Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

Abstract

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

1. Introduction

Many systems in physics, chemistry, biology, and information science have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. This complex dynamical behavior can be modeled by impulsive differential equations. The theory of impulsive differential systems has been developed by numerous mathematicians (see [1–9]). As to the stability theory and boundary value problems to impulsive differential equations, There have been extensive studies in this area. However, there are very few works on the qualitative theory of impulsive differential equations and impulsive semidynamical systems. Recently, Bonotto and Federson have given a version of the Poincaré-Bendixson Theorem for impulsive semidynamical systems in [10, 11]. As it is known, the method of Poincaré map plays an important role in the research of qualitative theory and is a natural means to study the existence of periodic solutions and its asymptotic stability. However, due to the complexity of the associated impulsive dynamic models, this approach has only been applied successfully to Raibert’s one-legged-hopper (see [12–14]) predator-prey models (see [15–18]), and so forth. The bifurcation theory for ordinary differential equations or smooth systems appeared during the last decades (see, e.g., [19]); however, little is known about the bifurcation theory of impulsive differential equations due to its complexity (see [20]). In this paper, we mainly study a certain scalar impulsive differential equations on Moebius stripe undergoing impulsive effects at fixed time: 𝑑𝑥()𝑑𝑡=𝑓𝑡,𝑥,𝑡≠𝜏_𝑘,Δ𝑥|_{𝑡=𝜏𝑘}=𝐼_𝑘(𝑥),𝑘∈ℤ⁺, (1) where 0≤𝜏_𝑘<𝜏_𝑘+1, 𝑘∈ℤ⁺ are fixed with 𝜏_𝑘→+∞ as, 𝑘→+∞, and Δ𝑥|_{𝑡=𝜏𝑘}=𝑥(𝜏⁺_𝑘)−𝑥(𝜏_𝑘). Hu and Han (see [20]) investigated the existence of periodic solutions and bifurcations of (1) under the assumptions that 𝑓(𝑡,𝑥) and 𝐼_𝑘(𝑥) are periodic; that is, the following assumption holds.

(H*) There exist a constant 𝑇>0, a positive integer 𝑞, and two mutual coprime positive integers 𝑚 and 𝑛 such that 𝑓(𝑡+𝑇,𝑥)=𝑓(𝑡,𝑥),∀𝑡∈ℝ⁺𝐼,𝑥∈ℝ,_𝑘+𝑞(𝑥)=𝐼_𝑘(𝑥),𝜏_𝑘+𝑞−𝜏_𝑘=𝑇,∀𝑘∈ℤ⁺𝑚𝑡,𝑥∈ℝ,_𝑘+𝑞−𝑡_𝑘=𝑛𝑇,𝑘≥1. (2) In this paper, we assume that the following conditions hold.

(H1) Assume that both 𝑓(𝑡,𝑥) and 𝑓_𝑥(𝑡,𝑥) are continuous scalar functions on ℝ×ℝ, 𝐼_𝑘(𝑥)∶ℝ→ℝ,𝑘∈ℤ are odd, continuous functions; that is, 𝐼_𝑘(−𝑥)=−𝐼_𝑘(𝑥),𝑘∈ℤ⁺.

(H2) There exists a constant 𝑇>0, a positive integer 𝑞 such that 𝑓()()𝐼𝑡+𝑇,−𝑥=−𝑓𝑡,𝑥,∀𝑡∈ℝ,𝑥∈ℝ,_𝑘+𝑞(𝑥)=𝐼_𝑘(𝑥),𝜏_𝑘+𝑞−𝜏_𝑘=𝑇,∀𝑘∈ℤ⁺,𝑥∈ℝ. (3) From (H2), we have 𝑓(𝑡,𝑥) is 2𝑇 periodic and 𝜏_𝑘+𝑞−𝜏_𝑘=2𝑇. Hence assumption (H*) holds naturally. However, we show some new and fruitful results of system (1) with the condition (H1)-(H2). For example, we obtain the existence and stability of 2𝑇 periodic solutions to system (1) by double-periodic bifurcation.

This paper is organized as follows. In Section 2, for the sake of self-containedness of the paper, we present some basic definitions of impulsive differential equations. In Section 3, we describe the scalar impulsive differential equations on Moebius stripe and define the Poincaré map. Then we prove several essential lemmas and give sufficient conditions to ensure the existence and stability of one-side and two-side orbits of impulsive differential equation on Moebius stripe. In Section 4, we are mainly concerned with the double-periodic bifurcation impulsive differential equations on Moebius stripe.

2. Preliminaries

For the sake of self-containedness of the paper, we present the basic definitions and notations of the theory of impulsive differential equations we need (see [1, 2, 8]). We also include some fundamental results which are necessary for understanding the theory.

Let ℝ, ℤ, and ℤ⁺ be the sets of real numbers, integers, and positive integers, respectively. Denote by 𝜃={𝜃_𝑖} a strictly increasing sequence of real numbers such that the set 𝔄 of indexes 𝑖 is an interval in ℤ.

Definition 1. A function 𝜙∶ℝ→ℝ^𝑛,𝑛∈ℝ, is from the set 𝑃𝐶(ℝ,𝜃) if (i)it is left continuous;(ii)it is continuous, except, possibly, points of 𝜃, where it has discontinuities of the first kind.

The last definition means that if 𝜙(𝑡)∈𝑃𝐶(ℝ,𝜃), then the right limit 𝜙(𝜃_𝑖+)=lim_𝑡→𝜃⁺_𝑖𝜙(𝑡) exists and 𝜙(𝜃_𝑖(−))=𝜙(𝜃_𝑖), where 𝜙(𝜃_𝑖−)=lim_𝑡→𝜃⁻_𝑖𝜙(𝑡), for each 𝜃_𝑖∈𝜃.

Definition 2. A function 𝜙∶ℝ→ℝ^𝑛 is from the set 𝑃𝐶¹(ℝ,𝜃) if 𝜙(𝑡), 𝜙^′(𝑡)∈𝑃𝐶(ℝ,𝜃), where the derivative at points of 𝜃 is assumed to be the left derivative.
In what follows, in this section, 𝐽∈ℝ is an interval in ℝ. For simplicity of notation, 𝜃 is not necessary a subset of 𝐽.

Definition 3. The solution 𝜙(𝑡) is stable if to any 𝜀>0 and 𝑡₀∈𝐽 there corresponds 𝛿(𝑡₀,𝜀)>0 such that for any other solution 𝜓(𝑡) of (1) with ‖𝜙(𝑡₀)−𝜓(𝑡₀)‖<𝛿(𝑡₀,𝜀) we have ‖𝜙(𝑡)−𝜓(𝑡)‖<𝜀 for 𝑡≥𝑡₀; the solution 𝜙(𝑡) is uniformly stable, if 𝛿(𝑡₀,𝜀) can be chosen independently of 𝑡₀.

Definition 4. The solution 𝜙(𝑡) is asymptotically stable if it is stable in the sense of Definition 3 and there exists a positive number 𝜅(𝑡₀) such that if 𝜓(𝑡) is any other solution of (1) with ‖𝜙(𝑡₀)−𝜓(𝑡₀)‖<𝜅(𝑡₀), then ‖𝜙(𝑡)−𝜓(𝑡)‖→0 as 𝑡→∞; if 𝜅(𝑡₀) can be chosen to be independent of 𝑡₀ and 𝜙(𝑡) is uniformly stable, then 𝜙(𝑡) is said to be uniformly asymptotically stable.

Definition 5. The solution 𝜙(𝑡) is unstable if there exist numbers 𝜀₀>0 and 𝑡₀∈𝐽 such that for any 𝛿>0 there exists a solution 𝑦_𝛿(𝑡), ‖𝜙(𝑡₀)−𝑦_𝛿(𝑡₀)‖<𝛿, of (1) such that either it is not continuable to ∞ or there exists a moment 𝑡₁, 𝑡₁>𝑡₀ such that ‖𝜙(𝑡₁)−𝑦_𝛿(𝑡₁)‖≥𝛿.
For any 𝑡₀∈ℝ, we assume that there exists a 𝑘∈ℤ⁺, such that 𝜏_𝑘−1<𝑡₀≤𝜏_𝑘; then the initial value problem (IVP) to first-order impulsive differential equations (1) is given as 𝑑𝑥()𝑑𝑡=𝑓𝑡,𝑥,𝑡≠𝜏_𝑘,Δ𝑥|_{𝑡=𝜏𝑘}=𝐼_𝑘(𝑥),𝑘∈ℤ⁺,𝑥𝑡⁺₀=𝑥₀. (4) In what followed, we use 𝑥(𝑡,𝑡₀,𝑥₀) to denote the solution of IVP (4).

