following theorem [10, Theorem  1], characterizing, for the critical values such that , the variation of the number of zeros with nonnegative real parts of , in terms of the order and sign of the first nonzero derivate of .

Theorem 1. Assume that Let be the number of zeros with nonnegative real parts of , and let be an integer such that and for all Then(a) keeps unchanged as increases along if is even,(b)when is odd, increases by one if , and decreases by one if , as increases along

This theorem facilitates the stability analysis with respect to the method used in [14] and extends to the complex coefficients setting a previous result which was only valid for real DDEs [15].

Hopf bifurcation theorem gives the conditions for the existence of local nontrivial periodic solutions (e.g., [4, 12, 13]). Basic conditions are the existence of a nonzero purely imaginary root of the characteristic equation, , that all other eigenvalues are not integer multiples of , and, in addition, it must hold that, if is the bifurcation parameter, the branch of eigenvalues which satisfies is such that , which is called the transversality condition.

3. Stability Analysis of the Second-Order Complex DDE

Consider the complex DDE (4), whereThe characteristic equation associated with (4) isso that for the function , as defined in (5), one hasSince is of higher order than , and since we assume , it also holds that . Thus, the conditions to apply Theorem 1 are satisfied.

The following lemma gives , the number of zeros with nonnegative real parts of when the delay is zero.

Lemma 2. Consider the complex numberIf andthen . Else, if then , and if then when orand when and

Proof. Consider the equationThen,If , there is a double root with real part . If , can be written asand the conclusion of the lemma follows.

Now consider the function defined in (7), and calculate its zeros. One getsWe will consider two different cases and several subcases.

Case 1 ().  Case 1(a):  Case 1(b):  Case 1(c):

Case 2 (). First, we assume that (Case 1).
If (Case 1(a)), then has four real roots, , such thatIf (Case 1(b)), then has two double real roots, , such thatIf (Case 1(c)), then has no real root, and therefore the stability of the zero solution of (4) does not change for any .
Consider now Case 1(a), where Substituting into (10), and separating the real and imaginary parts, one gets obtaining the following four sets of values of for which there are roots.
For and , one getsAs , then and By (24) and Therefore, in what followsSimilarly for and , we obtain the following set of values of for which there are roots,Sinceone has Therefore, according to Theorem 1, as is increased, the number of the characteristic roots with nonnegative real parts increases by two as passes through and decreases by two as passes through
If , that is, if the zero solution of (4) is stable for , as , there are stability switches when the delays are such thatSincethe intervals become smaller with increasing , so that eventually, for a certain ,Thus, the distribution of delays isand there is only a finite number of stability switches, with the system becoming unstable for
If or , the system is always unstable because and a distribution of delays for stability switches to occur is not possible.
After the study of the stability, we wonder what happens, when there are stability switches, in the critical delays Denote as the root of (10) satisfying According to Theorem 1, one hasBy (28), one gets that the transversality condition required by Hopf Theorem is satisfied. Therefore, a Hopf bifurcation occurs for these critical values.
Now we study Case 1(b), where are two real roots. Proceeding as before, there are two sets of critical values of delays and , corresponding to and , respectively, such that . Since , we consider the second derivative,By Theorem 1, since keeps unchanged as increases along Consequently, the stability of zero solution of (4) does not change for any
Finally, consider Case 2, whereThe function defined in (19) has no real root, and therefore the stability of the zero solution of (4) does not change for any Thus, the following theorem has been established.

Theorem 3. Consider the second-order complex delay equation (4). The following two cases may occur concerning its stability: (a) In this case, if , and the distribution of delays is , then the zero solution of (4) is asymptotically stable for and , and unstable for and Otherwise, if or , the zero solution of (4) is unstable for all  When there are stability switches, the critical delays , and , are Hopf bifurcation values for (4).(b) In this case, the stability of the zero solution of (4) does not change for any

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.