Stability Analysis of Complex-Valued Nonlinear Differential System

Abstract

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.

1. Introduction

The stability of differential system has been studied by many researchers, for example, [1–6] 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 [7–10]. 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 𝑓(𝑡,𝑧) 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^𝑛 [16].

To the best of our knowledge, there have been few reports about the analysis and synthesis of complex-valued differential system except [9, 17–19], 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 complex-valued 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 complex-valued 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.

2. Notations and Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Let Ω⊆ℂ^𝑛 be a neighbourhood of the origin, 𝐼=[𝑡₀,+∞), 𝑡₀≥0. 𝑊(𝑧)∈𝐶[Ω,ℝ], 𝑊(0)=0, 𝑉(𝑡,𝑧)∈𝐶[𝐼×Ω,ℝ], 𝑉(𝑡,0)=0. 𝐻[Ω,ℂ^𝑛] denote all holomorphic maps from Ω to ℂ^𝑛. Let 𝑧 and 𝑧^∗ be the conjugate and conjugate transpose of 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛)^𝑇∈ℂ^𝑛, respectively, and ‖𝑧‖=√𝑧^∗𝑧,𝑖=√−1.

Definition 1. 𝑊(𝑧) is called a complex positive definite function on Ω, if 𝑊(𝑧)≥0 for any 𝑧∈Ω, and 𝑊(𝑧)=0 if and only if 𝑧=0.

Definition 2. 𝑉(𝑡,𝑧) is called a positive definite function on 𝐼×Ω if there exists a complex positive definite function 𝑊(𝑧) such that 𝑉(𝑡,𝑧)≥𝑊(𝑧) for any 𝑧∈Ω, and 𝑉(𝑡,𝑧)=0 if and only if 𝑧=0.

In this paper, we will study the stability of complex-valued nonlinear differential system described by 𝑑𝑧((𝑡,𝑧𝑡𝑑𝑡=𝑓𝑡,𝑧))₀=𝑧₀∈ℂ^𝑛, (1) where 𝑡₀∈𝑅⁺, 𝑓∶𝐼×Ω↦ℂ^𝑛 is a complex-valued continuous map. 𝑧∶𝐼↦ℂ^𝑛 is a complex-valued differentiable map. Let 𝑧_𝑒 be an equilibrium point satisfying 𝑓(𝑡,𝑧_𝑒)=0; 𝑧(𝑡)=𝑧(𝑡,𝑡₀,𝑧₀) satisfies system (1).

Definition 3. System (1) is said to be stable at 𝑧_𝑒, if for all 𝜀>0, ∃𝛿(𝜀,𝑡₀) such that ‖𝑧₀−𝑧_𝑒‖<𝛿 implies 𝑧(𝑡)−𝑧_𝑒<𝜀,∀𝑡≥𝑡₀. (2)

Definition 4. System (1) is said to be asymptotically stable at 𝑧_𝑒 if it is stable and lim_𝑡→∞‖𝑧(𝑡)−𝑧_𝑒‖=0.

Lemma 5 (see [20]). Let 𝑔∈𝐶[𝑅⁺×𝑅⁺,𝑅], 𝑚∈𝐶[𝑅⁺,𝑅⁺], and 𝐷𝑚(𝑡)≤𝑔(𝑡,𝑚(𝑡)), for all 𝑡∈[𝑡₀,+∞)⧵𝑆, where 𝑆 is a countable subset of [𝑡₀,∞) and 𝐷 is any one of the Dini derivatives. Suppose that the maximal solution 𝑟(𝑡) of 𝑢̇(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑢𝑡₀=𝑢₀≥0,𝑡₀∈𝑅⁺ (3) exists on [𝑡₀,∞) and 𝑚(𝑡₀)≤𝑢₀. Then 𝑚(𝑡)≤𝑟(𝑡), 𝑡≥𝑡₀.

Lemma 6. For any positive definite continuous function 𝑊(𝑧) on Ω={𝑧∈ℂ^𝑛∣‖𝑧‖≤𝐻}, 𝐻>0, there exist two 𝐾 class functions 𝑎(⋅) and 𝑏(⋅) such that 𝑎(‖𝑧‖)≤𝑊(𝑧)≤𝑏(‖𝑧‖). (4)

The proof of Lemma 6 is similar to the case of positive definition functions defined in the real fields [21].