In [20], Hu and Han investigated system (1) under the assumption (H*) and obtained the following stability results for the periodic solutions.

Theorem 6 (see [20]). Let 𝑥(𝑡,𝑡₀,𝑥^∗₀) be a periodic solution of system (1) with period T. If 0<|𝑃^′(𝑥^∗₀)|<1 (>1),  then it is uniformly asymptotically stable (unstable), where 𝑃(𝑥₀)=𝑥(𝑡₀+𝑛𝑇⁺,𝑡₀,𝑥₀) is the Poincaré map of system (1).

3. Poincaré Map and Periodic Solutions

In this section, we describe the scalar impulsive differential equations on Moebius stripe and define the Poincaré map. Then we prove several essential lemmas and give sufficient conditions to ensure the existence and stability of one-side and two-side orbits (Figure 2) of impulsive differential equation on Moebius stripe.

Lemma 7. Assume that conditions (H1), (H2) hold. Suppose that 𝑥(𝑡,𝑡₀,𝑥₀) is a solution of (1) satisfying initial value 𝑥(𝑡⁺₀)=𝑥₀. Then −𝑥(𝑡+𝑇,𝑡₀,𝑥₀) is also a solution of (1), and −𝑥𝑡+𝑇,𝑡₀,𝑥₀=𝑥𝑡,𝑡₀,−𝑥𝑡₀+𝑇,𝑡₀,𝑥₀,𝑡∈ℝ. (5)

Proof. Let 𝜑(𝑡)≡−𝑥(𝑡+𝑇,𝑡₀,𝑥₀),𝜓(𝑡)≡𝑥(𝑡,𝑡₀,−𝑥(𝑡₀+𝑇,𝑡₀,𝑥₀)). Then for 𝑡≠𝜏_𝑘,𝑘∈ℤ, we have by (H2) that 𝑑𝜑(𝑡)𝑑𝑡=−𝑑𝑥𝑡+𝑇,𝑡₀,𝑥₀𝑑𝑡=−𝑓𝑡+𝑇,𝑥𝑡+𝑇,𝑡₀,𝑥₀=𝑓𝑡,−𝑥𝑡+𝑇,𝑡₀,𝑥₀=𝑓𝑡,𝜑(𝑡). (6) For 𝑡=𝜏_𝑘, 𝑘∈ℤ⁺, it follows from (H1), (H2) that 𝜏_𝑘+𝑇=𝜏_𝑘+𝑞, 𝑘∈ℤ⁺ and Δ𝜑|_{𝑡=𝜏𝑘}=−𝑥𝜏_𝑘+𝑇⁺,𝑡₀,𝑥₀+𝑥𝜏_𝑘+𝑇,𝑡₀,𝑥₀=−𝑥𝜏⁺_𝑘+𝑞,𝑡₀,𝑥₀+𝑥𝜏_𝑘+𝑞,𝑡₀,𝑥₀=−𝐼_𝑘+𝑞𝑥𝜏_𝑘+𝑞,𝑡₀,𝑥₀=−𝐼_𝑘𝑥𝜏_𝑘+𝑞,𝑡₀,𝑥₀=−𝐼_𝑘𝑥𝜏_𝑘+𝑇,𝑡₀,𝑥₀=𝐼_𝑘−𝑥𝜏_𝑘+𝑇,𝑡₀,𝑥₀=𝐼_𝑘𝜑𝜏_𝑘. (7) Thus, we proved that 𝜑(𝑡)≡−𝑥(𝑡+𝑇,𝑡₀,𝑥₀) is a solution of (1). On the other hand, it is obvious that 𝜑(𝑡)|_𝑡=𝑡0=−𝑥𝑡₀+𝑇,𝑡₀,𝑥₀=𝜓(𝑡)|_𝑡=𝑡0. (8) Hence, by uniqueness theorem we have that 𝜑(𝑡)≡𝜓(𝑡), 𝑡∈ℝ. This completes the proof.

Let 𝐷 denotes the stripe area on the plain {(𝑡,𝑥)∣(𝑡,𝑥)∈ℝ×ℝ} between two lines 𝑡=𝑡₀ and 𝑡=𝑡₀+𝑇; that is, 𝐷=(𝑡,𝑥)∣𝑡₀≤𝑡≤𝑡₀+𝑇,−∞<𝑥<+∞. (9)

Assume that 𝑥(𝑡,𝑡₀,𝑥₀) exists for all 𝑡∈[𝑡₀,+∞). Define 𝐿₀={(𝑡,𝑥(𝑡,𝑡₀,𝑥₀))∣𝑡₀≤𝑡≤𝑡₀+𝑇}. In general, we denote 𝐿_𝑘 (𝑘≥1) by 𝐿_𝑘=𝑡,𝑥𝑡,𝑡₀,−𝑥_𝑘∣𝑡₀≤𝑡≤𝑡₀+𝑇, (10) where 𝑥_𝑘=𝑥(𝑡₀+𝑇⁺,𝑡₀,−𝑥_𝑘−1), 𝑡≥𝑡₀.

It follows from Lemma 7 that 𝐿_𝑘 has the form 𝐿_𝑘=𝑡,(−1)^𝑘𝑥𝑡+𝑘𝑇,𝑡₀,𝑥₀∣𝑡₀≤𝑡≤𝑡₀+𝑇. (11)

We now introduce an equivalence relation ∼ on 𝐷 such that for (𝑡,𝑥),(𝑡^′,𝑥^′)∈𝐷(𝑡,𝑥)∼𝑡^′,𝑥^′iff𝑡−𝑡^′=𝑇,𝑥=−𝑥^′. (12) Then we denote the corresponding quotient space by 𝑀₂. From geometric point of view, 𝑀₂ is obtained by considering two elements (𝑡₀,𝑥) and (𝑡₀+𝑇,−𝑥) on 𝐷 as the same point (or sticking (𝑡₀,𝑥) and (𝑡₀+𝑇,−𝑥) together). Thus 𝑀₂ is a surface with only one side or the well-known Moebius stripe. Obviously, by Lemma 7 the union _𝑘∈ℤ+𝐿_𝑘=_𝑘∈ℤ+𝑡,(−1)^𝑘𝑥𝑡+𝑘𝑇,𝑡₀,𝑥₀∣𝑡₀≤𝑡≤𝑡₀+𝑇 (13) define a flow on 𝑀₂. From this point of view, we call (1) satisfying (H1) and (H2) an impulsive dynamical system on Moebius stripe (see Figure 1).

Figure 1 

Moebius stripe.

Figure 2 

Figure of one-side and two-side orbits.

Definition 8 (Poincaré Map). Let 𝑥(𝑡,𝑡₀,𝑥₀) be the solution of (IVP) (4). Assume that there exists an interval 𝐽 such that for any 𝑥₀∈𝐽, 𝑥(𝑡,𝑡₀,𝑥₀) exists on [𝑡₀,𝑡₀+𝑇]. A map 𝑃∶𝐽→ℝ is called a Poincaré map of system (1) if for any 𝑥₀∈𝐽𝑃𝑥₀=−𝑥𝑡₀+𝑇⁺,𝑡₀,𝑥₀. (14)

Definition 9. A closed curve 𝛾⁺(𝑥₀) is called a one-side periodic orbit on 𝑀₂ if 𝛾⁺(𝑥₀)=𝐿₀. And a closed curve 𝛾⁺(𝑥₀) is called a two-side periodic orbit on 𝑀₂ if 𝛾⁺(𝑥₀)=𝐿₀∪𝐿₁≠𝐿₁.

