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

The delayed friction equation x󸀠󸀠 (t) + ax󸀠 (t) + bx󸀠 (t − τ) + cx (t) = 0, (1) 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 andmodeling 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 x󸀠 (t) = px (t) + qx (t − τ) , (2)


Introduction
The delayed friction equation   () +   () +   ( − ) +  () = 0, where  > 0,  > 0, and  and  are nonnegative such that  +  > 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][6][7]).In [8], Wei and Zhang characterized the stability of the zero solution of the retarded equation with complex coefficients   () =  () +  ( − ) , 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 using the results developed in [10].
The aim of this paper is to characterize the stability of the zero solution of the equation where  > 0 is a constant delay and ,  are complex parameters, with  ̸ = 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.

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  ≥ 0 in the form where () 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 ,  ∈ C being, respectively, the highest order coefficients of () and ().Also, it is necessary that () and () have no roots on the imaginary axis simultaneously and that  = 0 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 function if  * ̸ = 0 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 Δ( * ,   ) = 0, the variation of the number of zeros with nonnegative real parts of Δ(, ), in terms of the order and sign of the first nonzero derivate of ( * ).
Theorem 1. Assume that Δ( * ,   ) = 0,  = 0, 1, 2, . . .Let () be the number of zeros with nonnegative real parts of Δ(, ), and let  be an integer such that  () ( * ) ̸ = 0 and   ( * ) = 0 for all  < .Then 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,  0 , that all other eigenvalues are not integer multiples of  0 , and, in addition, it must hold that, if  is the bifurcation parameter, the branch of eigenvalues () which satisfies (0) =  0 is such that Re(  (0)) ̸ = 0, which is called the transversality condition.

Stability Analysis of the Second-Order Complex DDE
Consider the complex DDE (4), where The characteristic equation associated with (4) is so that for the function Δ(, ), as defined in ( 5), one has Since () is of higher order than (), and since we assume  ̸ = 0, it also holds that (0) + (0) ̸ = 0. Thus, the conditions to apply Theorem 1 are satisfied.
The following lemma gives (0), the number of zeros with nonnegative real parts of Δ(, ) when the delay is zero.
Now consider the function () defined in (7), Mathematical Problems in Engineering 3 and calculate its zeros.One gets We will consider two different cases and several subcases.
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  + 1 and  − 1 , one gets As  + 1 = − Similarly for  + 2 and  − 2 , we obtain the following set of values of  for which there are roots, Since one 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  + ,1 and decreases by two as  passes through  + ,2 .If (0) = 0, that is, if the zero solution of ( 4) is stable for  = 0, as  + 0,1 <  + 0,2 , there are stability switches when the delays are such that (a) () keeps unchanged as  increases along   if  is even, (b) when  is odd, () increases by one if  *  () ( * ) > 0, and decreases by one if  *  () ( * ) < 0, as  increases along   .