3. Stability Criteria of Complex-Valued Autonomous System

In this section, we will study the stability of complex-valued autonomous system described by 𝑑𝑧(𝑧),𝑧𝑡𝑑𝑡=𝑓₀=𝑧₀∈ℂ^𝑛, (5) where 𝑓(𝑧)=(𝑓₁(𝑧),…,𝑓_𝑛(𝑧))^𝑇,𝑧_𝛼∈ℂ, 𝑓_𝛼(𝑧)∶Ω↦ℂ, 𝑓(𝑧_𝑒)=0,𝛼=1,…,𝑛.

By the substitutions of 𝑧_𝛼=𝑥_𝛼+𝑖𝑦_𝛼 and 𝑓_𝛼(𝑧)=𝑢_𝛼(𝑥,𝑦)+𝑖𝑣_𝛼(𝑥,𝑦), system (5) can be recast into the following real-valued system: ̇,̇,𝑥=𝑢𝑥,𝑦𝑦=𝑣𝑥,𝑦 (6) where 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇, 𝑦=(𝑦₁,…,𝑦_𝑛)^𝑇, 𝑢=(𝑢₁,…,𝑢_𝑛)^𝑇, and 𝑣=(𝑣₁,…,𝑣_𝑛)^𝑇. 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): 𝑥̇𝑦̇=𝐽𝑥_𝑒,𝑦_𝑒𝑥𝑦, (7) where 𝐽𝑥_𝑒,𝑦_𝑒=𝜕𝑢₁,…,𝑢_𝑛,𝑣₁,…,𝑣_𝑛𝜕𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_{𝑛(𝑥𝑒,𝑦𝑒)} (8) is the Jacobi matrix of (𝑢₁,…,𝑢_𝑛,𝑣₁,…,𝑣_𝑛) evaluated at 𝑧_𝑒=𝑥_𝑒+𝑖𝑦_𝑒. 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 𝐽(𝑥_𝑒,𝑦_𝑒) needs an explicit separation of the complex functions 𝑓_𝛼(𝑧) into its real part and imaginary part. This separation is not always expressible in an analytical form. We can evaluate 𝐽(𝑥_𝑒,𝑦_𝑒) by manipulating 𝑓(𝑧) directly without resorting to its real and imaginary parts. Next we will discuss two cases.