From Definitions 8 and 9, we can easily prove the following assertion.

Lemma 10. One of following alternatives is valid: (i)𝛾⁺(𝑥₀) is a one-side periodic orbit;(ii)𝑥₀ is a fixed point of 𝑃; that is, 𝑃(𝑥₀)=𝑥₀;(iii)𝑥(𝑡+𝑇,𝑡₀,𝑥₀)=−𝑥(𝑡,𝑡₀,𝑥₀), 𝑡∈ℝ.

Proof. We prove it from (i)⇒(ii)⇒(iii)⇒(i). Assume (i) is true; that is, 𝛾⁺(𝑥₀) is a one-side periodic orbit. Then by Definition 9 we have that −𝑥𝑡₀+𝑇⁺,𝑡₀,𝑥₀=𝑥𝑡⁺₀,𝑡₀,𝑥₀=𝑥₀, (15) that is, 𝑃(𝑥₀)=𝑥₀. Hence (ii) is valid.
Next, we suppose that (ii) is fulfilled; that is, −𝑥(𝑡₀+𝑇⁺,𝑡₀,𝑥₀)=𝑥₀. Then by Lemma 7 we know 𝑥𝑡+𝑇,𝑡₀,𝑥₀=𝑥𝑡,𝑡₀,−𝑥𝑡₀+𝑇⁺,𝑡₀,𝑥₀=−𝑥𝑡,𝑡₀,𝑥₀. (16)
Thus, (iii) is proved.
Finally, if (iii) is true, then 𝑥(𝑡₀+𝑇⁺,𝑡₀,𝑥₀)=−𝑥₀.  By the uniqueness of solution of IVP (4), we know 𝑥𝑡,𝑡₀,−𝑥𝑡₀+𝑇⁺,𝑡₀,𝑥₀=𝑥𝑡,𝑡₀,𝑥₀,𝑡∈𝑡₀,𝑡₀+𝑇. (17)
Thus we obtain that 𝛾⁺(𝑥₀) is a one-side periodic orbit. The proof is completed.

Similarly, as proof of Lemma 10, we have the following lemma.

Lemma 11. One of following alternatives is valid: (i)𝛾⁺(𝑥₀) is a two-side periodic orbit;(ii)𝑥₀ is a 2-periodic point of 𝑃; that is, 𝑃(𝑥₀)≠𝑥₀, 𝑃²(𝑥₀)=𝑥₀;(iii)𝑥(𝑡+2𝑇,𝑡₀,𝑥₀)=𝑥(𝑡,𝑡₀,𝑥₀), 𝑡∈ℝ. And there exists a 𝑡₀, such that 𝑥(𝑡₀+𝑇⁺,𝑡₀,𝑥₀)≠−𝑥₀.

Remark 12. From Lemmas 10 and 11, we see that a one-side periodic orbit must be a two-side periodic orbit since 𝑃𝑥₀=𝑥₀implies𝑃²𝑥₀=𝑃𝑃𝑥₀=𝑃𝑥₀=𝑥₀. (18) Nevertheless, the converse is not true.
From Remark 12, we give the definition of stability of the mentioned orbits.

Definition 13. Let 𝛾⁺(𝑥₀) be a periodic orbit of system (1) (one-side or two-side). Then 𝛾⁺(𝑥₀) of system (1) is called stable (asymptotically stable or unstable) if 𝛾⁺(𝑥₀) as a 2𝑇 periodic solution is stable (asymptotically stable or unstable).

Theorem 14. Assume 𝑥(𝑡,𝑡₀𝑥₀) is the solution of IVP (4) and let 𝑧(𝑡)=𝜕𝑥(𝑡,𝑡₀,𝑥₀)/𝜕𝑥₀. Then 𝑧(𝑡) is a solution to the following IVP of impulsive differential equations: 𝑑𝑧𝑑𝑡=𝑓_𝑥(𝑡,𝑥)𝑧,𝑡≠𝑡_𝑘,Δ𝑧|_{𝑡=𝜏𝑘}=𝐼^′_𝑘(𝑥),𝑘∈ℤ⁺,𝑧𝑡₀=1. (19)

Proof. Let 𝐽=(𝑡₀,+∞) and 𝐽_𝑘=(𝜏_𝑘−1,𝜏_𝑘], 𝑘∈ℤ⁺. Without losing generality, we assume that 𝑡₀∈𝐽_𝑗 for some 𝑗≥1. The solution of IVP 𝑑𝑥(),𝑥𝑡𝑑𝑡=𝑓𝑡,𝑥₊₀=𝑥₀ (20) can be expressed as 𝑥𝑡,𝑡₀,𝑥₀=𝑥₀+^𝑡_𝑡0𝑓𝑠,𝑥𝑠,𝑡₀,𝑥₀𝑑𝑠. (21) Differentiate between both sides of the above equation with respect to 𝑥₀, we have 𝜕𝑥𝑡,𝑡₀,𝑥₀𝜕𝑥₀=1+^𝑡_𝑡0𝑓_𝑥𝑠,𝑥𝑠,𝑡₀,𝑥₀⋅𝜕𝑥𝑠,𝑡₀,𝑥₀𝜕𝑥₀𝑑𝑠. (22) Let 𝑧(𝑡)=𝜕𝑥(𝑡,𝑡₀,𝑥₀)/𝜕𝑥₀, then for 𝑡∈[𝑡₀,𝜏_𝑗), 𝑧(𝑡) is the solution of IVP to ordinary differential equation 𝑑𝑧𝑑𝑡=𝑓_𝑥(𝑡,𝑥)𝑧,𝑡≠𝑡_𝑘,𝑧𝑡₀=1. (23)
Thus 𝑧(𝑡)=exp^𝑡_𝑡0𝑓_𝑥𝑠,𝑥𝑠,𝑡₀,𝑥₀𝑑𝑠. (24)
Since 𝑧(𝑡) is left continuous on [𝑡₀,∞), we have 𝑧𝜏_𝑗=exp^𝜏𝑗_𝑡0𝑓_𝑥𝑠,𝑥𝑠,𝑡₀,𝑥₀𝑑𝑠. (25)
For 𝑡∈𝐽_𝑗+1, 𝑥(𝑡,𝑡₀,𝑥₀) is a solution of system 𝑑𝑥(),𝑥𝜏𝑑𝑡=𝑓𝑡,𝑥_𝑗=𝑥₁, (26) where 𝑥₁=𝑥(𝜏_𝑗⁺,𝑡₀,𝑥₀)=𝑥(𝜏_𝑗,𝑡₀,𝑥₀)+𝐼_𝑗(𝑥(𝜏_𝑗,𝑡₀,𝑥₀)). Thus, we have 𝑥𝑡,𝑡₀,𝑥₀≡𝑥𝑡,𝜏_𝑗,𝑥₁=𝑥₁+^𝑡_𝜏𝑗𝑓𝑠,𝑥𝑠,𝜏_𝑗,𝑥₁𝑑𝑠,𝑡∈𝐽_𝑗+1. (27)
Similarly, we have 𝑡∈(𝜏_𝑗,𝜏_𝑗+1), 𝜕𝑥𝑡,𝑡₀,𝑥₀𝜕𝑥₁=𝜕𝑥𝑡,𝜏_𝑗,𝑥₁𝜕𝑥₁=exp^𝑡_𝜏𝑗𝑓_𝑥𝑠,𝑥𝑠,𝜏_𝑗,𝑥₁𝑑𝑠=exp^𝑡_𝜏𝑗𝑓_𝑥𝑠,𝑥𝑠,𝑡₀,𝑥₀𝑑𝑠. (28)
Note𝜕𝑥₁𝜕𝑥₀=𝜕𝑥𝜏_𝑗,𝑡₀,𝑥₀+𝐼_𝑗𝑥𝜏_𝑗,𝑡₀,𝑥₀𝜕𝑥𝜏_𝑗,𝑡₀,𝑥₀⋅𝜕𝑥𝜏_𝑗,𝑡₀,𝑥₀𝜕𝑥₀=1+𝐼^′_𝑗𝑥𝜏_𝑗,𝑡₀,𝑥₀𝜕𝑥𝜏_𝑗,𝑡₀,𝑥₀𝜕𝑥₀. (29) We obtain for 𝑡∈(𝜏_𝑗,𝜏_𝑗+1) that 𝜕𝑥𝑡,𝑡₀,𝑥₀𝜕𝑥₀=𝜕𝑥𝑡,𝜏_𝑗,𝑥₁𝜕𝑥₀=1+𝐼^′_𝑗𝑥𝜏_𝑗,𝑡₀,𝑥₀⋅exp^𝑡_𝑡0𝑓_𝑥𝑠,𝑥𝑠,𝑡₀,𝑥₀𝑑𝑠. (30)Deducing in a similar way, we get 𝜕𝑥𝑡,𝑡₀,𝑥₀𝜕𝑥₀=_{𝑡0<𝜏𝑘≤𝑡}1+𝐼^′_𝑘𝑥𝜏_𝑗,𝑡₀,𝑥₀⋅exp^𝑡_𝑡0𝑓_𝑥𝑠,𝑥𝑠,𝑡₀,𝑥₀𝑑𝑠, (31) where 𝑡∈𝐽. Then the proof is completed.

