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

Authors: M. Roales; F. Rodríguez
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The existence of stability switches and Hopf bifurcations for the second-order delay differential equationx′′t+ax′t-τ+bxt=0,t>0, with complex coefficients, is studied in this paper.


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## Body

## 1. Introduction

The delayed friction equation(1)x′′t+ax′t+bx′t-τ+cxt=0,where c>0, τ>0, and a and b are nonnegative such that a+b>0, 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 coefficients(2)x′t=pxt+qxt-τ,by 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 coefficients(3)x′t+ax′t-τ=bxt+cxt-τ,using the results developed in [10].The aim of this paper is to characterize the stability of the zero solution of the equation(4)x′′t+ax′t-τ+bxt=0,where τ>0 is a constant delay and a,b are complex parameters, with b≠0.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 τ≥0 in the form(5)Δλ,τ=Pλ+Qλe-λτ,where Pλ and Qλ are complex polynomial. To be able to apply the main result in [10], we will require the order of Pλ to be either higher than that of Qλ or, if they have the same order, that α>β, with α,β∈C being, respectively, the highest order coefficients of Pλ and Qλ. Also, it is necessary that Pλ and Qλ have no roots on the imaginary axis simultaneously and that λ=0 is not a root of (5): that is,(6)P0+Q0≠0.In the next section, it will be shown that all these conditions hold in our problem.As shown in [10], introducing the function(7)Fω=Piω2-Qiω2,if ω∗≠0 is a zero of Fω, then there are an infinite number of delays τj corresponding to ω∗ satisfying(8)Δiω∗,τj=0.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 τj such that Δiω∗,τj=0, the variation of the number of zeros with nonnegative real parts of Δλ,τ, in terms of the order and sign of the first nonzero derivate of Fω∗.Theorem 1.
Assume thatΔiω∗,τj=0,j=0,1,2,… Let Nτ be the number of zeros with nonnegative real parts of Δλ,τ, and let M be an integer such that FMω∗≠0 and Fmω∗=0 for all m<M. Then(a)
N(τ) keeps unchanged as τ increases along τj if M is even,(b)
whenM is odd, Nτ increases by one if ω∗FMω∗>0, and decreases by one if ω∗FMω∗<0, as τ increases along τj.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, λ0, that all other eigenvalues are not integer multiples of λ0, and, in addition, it must hold that, if α is the bifurcation parameter, the branch of eigenvalues λα which satisfies λ0=λ0 is such that Reλ′0≠0, which is called the transversality condition.