Case 1. 𝑓(𝑧) is holomorphic map, namely, 𝑓(𝑧)∈𝐻[Ω,ℂ^𝑛]. In this case, since 𝑓_𝛼(𝑧) is analytic function, 𝑓_𝛼(𝑧) satisfies Cauchy-Riemann equation, and the following equalities hold for 𝛼,𝑗=1,2,…,𝑛: 𝜕𝑢_𝛼𝜕𝑥_𝑗=𝜕𝑣_𝛼𝜕𝑦_𝑗,𝜕𝑢_𝛼𝜕𝑦_𝑗=−𝜕𝑣_𝛼𝜕𝑥_𝑗,𝜕𝑓_𝛼𝜕𝑧_𝑗=𝜕𝑢_𝛼𝜕𝑥_𝑗+𝑖𝜕𝑣_𝛼𝜕𝑥_𝑗. (9) Denote 𝜕𝑢≜𝜕𝑢𝜕𝑥¹,…,𝑢_𝑛𝜕𝑥₁,…,𝑥_𝑛,𝜕𝑢≜𝜕𝑢𝜕𝑦¹,…,𝑢_𝑛𝜕𝑦₁,…,𝑦_𝑛,𝜕𝑣≜𝜕𝑣𝜕𝑥¹,…,𝑣_𝑛𝜕𝑥₁,…,𝑥_𝑛,𝜕𝑣≜𝜕𝑣𝜕𝑦¹,…,𝑣_𝑛𝜕𝑦₁,…,𝑦_𝑛,𝜕𝑓≜𝜕𝑓𝜕𝑧¹,…,𝑓_𝑛𝜕𝑧₁,…,𝑧_𝑛. (10) It follows from (9) that 𝜕𝑢=𝜕𝑥𝜕𝑣,𝜕𝑦𝜕𝑢𝜕𝑦=−𝜕𝑣,𝜕𝑥𝜕𝑓=𝜕𝑧𝜕𝑢𝜕𝑥+𝑖𝜕𝑣.𝜕𝑥 (11) Thus 12𝐼_𝑛−𝑖𝐼_𝑛𝐼_𝑛𝑖𝐼_𝑛𝐽𝑥_𝑒,𝑦_𝑒𝐼_𝑛𝐼_𝑛𝑖𝐼_𝑛−𝑖𝐼_𝑛=12𝜕𝑢𝜕𝑥−𝑖𝜕𝑢𝜕𝑦𝜕𝑣𝜕𝑥−𝑖𝜕𝑣𝜕𝑦𝜕𝑢𝜕𝑥+𝑖𝜕𝑢𝜕𝑦𝜕𝑣𝜕𝑥+𝑖𝜕𝑣𝐼𝜕𝑦^𝑛𝐼_𝑛𝑖𝐼_𝑛−𝑖𝐼_𝑛=12𝜕𝑢𝜕𝑥−𝑖𝜕𝑢𝜕𝑦+𝑖𝜕𝑣+𝜕𝑥𝜕𝑣𝜕𝑦𝜕𝑢𝜕𝑥−𝑖𝜕𝑢𝜕𝑦−𝑖𝜕𝑣−𝜕𝑥𝜕𝑣𝜕𝑦𝜕𝑢𝜕𝑥+𝑖𝜕𝑢𝜕𝑦+𝑖𝜕𝑣−𝜕𝑥𝜕𝑣𝜕𝑦𝜕𝑢𝜕𝑥+𝑖𝜕𝑢𝜕𝑦−𝑖𝜕𝑣+𝜕𝑥𝜕𝑣=𝜕𝑦𝜕𝑢𝜕𝑥+𝑖𝜕𝑣00𝜕𝑥𝜕𝑢𝜕𝑥−𝑖𝜕𝑣𝑧𝜕𝑥=𝐶𝐽_𝑒, (12) where 𝐽(𝑥_𝑒,𝑦_𝑒) can be rewritten by 𝐽𝑥_𝑒,𝑦_𝑒=𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑥𝜕𝑢𝜕𝑦𝜕𝑣𝜕𝑦_{(𝑥𝑒,𝑦𝑒)},𝐶𝐽𝑧_𝑒=𝜕𝑓00𝜕𝑧𝜕𝑓𝜕𝑧_𝑧𝑒. (13) The equality (12) means that 𝐽(𝑥_𝑒,𝑦_𝑒) and 𝐶𝐽(𝑧_𝑒) are similar matrices; thus they have the same eigenvalues. Moreover, we notice that if 𝜆 is an eigenvalue of 𝜕𝑓/𝜕𝑧, then 𝜆 is an eigenvalue of 𝜕𝑓/𝜕𝑧. It turns out that the eigenvalues of 𝐽(𝑥_𝑒,𝑦_𝑒) are merely the eigenvalues of [𝜕𝑓/𝜕𝑧]_𝑧𝑒 if we do not consider the multiplicity of the eigenvalues. Therefore, the stability of (7) can be easily checked by examining whether the real part of the eigenvalues of 𝐽𝑧_𝑒=𝜕(𝑓₁,…,𝑓_𝑛)𝜕(𝑧₁,…,𝑧_𝑛)_𝑧𝑒 (14) is negative. We conclude the following results.

Theorem 7. For system (5) such that 𝑓(𝑧) is holomorphic,(i)if all of the eigenvalues of 𝐽(𝑧_𝑒) lie on the left half complex plane, system (5) is exponentially asymptotically stable at 𝑧_𝑒;(ii)if there exists an eigenvalue of 𝐽(𝑧_𝑒) lying on the right half complex plane, system (5) is not stable at 𝑧_𝑒;(iii)if there is no eigenvalue of 𝐽(𝑧_𝑒) 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 𝑧_𝑒 is not clear.

Corollary 8. For system (5) such that 𝑛=1 and 𝑓(𝑧) is holomorphic,(i)if the real part of 𝑓^′(𝑧_𝑒)<0, system (5) is exponentially asymptotically stable at 𝑧_𝑒;(ii)if the real part of 𝑓^′(𝑧_𝑒)>0, system (5) is not stable at 𝑧_𝑒;(iii)if the real part of 𝑓^′(𝑧_𝑒)=0, the stability of system (5) at 𝑧_𝑒 is not clear.