By Definitions 9 and (31), we conclude the following assertion.

Corollary 15. Assume that conditions (H1), (H2) hold. Then 𝑃^′𝑥₀=−_{𝑡0<𝜏𝑘≤𝑡0+𝑇}1+𝐼^′_𝑘𝑥𝜏_𝑘,𝑡₀,𝑥₀⋅exp^𝑡0_𝑡^+𝑇₀𝑓_𝑥𝑡,𝑥𝑡,𝑡₀,𝑥₀𝑑𝑡. (32)
As usual, one uses the notion 𝑃²(𝑥₀)=𝑃(𝑃(𝑥₀)). Then one has 𝑃²𝑥₀^′=_{𝑡0<𝜏𝑘≤𝑡0+2𝑇}1+𝐼^′_𝑘𝑥𝜏_𝑘,𝑡₀,𝑥₀⋅exp^𝑡0_𝑡^+2𝑇₀𝑓_𝑥𝑡,𝑥𝑡,𝑡₀,𝑥₀𝑑𝑡. (33)

Definition 16. 𝑥₀ is called a hyperbolic fixed point of 𝑃 if 𝑥₀=𝑃(𝑥₀) and 𝑃^′(𝑥₀)≠−1; the corresponding one-side periodic orbit 𝛾⁺(𝑥₀) is called hyperbolic one-side periodic orbit. If 𝛾⁺(𝑥₀) is a two-side periodic orbit with (𝑃²)^′(𝑥₀)≠1, then we call 𝛾⁺(𝑥₀) a hyperbolic two-side periodic orbit.

Theorem 17. Assume that the conditions (H1), (H2) hold. Let 𝛾⁺(𝑥₀) be a periodic orbit of system (1) and 𝐼^′_𝑘(𝑥(𝜏_𝑘,𝑡₀,𝑥₀))≠−1. Then (i) ∫^𝑡0_𝑡^+2𝑇₀𝑓_𝑥(𝑡,𝑥(𝑡,𝑡₀,𝑥₀))𝑑𝑡<−∑_{𝑡0<𝜏𝑘≤𝑡0+2𝑇}ln|1+𝐼^′_𝑘(𝑥(𝜏_𝑘,𝑡₀,𝑥₀))| implies 𝛾⁺(𝑥₀) is asymptotically stable,
(ii)∫^𝑡0_𝑡^+2𝑇₀𝑓_𝑥(𝑡,𝑥(𝑡,𝑡₀,𝑥₀))𝑑𝑡>−∑_{𝑡0<𝜏𝑘≤𝑡0+2𝑇}ln|1+𝐼^′_𝑘(𝑥(𝜏_𝑘,𝑡₀,𝑥₀))| implies 𝛾⁺(𝑥₀) is unstable.

Proof. If 𝛾⁺(𝑥₀) is a two-side periodic orbit; that is, 𝑥(𝑡,𝑡₀,𝑥₀) is a 2𝑇 periodic solution of (1). Since both (H1) and (H2) hold, we know that (1) is a periodic impulsive differential equation. Then by (33) and Theorem 6, the conclusion is straightforward.

Example 18. Consider the linear periodic impulsive differential equations on Moebius stripe as follows: 𝑑𝑥(𝑡)(𝑡)𝑑𝑡=𝑎𝑥+𝑏,𝑡≠𝜏_𝑘,Δ𝑥|_{𝑡=𝜏𝑘}=𝑐_𝑘𝑥𝜏_𝑘,𝑘∈ℤ⁺, (34) where 𝜏_𝑘<𝜏_𝑘+1 (𝑘≥1), 𝜏_𝑘→+∞, 𝑘→+∞, 𝑐_𝑘≠−1 and there exists a constant 𝑇>0, a positive integer 𝑞,  such that the following conditions are satisfied:(H^∼1)𝑎(𝑡+𝑇)=𝑎(𝑡) and 𝑏(𝑡+𝑇)=−𝑏(𝑡) for 𝑡∈ℝ; (H^∼2)𝑎(𝑡) and 𝑏(𝑡) are continuous; (H^∼3)𝑐_𝑘+𝑞=𝑐_𝑘, for all 𝑘∈ℤ⁺; (H^∼4)𝜏_𝑘+𝑞−𝜏_𝑘=𝑇, for all 𝑘∈ℤ⁺.

Assume that 𝑥(𝑡,𝑡₀,𝑥₀) is a one-side periodic solution of system (34), by the method of variation of constants formula (see [1]), we get𝑥𝑡,𝑡₀,𝑥₀=_{𝑡0<𝜏𝑘≤𝑡}1+𝑐_𝑘⋅exp^𝑡_𝑡0𝑎(𝑡)×𝑥𝑑𝑡₀+^𝑡_𝑡0exp−^𝑠_𝑡0𝑎(𝑢)𝑏(𝑠),𝑃𝑥𝑑𝑢𝑑𝑠₀=−_{𝑡0<𝜏𝑘≤𝑡0+𝑇}1+𝑐_𝑘⋅exp^𝑡0_𝑡^+𝑇₀𝑎(𝑡)×𝑥𝑑𝑡₀+^𝑡0_𝑡^+𝑇₀exp−^𝑡_𝑡0𝑎(𝑢)𝑏(𝑡),𝑃𝑑𝑢𝑑𝑡₂𝑥₀=_{𝑡0<𝜏𝑘≤𝑡0+2𝑇}1+𝑐_𝑘⋅exp^𝑡0_𝑡^+2𝑇₀𝑎(𝑡)×𝑥𝑑𝑡₀+^𝑡0_𝑡^+2𝑇₀exp−^𝑡_𝑡0𝑎(𝑢)𝑑𝑢𝑏(𝑡)𝑑𝑡. (35)(36)(37)

Let 𝐴=∏_{𝑡0<𝜏𝑘≤𝑡0+𝑇}(1+𝑐_𝑘)⋅exp∫^𝑡0_𝑡^+𝑇₀𝑎(𝑡)𝑑𝑡; therefore, we have the following theorem.

Theorem 19. Suppose that (𝐻^∼1–𝐻^∼4) are satisfied, then(i)there exists a unique one-side periodic orbit for system (34) if 𝐴≠−1, which is asymptotically stable (unstable) provided 0<|𝐴|<1 (|𝐴|>1),(ii)if 𝐴²≠1, (34) has no two-side periodic orbit. If 𝐴=1 all the trajectories are two-side periodic orbits expect for a unique one-side periodic orbit.

