Stability Analysis of Complex-valued Nonlinear Differential System

This paper studies the stability of complex-valued nonlinear differential system. The stability criteria of complex-valued nonlinear autonomous system are established. For the general complex-valued nonlinear non-autonomous system, the comparison principle in the context of complex fields is given. Those derived stability criteria not only provide a new method to analyze complex-valued differential system, but also greatly reduce the complexity of analysis and computation.


Introduction
The stability of differential system has been studied by many researchers, for example, [1][2][3][4][5][6] and references therein.However, the common setting adopted in aforementioned works is always in real number fields; namely, the objects of study are real-valued differential systems.The object of study in this paper is complex-valued differential system.Complex-valued differential system has also many potential applications in science and engineering.For example, quantum system and complex-valued neural networks are classical complex-valued differential system [7][8][9][10].In fact, equations of many other classical systems, such as Ginzburg-Landau equation [11], Orr-Sommerfeld equation [12], complex Riccati equation [13], and complex Lorenz equation [14], are considered in the complex number fields.The usual method analyzing complex-valued system is to separate it into real part and imaginary part and then recast it into an equivalent real-valued system (see [14,15] and references therein).But this method encounters two problems.One is that the dimension of the real-valued system is double that of complex-valued system, which leads to the difficulties on the analysis.Another is that this method needs an explicit separation of complex-valued function (, ) into its real part and imaginary part; however, this separation is not always expressible in an analytical form.An efficient way to analyze complex-valued system is to retain the complex nature of system and consider its properties on C  [16].
To the best of our knowledge, there have been few reports about the analysis and synthesis of complex-valued differential system except [9,[17][18][19], and there is no result so far about the stability of general complex-valued nonlinear differential system.In this paper, the stability criteria of complex-valued autonomous system are derived by the theory of several complex variable functions.Furthermore, the comparison principle of complex-valued nonautonomous differential system is given.Those derived stability criteria not only generalize some known results in literature but also greatly reduce the complexity of analysis and computation.As an application, the stability conditions of a class of complexvalued nonlinear systems are presented.Those problems are meaningful and challenging.
The remainder of the paper is organized as follows.In Section 2, the complex-valued system to be dealt with is formulated and several results about the complex-valued functions are presented.The stability criteria of complexvalued autonomous system are established in Section 3.Moreover, the comparison principle of complex-valued nonlinear system is given in Section 4. Based on the comparison principle, the stability conditions of a class of complex-valued systems are also presented in Section 4. The main points of the paper are illustrated by some examples in the corresponding sections.Finally, some conclusions are drawn in Section 5.

