Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems

1 Introduction

Homoclinic and heteroclinic bifurcations play an important role in dynamical systems. More and more mathematicians have devoted themselves to study the bifurcation problems of homoclinic or heteroclinic orbits. An overview of homoclinic and heteroclinic bifurcation is given in [1]. A heteroclinic loop consists of two distinct hyperbolic saddle points and two heteroclinic orbits connecting them. When there are two saddle points with different Morse indices, this heteroclinic loop is called a heterodimensional loop. Otherwise, the heteroclinic loop is called a equidimensional loop [2]. Numerical and explicit examples of a heteroclinic loop are given in [3,4].

There is rich and complex recurrent dynamics near homoclinic or heteroclinic orbits. Hence, a central task is to find all orbits that stay near the homoclinic orbits or heteroclinic loop for all times. There are two different approaches to treat those problems. The first approach is to use Poincare or first-return maps. The existence of these special orbits is equivalent to the existence of the fixed points of Poincare or first-return maps. These methods are called geometric approaches. The second is the analytical approach. The core is using Lyapunov-Schmidt reduction. The heart of Lyapunov-Schmidt method is the Fredholm property. Chow et al. studied the persistence of the homoclinic orbit of the Duffing equation by the Fredholm property [5]. Following this work, many people have helped to develop the analytical approach to homoclinic or heteroclinic bifurcation problems. In 1990, Lin investigated the existence of periodic or aperiodic solutions near the heteroclinic chains for systems of ordinary differential equations and delay equations by analytical approach. This method was generalized by Fiedler, Vanderbauwhede, Sandstede, and many others as Lin’s method [6]. The idea of Lin’s method is to construct a sequence of piecewise continuous solutions near the original heteroclinic chain, and the bifurcation function can be obtained from these solutions. If the bifurcation function has zeros, then there exist periodic or aperiodic solutions near the heteroclinic chain. Lin’s method can also be used in discrete dynamical systems, singularly perturbed systems, and numerical computation, cf. [7–9].

For the periodic or aperiodic solutions bifurcated from homoclinic orbit by analytical approach, refer [10–12]. Chow et al. [13] considered the equidimensional heteroclinic loop, which is the equilibria that form the heteroclinic loop that has the same dimension of the unstable manifold. Meanwhile, the authors considered non-degenerate heteroclinic orbit. They used geometric approach to seek homoclinic or periodic orbit bifurcated from a heteroclinic loop. Rademacher [14], studied the homoclinic orbit bifurcated from a heteroclinic loop with one equilibrium and one periodic orbit. They assumed that the unperturbed heteroclinic orbits are one- or two-dimensional. By exponential trichotomy and Lin’ method, they found 1-homoclinic orbits near the heteroclinic loop. Jin et al. [15] considered an equidimensional loop for high-dimensional systems. They used local coordinate systems in a neighborhood of a heteroclinic loop to construct the Poincare maps and the bifurcation equations and then obtained the coexistence and coexistence regions of the 1-homoclinic loop, 1-periodic orbit, 2-homoclinic loop, and 2-periodic orbit near the heteroclinic loop. Zhu and Sun [16] considered the same subject that is homoclinic and periodic orbits bifurcated from the heteroclinic cycle connecting saddle-foci and saddle. Bykov cycle is a special heteroclinic cycle between two hyperbolic equilibria of saddle types p1 and p2, where one of the connections is transverse and isolated. Labouriau and Rodrigues [17] considered a differential equation in a three-dimensional manifold having a heteroclinic cycle that consists of two saddle-foci of different Morse indices whose one-dimensional invariant manifolds coincide and whose two-dimensional invariant manifolds intersect transversely. So, the heteroclinic cycle is defined by the presence of the Bykov cycle. They showed the existence of mixed dynamics in the neighborhood of the Bykov cycle. In the recent work by Knlbloch [18], this subject was extended to higher dimensions using Lin’s method. Long and Xu [19] investigated the persistence of a heterodimensional loop under periodic perturbation. Under some conditions, the perturbed system can have a heteroclinic loop near the unperturbed heterodimensional loop. For more research results regarding the recurrent dynamic near heteroclinic loop, refer [20–24].