Proof. For the sake of convenience, we denote 𝐵₁=^𝑡0_𝑡^+𝑇₀exp−^𝑡_𝑡0𝑎(𝑢)𝑏(𝑡)𝐵𝑑𝑢𝑑𝑡,₂=^𝑡0_𝑡^+2𝑇₀exp−^𝑡_𝑡0𝑎(𝑢)𝑑𝑢𝑏(𝑡)𝑑𝑡. (38)
Then 𝑃𝑥₀=−𝐴𝑥₀+𝐵₁,𝑃²𝑥₀=𝐴²𝑥₀+𝐵₂. (39)
Obviously, 𝑃(𝑥₀)=𝑥₀ has a unique solution for any 𝑥₀∈ℝ if 𝐴≠−1, and 𝑃²(𝑥₀)=𝑥₀ has a unique solution for any 𝑥₀∈ℝ if 𝐴²≠1. Observing that any two-side periodic orbit obtained under the assumption 𝐴²≠1 must be a one-side periodic orbit since 𝐴²≠1 implies 𝐴≠−1, together with Remark 12, we have (34) has no two-side periodic orbit.
It follows from (36) that 𝑃^′(𝑥₀)=−𝐴. Then by Theorem 6 we have the one-side orbit is asymptotically stable (unstable) provided 0<|𝐴|<1 (|𝐴|>1).
Next, let 𝐴=1. By taking (36) and (37) into account, we have 𝑃𝑥₀=−𝑥₀−^𝑡0_𝑡^+𝑇₀exp−^𝑡_𝑡0𝑎(𝑢)𝑑𝑢𝑏(𝑡)𝑑𝑡≡−𝑥₀−𝐵₁,𝑃²𝑥₀=𝑥₀+^𝑡0_𝑡^+2𝑇₀exp−^𝑡_𝑡0𝑎(𝑢)𝑑𝑢𝑏(𝑡)𝑑𝑡≡𝑥₀+𝐵₂. (40)
Suppose that 𝑃 has a unique fixed point 𝑥^∗₀=−𝐵₁/2, from the above we have −𝐵₁/2+𝐵₂=−𝐵₁/2, then 𝐵₂≡0 and 𝑃²(𝑥₀)=𝑥₀. So by taking Lemma 11, 𝛾⁺(𝑥₀) is a two-side periodic orbit if 𝑥₀≠𝑥^∗₀.
The proof is ended.

Remark 20. If 𝑐_𝑘≡0,𝑘∈ℤ⁺, in (34); that is, (34) reduces to an ordinary differential equation. We see that 𝐴=exp∫^𝑡0_𝑡^+𝑇₀𝑎(𝑡)𝑑𝑡. Hence 𝐴≠−1 holds automatically, and therefore (34) always has a unique one-side periodic orbit.

Corollary 21. Let (H^∼1--H^∼4)  be fulfilled and 𝐴=1. Then 𝐵₂=^𝑡0_𝑡^+2𝑇₀exp−^𝑡_𝑡0𝑎(𝑢)d𝑢𝑏(𝑡)𝑑𝑡=0. (41)

Now we are in position to consider nonlinear impulsive system on Meobius stripe. To explore the uniqueness of one-side periodic orbit, we induce the following condition.

(H3) Operator 𝐵_𝑘∶ℝ→ℝ, 𝐵_𝑘(𝑥)=𝑥+𝐼_𝑘(𝑥) is strictly increasing, for all 𝑘∈ℤ⁺.

Theorem 22. Suppose that conditions (H1)–(H3) hold, then(i)system (1) has at most one one-side periodic orbit; (ii)if any solution 𝑥(𝑡,𝑡₀,𝑥₀) of (1) with |𝑥₀|≤|𝑃(0)| is well defined on 𝑡∈[𝑡₀,𝑡₀+𝑇], then system (1) must has a unique one-side periodic orbit.

Proof. We first prove that system (1) cannot have two one-side periodic orbits. Suppose 𝛾⁺₁(𝑥^∗₀)∶𝑥=𝑥(𝑡,𝑡₀,𝑥^∗₀), 𝑡∈[𝑡₀,𝑡₀+𝑇] and 𝛾⁺₂(𝑥₀)∶𝑥=𝑥(𝑡,𝑡₀,𝑥₀), 𝑡∈[𝑡₀,𝑡₀+𝑇] are two one-side periodic orbits system (1). Then 𝑥𝑡₀+𝑇,𝑡₀,𝑥^∗₀=−𝑥^∗₀,𝑥𝑡₀+𝑇,𝑡₀,𝑥₀=−𝑥₀. (42) Without losing generality, we assume 𝑥₀>𝑥^∗₀, then it follows from uniqueness theorem of ordinary differential equations that𝑥=𝑥(𝑡,𝑡₀,𝑥₀) and 𝑥(𝑡,𝑡₀,𝑥^∗₀) cannot intersect when 𝑡 is not an impulsive time. Therefore we have 𝑥𝑡,𝑡₀,𝑥₀>𝑥𝑡,𝑡₀,𝑥^∗₀,𝑡₀≤𝑡≤𝜏₁. (43) Note 𝐵_𝑘(𝑥)=𝑥+𝐼_𝑘(𝑥) is strictly increasing, we get 𝑥𝜏⁺₁,𝑡₀,𝑥₀>𝑥𝜏⁺,𝑡₀,𝑥^∗₀. (44) In a similar way, we can prove that 𝑥(𝑡,𝑡₀,𝑥₀)>𝑥(𝑡,𝑡₀,𝑥^∗₀), 𝑡₀≤𝑡≤𝑡₀+𝑇. That is, the curve {(𝑡,𝑥)∣𝑥=𝑥(𝑡,𝑡₀,𝑥₀),𝑡₀≤𝑡≤𝑡₀+𝑇} always stays above curve {(𝑡,𝑥)∣𝑥=𝑥(𝑡,𝑡₀,𝑥^∗₀),𝑡₀≤𝑡≤𝑡₀+𝑇}. This contradicts (42). We put it in another way that 𝑥₀>𝑥^∗₀⟹𝑃𝑥₀>𝑃𝑥^∗₀. (45) Thus, (1) has at most a one-side periodic orbit.
Further, let the solution 𝑥(𝑡,𝑡₀,𝑥₀) of system (1) be all defined on 𝑡∈[𝑡₀,𝑡₀+𝑇]. If 𝑃(0)=0, the conclusion is proved. We assume that 𝑃(0)>0, then we know 𝑃(𝑥₀)<𝑃(0) if 0<𝑥₀≤𝑃(0). Note 𝑃²(0)−𝑃(0)=𝑃(𝑃(0))−𝑃(0)<𝑃(0)−𝑃(0)=0. (46) We obtain that 𝑃(𝑥₀)−𝑥₀ have opposite signs between 𝑥₀=0  and 𝑥₀=𝑃(0), and then it follows from the continuity of 𝑃 that there exists 𝑥^∗₀∈(0,𝑃(0)) such that 𝑃(𝑥^∗₀)=𝑥^∗₀. Similarly, we can prove 𝑃 has a fixed point in the case of 𝑃(0)<0. The proof is completed.

Theorem 23. Assume that conditions (H1)–(H3) hold. Furthermore, suppose there exists a positive number 𝑁 such that 𝑓(𝑡,𝑁)≤0,𝑓(𝑡,−𝑁)≥0,𝑡∈𝑡₀,𝑡₀+𝑇,−2𝑁≤𝐼_𝑘(𝑁)≤0,∀𝑘∈ℤ⁺. (47)(48) Then (1) has a unique one-side periodic orbit.

Proof. From (47) we have that 𝑥(𝑡,𝑡₀,𝑥₀) will stay inside [−𝑁,𝑁] for 𝑡≠𝜏_𝑘,𝑘∈ℤ⁺. On the other hand, by (H3), we have that −𝑁+𝐼_𝑘(−𝑁)≤𝑥(𝜏_𝑘)+𝐼_𝑘(𝑥(𝜏_𝑘))≤𝑁+𝐼_𝑘(𝑁) for −𝑁≤𝑥(𝜏_𝑘)≤𝑁. Then it follows from (48) that −𝑁≤𝑁+𝐼_𝑘(𝑁)≤𝑁≤𝑁,−𝑁≤−𝑁+𝐼_𝑘(−𝑁)=−𝑁−𝐼_𝑘(𝑁)≤𝑁 (49)
(see Figure 3).

Figure 3 

A one-side periodic orbit.

Thus, |𝑃(0)|=−𝑥𝑡₀+𝑇,0=𝑥𝑡₀+𝑇,0≤𝑁. (50) This implies that 𝑃(𝑥₀) is well defined for |𝑥₀|≤|𝑃(0)|. By Theorem 22, we obtain that (1) has a unique one-side periodic orbit.