Case 2. 𝑓(𝑧) is not holomorphic. In particular, we consider the case that 𝑓(𝑧)=𝑔(𝑧,𝑧), where 𝑔(⋅,⋅)∈𝐻[Ω×Ω,ℂ^2𝑛].

The motivation discussing this case is that some well-known 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 𝑓(𝑧) is not holomorphic in 𝑧, 𝑓_𝛼(𝑧) does not satisfy Cauchy-Riemann equation again. But we have the following equalities: 𝑢_𝛼=1𝑥,𝑦2𝑔_𝛼(𝑧,𝑧)+𝑔_𝛼(𝑧,𝑧,𝑣_𝛼)𝑖𝑥,𝑦=−2𝑔_𝛼(𝑧,𝑧)−𝑔_𝛼(𝑧,𝑧),𝜕𝑢_𝛼𝜕𝑥_𝑗=12𝜕𝑔_𝛼𝜕𝑧_𝑗+𝜕𝑔_𝛼𝜕𝑧_𝑗+𝜕𝑔_𝛼𝜕𝑧_𝑗+𝜕𝑔_𝛼𝜕𝑧_𝑗,𝜕𝑢_𝛼𝜕𝑦_𝑗=𝑖2𝜕𝑔_𝛼𝜕𝑧_𝑗−𝜕𝑔_𝛼𝜕𝑧_𝑗+𝜕𝑔_𝛼𝜕𝑧_𝑗−𝜕𝑔_𝛼𝜕𝑧_𝑗,𝜕𝑣_𝛼𝜕𝑥_𝑗𝑖=−2𝜕𝑔_𝛼𝜕𝑧_𝑗+𝜕𝑔_𝛼𝜕𝑧_𝑗−𝜕𝑔_𝛼𝜕𝑧_𝑗−𝜕𝑔_𝛼𝜕𝑧_𝑗,𝜕𝑣_𝛼𝜕𝑦_𝑗=12𝜕𝑔_𝛼𝜕𝑧_𝑗−𝜕𝑔_𝛼𝜕𝑧_𝑗−𝜕𝑔_𝛼𝜕𝑧_𝑗+𝜕𝑔_𝛼𝜕𝑧_𝑗. (15) From (15), we have 𝑃⁻¹𝐽(̃𝑥_𝑒,𝑦_𝑒)𝑃=𝐽(̃𝑧_𝑒), where 𝑃⁻¹=12(𝑖−1)𝐼_𝑛𝑖𝐼_𝑛(𝑖−1)𝐼_𝑛𝐼_𝑛,𝑃=−𝑖𝐼_𝑛−𝐼_𝑛−(1+𝑖)𝐼_𝑛(1+𝑖)𝐼_𝑛,𝐽̃𝑥_𝑒,𝑦_𝑒=𝜕𝑢₁,…,𝑢_𝑛𝜕𝑥₁,…,𝑥_𝑛𝜕𝑢₁,…,𝑢_𝑛𝜕𝑥₁,…,𝑥_𝑛𝜕𝑣₁,…,𝑣_𝑛𝜕𝑦₁,…,𝑦_𝑛𝜕𝑣₁,…,𝑣_𝑛𝜕𝑦₁,…,𝑦_{𝑛(𝑥𝑒,𝑦𝑒)},𝐽̃𝑧_𝑒=𝜕𝑔₁,…,𝑔_𝑛𝜕𝑧₁,…,𝑧_𝑛𝜕𝑔−𝑖¹,…,𝑔_𝑛𝜕𝑧₁,…,𝑧_𝑛𝑖𝜕𝑔₁,…,𝑔_𝑛𝜕𝑧₁,…,𝑧_𝑛𝜕𝑔₁,…,𝑔_𝑛𝜕𝑧₁,…,𝑧_𝑛𝑧𝑒. (16) Hence, 𝐽(̃𝑥_𝑒,𝑦_𝑒) and 𝐽(̃𝑧_𝑒) are similar matrices; they have the same eigenvalues. We conclude the following:

Theorem 9. For system (5) with 𝑓(𝑧)=𝑔(𝑧,𝑧),(i)if all of eigenvalues of 𝐽(̃𝑧_𝑒) lie on the left half complex plane, then system (5) is exponentially asymptotically stable at 𝑧_𝑒;(ii)if there exists an eigenvalue of 𝐽(̃𝑧_𝑒) lying on the right half complex plane, then system (5) is not stable at 𝑧_𝑒;(iii)if there is no eigenvalue of 𝐽(̃𝑧_𝑒) 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 𝑧_𝑒 is not clear.