Notations and Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
The proof of Lemma 6 is similar to the case of positive definition functions defined in the real fields [21].
By the substitutions of   =   +   and   () =   (, )+V  (, ), system (5) can be recast into the following real-valued system: where  = ( 1 , . . .,   )  ,  = ( 1 , . . .,   )  ,  = ( 1 , . . .,   )  , and The homomorphism between the complex-valued system ( 5) and the real-valued system (6) allows us to employ the existing methods to analyze the stability of the complex system (5).We linearize system ( 6): where is the Jacobi matrix of ( 1 , . . .,   , V 1 , . . ., V  ) evaluated at   =   +   .Therefore, judging the stability of a nonlinear system is reduced to the problem of finding the eigenvalues of its related Jacobi matrix.The evaluation of  (  ,   ) needs an explicit separation of the complex functions   () into its real part and imaginary part.This separation is not always expressible in an analytical form.We can evaluate (  ,   ) by manipulating () directly without resorting to its real and imaginary parts.Next we will discuss two cases.
Theorem 7.For system (5) such that () is holomorphic, (i) if all of the eigenvalues of (  ) lie on the left half complex plane, system (5) is exponentially asymptotically stable at   ; (ii) if there exists an eigenvalue of (  ) lying on the right half complex plane, system (5) is not stable at   ; (iii) if there is no eigenvalue of (  ) lying on the right half complex plane, and there are a pair of eigenvalues lying on the imaginary axis at least, the stability of system (5) at   is not clear.
Corollary 8.For system (5) such that  = 1 and () is holomorphic, (i) if the real part of   (  ) < 0, system (5) is exponentially asymptotically stable at   ; (ii) if the real part of   (  ) > 0, system (5) is not stable at   ; (iii) if the real part of   (  ) = 0, the stability of system (5) at   is not clear.
Case 2. () is not holomorphic.In particular, we consider the case that () = (, ), where The motivation discussing this case is that some wellknown complex-valued systems, such as complex Lorenz system [14], complex Chen system [15] and complex Lü system [15], belong to this case.For the details see Example 13.In this case, since () is not holomorphic in ,   () does not satisfy Cauchy-Riemann equation again.But we have the following equalities: From ( 15), we have  −1 J(  ,   ) = J(  ), where Hence, J(  ,   ) and J(  ) are similar matrices; they have the same eigenvalues.We conclude the following: Theorem 9.For system (5) with () = (, ), (i) if all of eigenvalues of J(  ) lie on the left half complex plane, then system (5) is exponentially asymptotically stable at   ; (ii) if there exists an eigenvalue of J(  ) lying on the right half complex plane, then system (5) is not stable at   ; (iii) if there is no eigenvalue of J(  ) lying on the right half complex plane, and there are a pair of eigenvalues lying on the imaginary axis at least, then the stability of system (5) at   is not clear.
Remark 10.Theorems 7 and 9 provide a new method to analyze complex-valued differential system.
Example 12. Consider the stability of the trivial solution of a class complex-valued Riccati equation defined by where  is a positive integer,   ∈ C,  = 1, . . ., .
Remark 14.The common method to analyze the complexvalued systems is very tedious and complex.Theorem 7 not only generalizes some known results in the literature [9], but also greatly reduces the complexity of analysis and computation.
Example 15.In 2007, Mahmoud presented complex-valued Chen system [15] described by where , , and  are positive real parameters;  1 ,  2 ∈ C are complex state variables; and is not holomorphic because of the existence of  1 and  2 .For simplicity, we discuss only the stability of the origin.According to (16), we have By Theorem 9, it is easy to conclude that the origin of system ( 22) is exponentially asymptotically stable when  > 2.
Remark 16.It is not necessary to analyze the stability of system ( 22) by decoupling it into real and imaginary parts just like [15].The conclusion in Example 15 is just the existing result in [15].Similarly, the stability of complex-valued Lorenz system and Lü system can be derived by Theorem 9.

Stability of Complex-Valued Nonautonomous System
In this section, the comparison principle of system (1) is presented.
Proof.We just prove the case that the trivial solution of system (3) is stable.Since (, ) is a positive definite function, by Lemma 6, there exists a  class function (⋅) such that  (, ) ≥  (‖‖) . (25) Considering the stability of the trivial solution of (3), for all  > 0, ∃ * (,  0 ), when 0 <  0 <  * , the solution of (3) satisfies Considering the continuity of (, ) and the fact (, 0) = 0, ∃(,  0 ), when ‖ 0 ‖ < , we have 0 < ( 0 ,  0 ) ≜  0 <  0 <  * .Let () = (,  0 ,  0 ) be the solution of system (1), then  Remark 19.Just as Example 13, we can separate (29) into real and imaginary parts and rewrite it as a 4-dimension real time-varying system and then use the comparison principle in the context of real fields to judge the stability of (29), but the computation is more complex and lengthy.In fact, Theorem 17 is the generalization of the real comparison principle in the complex fields, we can judge directly the stability properties of complex-valued differential system by Theorem 17.
Generally speaking, the construction of the complex positive definite function (, ) is more difficult than the case of real system.Next we will give the constructing method of complex Lyapunov function of a class complex-valued system defined by ż =  (, ) [ℎ () +  (, )] , where ℎ  (0) ̸ = 0, and ℎ(0) = 0 if and only if  = 0.
Remark 22.The asymptotic stability of system (39) can not be judged by Theorem 3.2 of the paper [18], because condition (3.13) of the theorem can not be satisfied, so our stability criterion is more effective.

Conclusion
This paper has discussed the stability of complex-valued differential system.The stability criteria of complex-valued autonomous system has been established.For the general complex-valued non-autonomous system, the comparison principle in the context of complex fields have been derived.Those derived stability criteria not only provide a new method to analyze complex-valued differential system, but also greatly reduce the complexity of analysis and computation.Based on the new results, the stability conditions of a class complex-valued nonlinear system have been presented.