4. Double-Period Bifurcation

In this section, we mainly discuss the bifurcation on periodic orbits. If system (1) has a one-side periodic orbit, without losing generality, we may assume that 𝑓(𝑡,0)=0; that is, 𝑥=0 is the one-side periodic orbit. Actually, if 𝑥(𝑡) is a one-side periodic orbit, then we let 𝑦=𝑥−𝑥(𝑡); therefore there exists a transformation of system (1) that 𝑑𝑦(𝑡)((𝑡𝑑𝑡=𝑓𝑡,𝑦+𝑥−𝑓𝑡,𝑥))≡𝑔𝑡,𝑦,𝑡≠𝜏_𝑘,Δ𝑦|_{𝑡=𝜏𝑘}=𝐼_𝑘𝑦+𝑥𝜏_𝑘−𝐼_𝑘𝑥𝜏_𝑘≡ℎ_𝑘𝑦,𝑘∈ℤ⁺, (51)

By (H2) and Lemma 10, we know 𝑔(𝑡+𝑇,−𝑦)=−𝑔(𝑡,𝑦), 𝑔(𝑡,0)=0, ℎ_𝑘+𝑞(𝑦)=ℎ_𝑘(𝑦), and 𝜏_𝑘+𝑞−𝜏_𝑘=𝑇, for all 𝑘∈ℤ⁺.

Next, we consider the following perturbed system of system (1): 𝑑𝑥()𝑑𝑡=𝐹𝑡,𝑥,𝜀,𝑡≠𝜏_𝑘,Δ𝑥|_{𝑡=𝜏𝑘}=𝐼^∼_𝑘𝑥𝜏_𝑘,𝜀,𝑘∈ℤ⁺, (52) where 𝐹∶ℝ×ℝ×ℝ→ℝ is 𝐶³ with respect to 𝑥, continuously differentiable with respect to 𝜀. 𝐼^∼_𝑘∶ℝ×ℝ→ℝ is 𝐶³(𝑘∈ℤ⁺) with respect to 𝑥. Moreover, we suppose 𝐹(𝑡+𝑇,−𝑥,𝜀)=−𝐹(𝑡,𝑥,𝜀), 𝐼^∼_𝑘(𝑥,𝜀)=𝐼^∼_𝑘+𝑞(𝑥,𝜀), 𝐼^∼_𝑘(−𝑥)=−𝐼^∼_𝑘(𝑥), for all 𝑘∈ℤ⁺, where 𝜏_𝑘+𝑞−𝜏_𝑘=𝑇. For 𝜀=0, we have 𝐹(𝑡,𝑥,0)=𝑓(𝑡,𝑥), 𝐼^∼_𝑘(𝑥,0)=𝐼_𝑘(𝑥). These assumptions mean (H1) and (H2) hold for 𝐹 and 𝐼^∼_𝑘, for all 𝑘∈ℤ⁺. Furthermore, assume that 𝑥+𝐼^∼_𝑘(𝑥,𝜀) is strictly increasing, then by Theorem 22 we have that system (52) has at most a one-side periodic orbit.

Suppose that (1) has a one-side periodic orbit and 𝑓(𝑡,0)=0. Then by using implicit function theorem in the Poincaré map of system (52), we know that system (52) has a one-side periodic orbit when |𝜀| is sufficiently small. Now let 𝑥^∗(𝑡,𝜀) be the solution of system (52) and 𝑦(𝑡,𝜀)=𝑥(𝑡,𝜀)−𝑥^∗(𝑡,𝜀). Then we can get a transformation of system (52): 𝑑𝑦𝑑𝑡=𝐹𝑡,𝑦+𝑥^∗,𝜀−𝐹𝑡,𝑥^∗,𝜀=𝐺𝑡,𝑦,𝜀,𝑡≠𝜏_𝑘,Δ𝑦|_{𝑡=𝜏𝑘}=𝐼^∼_𝑘𝑦+𝑥^∗𝜏_𝑘−,𝜀,𝜀_∼𝐼_𝑘𝑥^∗𝜏_𝑘,𝜀,𝜀=𝐻_𝑘𝑦,𝜀,𝑘∈ℤ⁺. (53)

By Taylor’s formula, we have 𝐺𝑡,𝑦,𝜀=𝐴₁(𝑡,𝜀)𝑦+𝐴₂(𝑡,𝜀)𝑦²+𝐴₃(𝑡,𝜀)𝑦³+𝑜𝑦³,𝐻_𝑘𝑦,𝜀=𝐵_𝑘1(𝜀)𝑦+𝐵_𝑘3(𝜀)𝑦³+𝑜𝑦³, (54) where 𝐴_𝑖()=1𝑡,𝜀𝜕𝑖!^𝑖𝐹𝜕𝑥^𝑖𝑡,𝑥^∗,𝐵,𝜀_𝑘𝑗(𝜀)=1𝜕𝑗!^𝑗∼𝐼_𝑘𝜕𝑥^𝑗𝑥^∗,𝐴,𝜀_𝑖(𝑡+𝑇,𝜀)=(−1)^𝑖−1𝐴_𝑖(𝑡,𝜀), (55) for 𝑘≥1, 𝑖=1,2,3; 𝑗=1,3.

If 𝜀=0, then 𝑥^∗=0. So 𝐴_𝑖(𝑡,0)=(1/𝑖!)(𝜕^𝑖𝑓/𝜕𝑥^𝑖)(𝑡,0) and 𝐵_𝑘𝑗(0)=(1/𝑗!)𝐼_𝑘^(𝑗)(0), for 𝑘≥1, 𝑖=1,2,3, 𝑗=1,3.

Suppose that 𝑦(𝑡,𝑦₀,𝜀)(𝑡≥𝑡₀) is the solution of system (53) with the initial value 𝑦(𝑡⁺₀,𝑦₀,𝜀)=𝑦₀, 𝑃(𝑦₀,𝜀) is the Poincaré map of system (53). Note 𝑃^∼𝑦₀,𝜀=𝑦𝑡₀+2𝑇⁺,𝑦₀,𝜀=𝑃²𝑦₀,𝜀. (56) Without losing generality, let 𝑥^∗(𝑡,𝜀)=0 is a nonhyperbolic solution. That is, 𝑃(0,0)=0 and (𝜕𝑃/𝜕𝑦₀)(0,0)=−1.

Noting that 𝑃(0,𝜀)=0, then by Taylor’s formula, we have 𝑃𝑦₀,𝜀=𝐴₁(𝜀)𝑦₀+𝐴₂(𝜀)𝑦²₀+𝐴₃(𝜀)𝑦³₀+𝑜𝑦³₀, (57) where 𝐴₁(𝜀)=(𝜕𝑃/𝜕𝑦₀)(0,𝜀), 𝐴₂(𝜀)=(1/2)(𝜕²𝑃/𝜕𝑦²₀)(0,𝜀), and 𝐴₃(𝜀)=(1/6)(𝜕³𝑃/𝜕𝑦³₀)(0,𝜀).

Theorem 24. Suppose that 𝑓(𝑡,0)=0 and 𝑥=0 is a one-side periodic orbit of system (52)_ε=0 with 𝑃(0,0)=0 and (𝜕𝑃/𝜕𝑦₀)(0,0)=−1. Let 𝑎^∗₃=(1/6)𝑃^∼′′′(0,0). If 𝑎^∗₃≠0, then for |𝜀| sufficiently small and [𝐴₁(𝜀)+1]𝑎^∗₃>0 (≤0) implies that system (52) has a unique (no) two-sides periodic orbit near 𝑥=0, except for a one-side periodic orbit 𝑥^∗(𝑡,𝜀).