Remark 10. Theorems 7 and 9 provide a new method to analyze complex-valued differential system.

Remark 11. Corollary 8 is the existing results in [9].

Example 12. Consider the stability of the trivial solution of a class complex-valued Riccati equation defined by 𝑑𝑧𝑑𝑡=𝑧^𝑛−𝑝_𝑛−1𝑧^𝑛−1+⋯+𝑝₁𝑧, (17) where 𝑛 is a positive integer, 𝑝_𝑗∈ℂ,𝑗=1,…,𝑛.

In this example, 𝑓(𝑧)=𝑧^𝑛−𝑝_𝑛−1𝑧^𝑛−1+⋯+𝑝₁𝑧, 𝑓^′(0)=𝑝₁; by Corollary 8, if the real part of 𝑝₁ is negative, system (17) is exponentially asymptotically stable at 𝑧_𝑒=0; if the real part of 𝑝₁ is positive, system (17) is not stable at 𝑧_𝑒=0.

Example 13. Consider the following system: 𝑧̇₁=𝑧²₁−𝑖𝑧₁+𝑧₂,𝑧̇₂=−2𝑧₁−𝑧₂,𝑧(0)=𝑧₀∈ℂ. (18) System (18) has two equilibrium points 𝑃₁(0,0) and 𝑃₂(2+𝑖,−4−2𝑖), and 𝐽𝑃₁=𝑃−𝑖1−2−1,𝐽₂=4+𝑖1−2−1. (19) The eigenvalues of 𝐽(0,0) and 𝐽(2+𝑖,−4−2𝑖) are 𝜆₁=−432/1331−899/467𝑖, 𝜆₂=−899/1331+432/467𝑖 and 𝜆₃=1575/439+1313/1195𝑖, 𝜆₄=−841/1431−118/1195𝑖, respectively. By Theorem 7, system (18) is exponentially asymptotically stable at 𝑃₁ and not stable at 𝑃₂.

An alternative, but tedious, approach to the problem is to separate (18) into real and imaginary parts and rewrite it as (6): 𝑥̇₁=𝑥²₁+𝑥₂+𝑦₁−𝑦²₁,𝑥̇₂=−2𝑥₁−𝑥₂,𝑦̇₁=2𝑥₁𝑦₁−𝑥₁+𝑦₂,𝑦̇₂=−2𝑦₁−𝑦₂. (20) The equilibrium points of system (20) are 𝑃₁(0,0,0,0) and 𝑃₂(2,−4,1,−2). To judge the stability of those equilibrium points, we have to compute the eigenvalue of Jacobi matrix: 𝐽𝑥,𝑦=2𝑥₁11−2𝑦₁0−2−1002𝑦₁−102𝑥₁100−2−1. (21) By calculating, it can be found that 𝑃₁ are stable and 𝑃₂ are not stable; it is not contradictory; in fact, 𝑃₁ is just the point (0,0) and 𝑃₂ is just (2+𝑖,−4−2𝑖) on ℂ².

Remark 14. The common method to analyze the complex-valued 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 𝑧̇₁=𝛼𝑧₂−𝑧₁,𝑧̇₂=𝛾−𝛼𝑧₁−𝑧₁𝑧₃+𝛾𝑧₂,𝑧̇₃=12𝑧₁𝑧₂+𝑧₁𝑧₂−𝛽𝑧₃, (22) where 𝛼, 𝛽, and 𝛾 are positive real parameters; 𝑧₁, 𝑧₂∈ℂ are complex state variables; and 𝑧₃∈ℝ is real state variables. Obviously, 𝑓(𝑧)=(𝛼(𝑧₂−𝑧₁), (𝛾−𝛼)𝑧₁−𝑧₁𝑧₃+𝛾𝑧₂, (𝑧₁𝑧₂+𝑧₁𝑧₂)/2−𝛽𝑧₃) is not holomorphic because of the existence of 𝑧₁ and 𝑧₂. For simplicity, we discuss only the stability of the origin. According to (16), we have 𝐽̃(0)=𝐴00𝐴,where𝐴=−𝛼𝛼0𝛾−𝛼𝛾000−𝛽. (23) By Theorem 9, it is easy to conclude that the origin of system (22) is exponentially asymptotically stable when 𝛼>2𝛾.