## 3. Stability Analysis of the Second-Order Complex DDE

Consider the complex DDE (4), where(9)a=a1+ia2,b=b1+ib2.The characteristic equation associated with (4) is(10)λ2+aλe-λτ+b=0,so that for the function Δλ,τ, as defined in (5), one has(11)Pλ=λ2+b,Qλ=aλ.Since Pλ is of higher order than Qλ, and since we assume b≠0, it also holds that P0+Q0≠0. Thus, the conditions to apply Theorem 1 are satisfied.The following lemma givesN0, the number of zeros with nonnegative real parts of Δλ,τ when the delay is zero.Lemma 2.
Consider the complex number(12)z=A,B=a12-a22-4b1,2a1a2-4b2.If z≠0 and(13)a1<B2-A+z/2then N0=1. Else, if a1≤0 then N(0)=2, and if a1>0 then N0=0 when z=0 or(14)a1>B2-A+z/2,and N0=1 when z≠0 and(15)a1=B2-A+z/2.Proof.
Consider the equation(16)Δλ,0=Pλ+Qλ=λ2+aλ+b=0.Then,(17)λ=-a±a2-4b2=-a1+ia2±a12-a22+2ia1a2-4b1-4ib22=-a1+ia2±z2.If z=0, there is a double root with real part -a1/2. If z≠0, λ can be written as(18)λ=-a1+ia2±B/2-A+z/2+i-A+z/22,and the conclusion of the lemma follows.Now consider the functionFω defined in (7), (19)Fω=-ω2+b12+b22-a2ω2=ω4-a2+2b1ω2+b2,and calculate its zeros. One gets(20)ω±2=a2+2b1±a2+2b12-4b22.We will consider two different cases and several subcases.Case 1 (a2+2b1>0).
Case 1(a):|a|2+2b12>4|b|2.
Case 1(b):a2+2b12=4b2.
Case 1(c):|a|2+2b12<4|b|2.Case 2 (a2+2b1≤0).
First, we assume thata2+2b1>0 (Case 1).
If|a|2+2b12>4|b|2 (Case 1(a)), then Fω has four real roots, ω1+,ω2+,ω1-,ω2-, such that(21)ω1+>ω2+>0>ω2->ω1-,ω1+=-ω1-,ω2+=-ω2-.
Ifa2+2b12=4b2 (Case 1(b)), then Fω has two double real roots, ω1+,ω1-, such that(22)ω1+>0>ω1-,ω1+=-ω1-.
Ifa2+2b12<4b2 (Case 1(c)), then Fω has no real root, and therefore the stability of the zero solution of (4) does not change for any τ>0.
Consider now Case1(a), where ω1+>ω2+>0>ω2->ω1-. Substituting λ=iω into (10), and separating the real and imaginary parts, one gets (23)-ω2+a1ωsin⁡ωτ-a2ωcos⁡ωτ+b1=0,b2+a1ωcos⁡ωτ+a2ωsin⁡ωτ=0,obtaining the following four sets of values of τ for which there are roots.
Forω1+ and ω1-, one gets(24)cos⁡ω1+τ=-ω1+2+b1a2-b2a1a2ω1+,sin⁡ω1+τ=ω1+2-b1a1-b2a2a2ω1+,cos⁡ω1-τ=-ω1-2+b1a2-b2a1a2ω1-,sin⁡ω1-τ=ω1-2-b1a1-b2a2a2ω1-.As ω1+=-ω1-, then cos⁡ω1+τ=cos⁡ω1-τ and sin⁡ω1+τ=-sin⁡ω1-τ. By (24) cos⁡ω1+τ=-cos⁡ω1-τ and sin⁡ω1+τ=-sin⁡ω1-τ. Therefore, cos⁡ω1+τ=0 in what follows(25)τn,1+=π/2ω1++nπω1+τn,1-=-π/2ω1--nπω1-.τn,1+=τn,1-,n=0,1,2,…,
Similarly forω2+ and ω2-, we obtain the following set of values of τ for which there are roots,(26)τn,2+=τn,2-=π/2ω2++nπω2+=-π/2ω2--nπω2-,n=0,1,2,….
Since(27)F′ω1+=-F′ω1-=2ω1+2ω1+2-a2+2b1>0,F′ω2+=-F′ω2-=2ω2+2ω2+2-a2+2b1<0,one has (28)ω1+F′ω1+>0ω1-F′ω1->0ω2+F′ω2+<0ω2-F′ω2-<0.
Therefore, according to Theorem1, as τ is increased, the number of the characteristic roots with nonnegative real parts increases by two as τ passes through τn,1+ and decreases by two as τ passes through τn,2+.
IfN0=0, that is, if the zero solution of (4) is stable for τ=0, as τ0,1+<τ0,2+, there are stability switches when the delays are such that(29)τ0,1+<τ0,2+<τ1,1+<τ1,2+<⋯.Since(30)τn+1,1+-τn,1+=π/2ω1+<π/2ω2+=τn+1,2+-τn,2+,the intervals become smaller with increasing n, so that eventually, for a certain k≥1,(31)τk-1,1+<τk,1+≤τk-1,2+.Thus, the distribution of delays is(32)τ0,1+<τ0,2+<τ1,1+<τ1,2+<⋯<τk-1,1+<τk,1+≤τk-1,2+<τk+1,1+<⋯,and there is only a finite number of stability switches, with the system becoming unstable for τ>τk-1,1+.
IfN0=1 or N0=2, the system is always unstable because τ0,1+<τ0,2+ 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 delaysτ=τn,j±. Denote λτ=ατ+iωτ as the root of (10) satisfying ατn,j±=0,ωτn,j±=ωn,j±,j=1,2. According to Theorem 1, one has(33)sgn⁡dReλτn,j±dτ=sgn⁡ωn,j±sgn⁡F′ωn,j±.By (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 Case1(b), where ω1+>0>ω1- are two real roots. Proceeding as before, there are two sets of critical values of delays τn+ and τn-, corresponding to ω1+ and ω1-, respectively, such that τn+=τn-,n=1,2,…. Since F′ω1+=F′ω1-=0, we consider the second derivative,(34)F′′ω1±=4a2+2b1≠0.By Theorem 1, since M=2,N(τ) keeps unchanged as τ increases along τn. Consequently, the stability of zero solution of (4) does not change for any τ>0.
Finally, consider Case2, where(35)a2+2b1≤0.The function Fω defined in (19) has no real root, and therefore the stability of the zero solution of (4) does not change for any τ>0. 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)
a2+2b12>4b2. In this case, if N0=0, and the distribution of delays is 0<τ0,1+<τ0,2+<⋯<τk-1,1+<τk,1+≤τk-1,2<τk+1,1+<⋯, then the zero solution of (4) is asymptotically stable for τ∈0,τ0,1+ and τ∈⋃n=0k-2τn,2+,τn+1,1+, and unstable for τ∈⋃n=0k-2τn,1+,τn,2+ and τ>τk-1,1+. Otherwise, if N0=1 or N0=2, the zero solution of (4) is unstable for all τ≥0.When there are stability switches, the critical delaysτ=τn,1±,n=0⋯k-1, and τ=τn,2±,n=0⋯k-2, are Hopf bifurcation values for (4).(b)
a2+2b12≤4b2. In this case, the stability of the zero solution of (4) does not change for any τ>0.



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