Proof. As before, we obtain that 𝑃^∼′(0,𝜀)=[𝑃^′(0,𝜀)]²=𝐴²₁(𝜀):𝑃^∼𝑦₀,𝜀=𝑃𝑃𝑦₀=,𝜀,𝜀𝐴₁(𝜀)𝑃𝑦₀,𝜀+𝐴₂(𝜀)𝑃²𝑦₀+,𝜀𝐴₃(𝜀)𝑃³𝑦₀,𝜀+𝑜𝑃³𝑦₀=,𝜀𝐴²₁(𝜀)𝑦₀+𝐴²₁(𝜀)+𝐴₁(𝜀)𝐴₂(𝜀)𝑦²₀+𝐴₁𝐴₃(𝜀)+2𝐴²₂(𝜀)+𝐴²₁(𝜀)𝐴₃(𝜀)𝑦³₀+𝑜𝑦³₀. (58)
By our assumption, we have 𝐴₁(𝜀)=−1+𝐴^′₁(0)𝜀+𝑜(𝜀). Therefore, 𝑑^∼𝑦₀=𝑃,𝜀²𝑦₀,𝜀−𝑦₀𝑦₀=𝑑₀(𝜀)+𝑑₁(𝜀)𝑦₀+𝑑₂(𝜀)𝑦²₀+𝑜𝑦²₀, (59) where 𝑑₀(𝜀)=𝐴²₁(𝜀)−1=−2𝐴₁(𝜀)+1+𝑜𝐴₁(𝜀),𝑑+1₁(𝜀)=𝐴²₁(𝜀)+𝐴₁(𝜀)𝐴₂(𝜀)=𝑂𝐴₁(𝜀)+1=−𝐴^′₁(0)𝐴₂(0)(𝜀),𝑑𝜀+𝑜₂(𝜀)=𝐴₁(𝜀)𝐴₃(𝜀)+2𝐴²₂(𝜀)+𝐴²₁(𝜀)𝐴₃(𝜀)=−2𝐴²₂(0)+𝐴₃(0)+𝑜(1)=𝑎^∗₃+𝑜(1). (60) By the implicit function theorem, there exists a unique function 𝑦₀=𝑦₁(𝜀), 𝑦₁(0)=0 such that (𝜕𝑑^∼/𝜕𝑦₀)(𝑦₁(𝜀),𝜀)=0. Therefore, for |𝜀| sufficiently small, there is a unique extremal point 𝑦₀=𝑦₁(𝜀) near 𝑥=0. Moreover, the function 𝑑^∼(𝑦₀,𝜀) takes its minimum (maximum) Δ(𝜀)≡𝑑^∼(𝑦₁(𝜀),𝜀) only if 𝑎^∗₃>0 (<0): Δ(𝜀)=𝑑₀(𝜀)+𝑜(𝜀)=𝐴²₁(𝜀)−1=−2𝐴₁(𝜀)+1+𝑜(𝜀). (61) Without loss of generality, we can let 𝑎^∗₃=𝑑₂(0)>0 and then 𝑦₀=0 is the minimum point of 𝑑(𝑦₀,0). So there exists 𝜀₀>0, such that 𝑑^∼±𝜀₀𝜕,0>0,^∼𝑑𝜕𝑦₀𝜀₀𝜕,0>0,^∼𝑑𝜕𝑦₀−𝜀₀,0<0, (62) and for |𝑦₀|≤𝜀₀, (𝜕𝑑^2∼/𝜕𝑦²₀)(𝑦₀,0)>0 exists. Therefore, there exists a 𝛿₀, such that, for |𝜀|≤𝛿₀, we have 𝑑^∼±𝜀₀𝜕,𝜀>0,^∼𝑑𝜕𝑦₀𝜀₀𝜕,𝜀>0,^∼𝑑𝜕𝑦₀−𝜀₀,𝜀<0. (63) For |𝜀|≤𝛿₀ and |𝑦₀|≤𝜀₀, we have 𝜕𝑑^2∼𝜕𝑦²₀𝑦₀,𝜀>0. (64) From (64), for any |𝜀|≤𝛿₀, we have Δ(𝜀)=min𝑑_{|𝑦0|≤𝜀0}^∼𝑦₀,𝜀,−𝜀₀<𝑦₁(𝜀)<𝜀₀. (65) And for 𝑦₀∈(−𝜀₀,𝑦₁(𝜀))(∈(𝑦₁(𝜀),𝜀₀)), 𝜕𝑑^∼𝜕𝑦₀𝑦₀,𝜀<0(>0). (66)
If Δ(𝜀)>0, then for all |𝜀|≤𝛿₀ and |𝑥₀|<𝜀₀, we have 0<Δ(𝜀)≤𝑑^∼.
If Δ(𝜀)=0, then 𝑦₀=𝑦₁(𝜀) is the unique solution of function 𝑑^∼.
If Δ(𝜀)<0, then there exist a unique 𝑦₁(𝜀) and a unique 𝑦₂(𝜀), such that 𝑑^∼𝑦_𝑖(𝜀)𝜕,𝜀=0,^∼𝑑𝜕𝑦₀𝑦_𝑖(𝜀),𝜀≠0,𝑖=1,2. (67) Thus system (52) has two (no) two-side periodic orbits if 𝑎^∗₃Δ(𝜀)<0 (≥0). The conclusion is completed (see Figures 4, 5, and 6).

Figure 4 

Figure 5 

Figure 6 

Now we shall calculate 𝐴₁(0) and 𝑎^∗₃ in the simplest case, let 𝑞=1. For 𝑞>1 we can calculate them in the same way. In this case, 𝐼_𝑘≡𝐼 and 𝐵_𝑘𝑖(𝑦₀,𝜀)=𝐵_𝑖(𝑦₀,𝜀), 𝑖=1,3. Suppose 𝑦(𝑡,𝑦₀,𝜀) (𝑡≤𝑡₀) is the solution to system (53) with initial value 𝑦(𝑡₀+,𝑦₀,𝜀)=𝑦₀. For 𝑦(𝑡,0,𝜀)=0, let 𝑦𝑡,𝑦₀,𝜀=𝜑₁(𝑡,𝜀)𝑦₀+𝜑₂(𝑡,𝜀)𝑦²₀+𝜑₃(𝑡,𝜀)𝑦³₀+𝑜𝑦³₀,𝑡≤𝑡₀. (68)

Then for 𝑡∈[𝑡₀,𝑡₀+𝑇], taking 𝑦(𝑡,𝑦₀,𝜀) into system (53), we can obtain 𝜑₁, 𝜑₂, and 𝜑₃ satisfying the following equations: 𝜑^′₁(𝑡,𝜀)=𝐴₁(𝑡,𝜀)𝜑₁(),𝜑𝑡,𝜀_′2(𝑡,𝜀)=𝐴₁(𝑡,𝜀)𝜑₂(𝑡,𝜀)+𝐴₂(𝑡,𝜀)𝜑²₁(),𝜑𝑡,𝜀_′3(𝑡,𝜀)=𝐴₁(𝑡,𝜀)𝜑₃(𝑡,𝜀)+2𝐴₂(𝑡,𝜀)𝜑₁(𝑡,𝜀)𝜑₂(𝑡,𝜀)+𝜑₃(𝑡,𝜀)𝜑³₁(𝑡,𝜀). (69) For 𝑦(0,𝑦₀,𝜀)=𝑦₀, we know 𝜑₁(0,𝜀)=1,𝜑₂(0,𝜀)=𝜑₃(0,𝜀)=0. (70)