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.

4. Stability of Complex-Valued Nonautonomous System

In this section, the comparison principle of system (1) is presented.

Theorem 17. Assume that there exists a positive definite function 𝑉(𝑡,𝑧)∈𝐶[𝐼×Ω,ℝ⁺] such that 𝑉(𝑡,𝑧) is locally Lipschitzian in 𝑧, and 𝐷⁺𝑉(𝑡,𝑧(𝑡))₍₁₎≤𝑔(𝑡,𝑉(𝑡,𝑧(𝑡))),𝑡∈𝐼⧵𝑆, (24) where 𝐼=[𝑡₀,+∞), 𝑆 is a countable subset of 𝐼, 𝑔∈𝐶[𝐼×𝑅⁺,𝑅], and Ω⊆ℂ^𝑛 is a neighbourhood of the origin. Then the stability properties of the trivial solution of (3) with 𝑔(𝑡,0)=0 imply the corresponding stability properties of the trivial solution of (1) such that 𝑓(𝑡,0)=0.

Proof. We just prove the case that the trivial solution of system (3) is stable.
Since 𝑉(𝑡,𝑧) is a positive definite function, by Lemma 6, there exists a 𝐾 class function 𝑎(⋅) such that 𝑉(𝑡,𝑧)≥𝑎(‖𝑧‖). (25) Considering the stability of the trivial solution of (3), for all𝜀>0, ∃𝛿^∗(𝜀,𝑡₀), when 0<𝑢₀<𝛿^∗, the solution of (3) satisfies 𝑢𝑡,𝑡₀,𝑢₀<𝑎(𝜀). (26) Considering the continuity of 𝑉(𝑡,𝑧) and the fact 𝑉(𝑡,0)=0, ∃𝛿(𝜀,𝑡₀), when ‖𝑧₀‖<𝛿, we have 0<𝑉(𝑡₀,𝑧₀)≜𝑉₀<𝑢₀<𝛿^∗. Let 𝑧(𝑡)=𝑧(𝑡,𝑡₀,𝑧₀) be the solution of system (1), then 𝐷⁺𝑉₍₁₎()𝑉𝑡≤𝑔𝑡,𝑉,𝑡∈𝐼⧵𝑆,₀,𝑧₀=𝑉₀. (27) By Lemma 5, we have 𝑎(‖𝑧(𝑡)‖)≤𝑉(𝑡,𝑧(𝑡))≤𝑢𝑡,𝑡₀,𝑢₀<𝑎(𝜀), (28) so ‖𝑧(𝑡)‖<𝜀; thus the trivial solution of (1) is stable. The proof is completed.

Example 18. Consider the stability of the trivial solution of the following system: 𝑧̇₁=(−3+8sin𝑡)𝑧₁+(sin𝑡)𝑧₂,𝑧̇₂=(cos𝑡)𝑧₁+(−3+8sin𝑡)𝑧₂,𝑧(0)=𝑧₀∈ℂ². (29) Let 𝑉(𝑡,𝑧)=(1/2)(‖𝑧₁‖²+‖𝑧₂‖²), then 𝑑𝑉𝑑𝑡₍₂₉₎=12𝑧̇₁𝑧₁+𝑧₁𝑧̇‌₁+𝑧̇₂𝑧₂+𝑧₂𝑧̇‌₂=(−3+8sin𝑡)𝑧₁²+𝑧₂²+(sin𝑡+cos𝑡)Re𝑧₁𝑧₂≤(−3+8sin𝑡)𝑧₁²+𝑧₂²+2𝑧₁𝑧₂≤(−3+8sin𝑡)𝑧₁²+𝑧₂²+𝑧₁²+𝑧₂²=(−2+8sin𝑡)𝑧₁²+𝑧₂²=2(−2+8sin𝑡)𝑉≜𝑔(𝑡,𝑉). (30) Since the solution of system ̇()()𝑢(0)𝑢=𝑔𝑡,𝑢=2−2+8sin𝑡𝑢,=𝑢₀∈ℝ (31) is 𝑢(𝑡)=𝑢₀exp{−16−4𝑡−16cos𝑡}, the trivial solution of system (29) is also asymptotically stable by Theorem 17.

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 𝑉(𝑡,𝑧) 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=𝐺(𝑡,𝑧)ℎ(𝑧)+𝑔(𝑡,𝑧), (32) where 𝐺∈𝐶[𝐼×Ω,ℝ⁺], 𝑔∈𝐶[𝐼×Ω,ℂ], 𝐼=[𝑡₀,+∞), 𝑡₀≥0, Ω⊆ℂ is a neighbourhood of the origin, ℎ∈𝐻[Ω,ℂ], ℎ^′(0)≠0, and ℎ(0)=0 if and only if 𝑧=0.

System (32) has been discussed to study the asymptotic stability of complex-valued Riccati equation 𝑧̇=𝑞(𝑡,𝑧)−𝑝(𝑡)𝑧² (33) in [18]. Next we will give the exponentially asymptotic stability condition of system (32) by Theorem 17. Let 𝑟(𝑧)=𝑧ℎ^′(0)−ℎ(𝑧)(𝑧)−ℎ𝑧ℎ,𝑧∈Ω,𝑧≠0,_′′(0)2ℎ^′(0),𝑧=0. (34) It can be verified that 𝑟(𝑧) is analytic function in Ω, so we can take 𝑤(𝑧)=𝑧exp^𝑧₀𝑟(𝑡)𝑑𝑡 (35) and 𝑉(𝑧)=‖𝑤(𝑧)‖, then when ℎ(𝑧)≠0, 𝑑𝑉𝑑𝑡²(𝑧)₍₃₂₎=𝑑𝑤(𝑧)𝑑𝑡𝑤(𝑧𝑤̇(𝑧))=2Re𝑤(𝑧)𝑧̇=2Re𝑤(𝑧)𝑤(𝑧)𝑧⁻¹+𝑟(𝑧)𝑧̇=2𝑉²(𝑧)Reℎ^′(0)ℎ⁻¹(𝑧)𝑧̇, (36) thus 𝑑𝑉(𝑧𝑑𝑡)₍₃₂₎=𝑉(𝑧)Reℎ^′(0)ℎ⁻¹(𝑧)𝑧̇()𝑉(𝑧)ℎ=𝐺𝑡,𝑧×Re^′(0)ℎ⁻¹(𝑧ℎ(𝑧)(()𝑉(𝑧)ℎ)+𝑔𝑡,𝑧)=𝐺𝑡,𝑧Re^′(0𝑔())1+𝑡,𝑧ℎ(𝑧). (37) By Theorem 17, when 𝑉̇(𝑧)|₍₃₂₎≤−𝑐𝑉(𝑧),𝑐>0, namely, 𝐺(𝑡,𝑧)Reℎ^′(0𝑔())1+𝑡,𝑧ℎ(𝑧)≤−𝑐, (38) the trivial solution of system (32) is exponentially asymptotically stable. So we have the following result.

Theorem 20. The trivial solution of system (32) is exponentially asymptotically stable if ∃𝑐>0 such that the inequality (38) holds for almost everywhere (𝑡,𝑧)∈𝐼×Ω.

Example 21. Consider complex-valued Riccati equation defined by 𝑑𝑧𝑑𝑡=−𝑒^−𝑡𝜌+𝑡𝑧+𝑧²,𝑧(0)=𝑧₀∈ℂ, (39) where 𝑡≥0 and ‖𝑧‖<𝜌,𝜌>0. In this example, 𝐺(𝑡,𝑧)=𝑒^−𝑡, ℎ(𝑧)=−𝑧²−𝜌𝑧, 𝑔(𝑡,𝑧)=−𝑧𝑡, and when ℎ(𝑧)≠0𝐺(𝑡,𝑧)Reℎ^′(0𝑔())1+𝑡,𝑧ℎ(𝑧)𝑡<−𝜌Re1+𝜌+𝑧=−𝜌1+𝜌+Re𝑧𝜌+𝑧²𝑡<−𝜌. (40) So the trivial solution of (39) is exponentially asymptotically stable.

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.

5. 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.

Acknowledgments

The authors would like to thank the editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper. This work is supported by the NNSF of China under Grants 61174039 and 61203128 and the Connotative Construction Project of Shanghai University of Engineering Science under Grants nhky-2012-13.