From (69) and (70), we have 𝜑₁(𝑡,𝜀)=exp^𝑡_𝑡0𝐴₁()𝜑𝑢,𝜀𝑑𝑢,₂(𝑡,𝜀)=𝜑₁(𝑡,𝜀)^𝑡_𝑡0𝐴₂(𝑠,𝜀)𝜑₁()𝜑𝑠,𝜀𝑑𝑠,₃(𝑡,𝜀)=𝜑₁()×𝑡,𝜀_𝑡𝑡02𝐴₂(𝑠,𝜀)𝜑₂(𝑠,𝜀)+𝐴₃(𝑠,𝜀)𝜑²₁(𝑠,𝜀)𝑑𝑠. (71) For 𝑡₀<𝑡<𝑡₀+𝑇, as we know, we get 𝑦𝑡₀+𝑇⁺,𝑦₀,𝜀=1+𝐵₁(𝜀)𝑦𝑡₀+𝑇,𝜀+𝐵₂(𝜀)𝑦²𝑡₀+𝑇,𝜀+𝐵₃(𝜀)𝑦³𝑡₀𝑦+𝑇,𝜀+𝑜³𝑡₀+𝑇,𝜀=𝜑₁𝑡₀+𝑇⁺,𝜀𝑦₀+𝜑₂𝑡₀+𝑇⁺,𝜀𝑦²₀+𝜑₃𝑡₀+𝑇⁺,𝜀𝑦³₀+𝑜𝑦³₀, (72) where 𝜑₁𝑡₀+𝑇⁺,𝜀=1+𝐵₁(𝜀)𝜑₁𝑡₀,𝜑+𝑇,𝜀₂𝑡₀+𝑇⁺,𝜀=1+𝐵₁(𝜀)𝜑₂𝑡₀+𝑇,𝜀+𝐵₂(𝜀)𝜑²₁𝑡₀,𝜑+𝑇,𝜀₃𝑡₀+𝑇⁺,𝜀=1+𝐵₁(𝜀)𝜑₃𝑡₀+𝑇,𝜀+2𝐵₂(𝜀)𝜑₁𝑡₀+𝑇,𝜀𝜑₂𝑡₀+𝑇,𝜀+𝐵₃(𝜀)𝜑³𝑡₀+𝑇,𝜀. (73) Clearly, 𝐴₁(𝜀)=−𝜑₁(𝑡₀+𝑇⁺,𝜀), 𝐴₂(𝜀)=−𝜑₂(𝑡₀+𝑇⁺,𝜀), and 𝐴₃(𝜀)=−𝜑₃(𝑡₀+𝑇⁺,𝜀).

Moreover, we know ^𝑡0_𝑡^+𝑇₀𝐴₂(𝑠,𝜀)𝜑₂()=𝑠,𝜀𝑑𝑠_{𝑡0𝑡+𝑇0}𝐴₂(𝑠,𝜀)𝜑₁()×𝑠,𝜀_𝑠𝑡0𝐴₂(𝑢,𝜀)𝜑₁()=1𝑢,𝜀𝑑𝑢𝑑𝑠2^𝑡0_𝑡^+𝑇₀𝐴₂(𝑢,𝜀)𝜑₁(𝑢,𝜀)𝑑𝑢². (74) Denote Φ(𝜀)=∫^𝑡0_𝑡^+𝑇₀𝐴₂(𝑡,𝜀)𝜑₁(𝑡,𝜀)𝑑𝑡 and 𝜙₁(𝜀)=𝜑₁(𝑡₀+𝑇,𝜀). Then, 𝜑₂𝑡₀+𝑇,𝜀=𝜙₁(𝜀)Φ(𝜀),𝜑₃𝑡₀+𝑇,𝜀=𝜙₁(𝜀)Φ²(𝜀)+^𝑡0_𝑡^+𝑇₀𝐴₃(𝑠,𝜀)𝜑²₁(𝑠,𝜀)𝑑𝑠. (75) Then we can obtain 𝐴₁(𝜀)=−1+𝐵₁(𝜀)exp^𝑡0_𝑡^+𝑇₀𝐴₁(𝑢,𝜀)𝑑𝑢,𝐴₂(𝜀)=−1+𝐵₁(𝜀)𝜙₁(𝜀)Φ(𝜀),𝐴₃(𝜀)=−1+𝐵₁(𝜀)𝜙₁(𝜀)×Φ²(𝜀)+^𝑡0_𝑡^+𝑇₀𝐴₃(𝑠,𝜀)𝜑²₁(𝑠,𝜀)𝑑𝑠−𝐵₃(𝜀)𝜙³₁(𝜀). (76) For 𝐴₁(0)=−1, we can have 𝜙₁(0)=1/(1+𝐵₁(0)). Then 𝐴₂(0)=−Φ(0),𝐴₃(0)=−Φ²(0)−𝐵₃(0)1+𝐵₁(0)³−Δ(0), (77)(78) where Δ(𝜀)=∫^𝑡0_𝑡^+𝑇₀𝐴₃(𝑠,𝜀)𝜑²₁(𝑠,𝜀)𝑑𝑠=∫^𝑡0_𝑡^+𝑇₀𝐴₃(𝑠,𝜀)exp[2∫^𝑠_𝑡0𝐴₁(𝑢,𝜀)𝑑𝑢]𝑑𝑠. Therefore, 𝑎^∗₃=−2𝐴²₂(0)+𝐴₃(0)=2Δ(0)+2𝐵₃(0)1+𝐵₁(0)³. (79)

By considering (76)–(79), we can easily have the following theorem when 𝑞=1.

Theorem 25. Suppose that 𝑓(𝑡,0)=0 and 𝑥=0 is a one-side periodic solution of system (52)_ε=0 with 𝑃(0,0)=0 and (𝜕𝑃/𝜕𝑦₀)(0,0)=−1. Let 𝐴₁(𝜀)=−1+𝐼^∼′𝑥^∗,𝜀exp^𝑡0_𝑡^+𝑇₀𝐹_𝑥𝑡,𝑥^∗𝑎,𝜀𝑑𝑡,_∗3=16^𝑡0_𝑡^+2𝑇₀𝑓_𝑥^′′′(𝑠,0)𝑒^{2∫𝑠𝑡0𝑓𝑥(𝑢,0)𝑑𝑢}2𝑑𝑠+^∼𝐼^′′′(0)1+𝐼^∼′(0)³. (80) If 𝑎^∗₃≠0, then for |𝜀| sufficiently small, [𝐴₁(𝜀)+1]𝑎^∗₃>0 (≤0) implies that system (52) has a unique (no) two-side periodic orbit of near 𝑥=0, except for a one-side periodic orbit 𝑥^∗(𝑡,𝜀).

By virtue of Theorem 25, we can have the following conclusion.

Corollary 26. (i) Let 𝐴₁(0)=−1, 𝑎^∗₃>0 (<0). Then 𝑥=0 is a nonhyperbolic one-side periodic orbit of system (48)(𝜀=0), which is asymptotically stable (unstable). (ii) Let 𝐴₁(0)=−1, [𝐴₁(𝜀)+1]𝑎^∗₃>0, 0<|𝜀|≪1. Then (𝐴₁(𝜀)+1)<0 (>0), 𝑥^∗(𝑡,𝜀) is a hyperbolic one-side periodic orbit of system (48) (𝜀=0), which is asymptotically stable (unstable. Moreover, the two-side periodic orbit is unstable (asymptotically stable) near 𝑥=0.

Finally, we give an example to illustrate it.

Example 27. Consider 𝑑𝑥𝑑𝑡=𝜀²𝑥+|𝜀|(sin𝑡)𝑥²−(1−2cos2𝑡)𝑥³,𝑡≠𝑘𝜋,Δ𝑥|_{𝑡=𝑘𝜋}=−𝜀𝑥(𝑘𝜋)+𝑏₃+𝜀𝑥³(𝑘𝜋),𝑘∈ℤ⁺, (81) where |𝜀|>0. It is obvious that for |𝜀|>0 sufficiently small, 𝑥−𝜀𝑥+(𝑏₃+𝜀)𝑥³ is strictly increasing, and then 𝑥^∗(𝑡,𝜀)=0 is the unique 𝜋-periodic solution. By direct computation, we have 𝐴₁(𝑡,𝜀)=𝜀², 𝐴₂(𝑡,𝜀)=|𝜀|(sin𝑡), 𝐴₃(𝑡,𝜀)=2cos2𝑡−1, 𝐵₁(𝜀)=−𝜀, and 𝐵₃(𝜀)=𝑏₃+𝜀. Therefore, 𝐴₁(𝜀)+1=(𝜀−1)𝑒^𝜀2𝜋+1=𝜀+𝑜(𝜀), 𝑎^∗₃=2𝑏₃−𝜋. It follows from Theorem 25 that system (81)  has two (no) 2𝜋-periodic solution of near 𝑥=0 if |𝜀| is sufficiently small and 𝑎^∗₃𝜀>0 (≤0).

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11271261), Shanghai Natural Science Foundation (no. 12ZR1421600), and Shanghai Educational Committee Project (10YZ74).