Based on the above background, we apply Lin’s method to investigate periodic or homoclinic orbits near the heterodimensional loop under periodic perturbation for a high-dimensional system. We consider the following autonomous differential equation:

and its periodic perturbed equation is as follows:

where x ∈ R n , μ = ( μ 1 , μ 2 ) ∈ R 2 and we give the following assumptions:

1. f ∈ C 3 .

2. The unperturbed equation (1.1) has two distinct hyperbolic equilibria P + and P − . Namely, f ( P ± ) = 0 and the eigenvalues of D f ( P ± ) lie off the imaginary axis, where D denotes the derivative operator.

3. The unperturbed equation (1.1) has two heteroclinic solutions γ 1 ( t ) and γ 2 ( t ) , which are asymptotic to equilibria P + and P − , respectively. That is, γ i ˙ ( t ) = f ( γ i ( t ) ) , i = 1 , 2 and

   lim t → + ∞ γ 1 ( t ) = P + , lim t → − ∞ γ 1 ( t ) = P − , lim t → + ∞ γ 2 ( t ) = P − , lim t → − ∞ γ 2 ( t ) = P + .

4. g j ∈ C 3 , g j ( P ± , μ , t ) = 0 , g j ( x , 0 , t ) = 0 , and g j ( x , μ , t + 2 ) = g j ( x , μ , t ) .

5. dim ( W s ( P + ) ) = d + , dim ( W s ( P − ) ) = d − , where W s ( P + ) and W s ( P − ) are the stable manifolds of the equilibria P + and P − , respectively.

6. dim ( T γ 1 ( 0 ) W s ( P + ) ⋂ T γ 1 ( 0 ) W u ( P − ) ) = d 1

   and

   dim ( T γ 2 ( 0 ) W s ( P − ) ⋂ T γ 2 ( 0 ) W u ( P + ) ) = d 2 ,

   where T γ i ( 0 ) W s ⁄ u ( P ± ) is the tangent spaces of the corresponding invariant manifolds at γ i ( 0 ) and d i > 1 , i = 1 , 2.

From ( H 3 ) , we know that the unperturbed equation (1.1) has a heteroclinic loop Γ . Refer Figure 1, where

Figure 1

Heteroclinic loop Γ .

From ( H 5 ) , the hyperbolic equilibria P + and P − can have different saddle point indices or Morse indices when d + ≠ d − [14,18]. Long and Yang [25] defined the splitting index of heteroclinic orbit in the unperturbed heteroclinic chain. Analogously, we define splitting index of heteroclinic orbit γ i as S ( γ i ) , which are expressed as follows:

The variational equation of (1.1) along the heteroclinic orbit γ i is:

Since γ i ( t ) is the heteroclinic solution of equation (1.1), γ i ( t ) is situated at the intersection of the stable manifold and the unstable manifold. From ( H 6 ) , we know equation (1.3) has d i ( d i > 1 ) linearly independent bounded solutions, i = 1 , 2. And the dimension of the intersection of the correspondence stable manifold and unstable manifold is d i . If the intersection is non-transversal, the bifurcation phenomenon will occur in the original heteroclinic loop under periodic perturbation. By the definition of the splitting index of heteroclinic orbit γ i and a simple calculation, we have the codimension of γ i as

for i = 1 , 2. When d i + ( − 1 ) i s > 0 , the intersection of the stable and unstable manifolds is non-transversal, i = 1 , 2. The heteroclinic loop has high codimension in this situation. Therefore, under small perturbation, the heteroclinic loop is broken. So, in this study, we are mainly concerned with the heterodimensional loop bifurcation of high codimension under periodic perturbation. We will apply Lin’s method to construct the periodic solution and homoclinic solution near the unperturbed heteroclinic loop.

The structure of this study is organized as follows. In Section 2, we first study the variational equation of (1.1) along the degenerate heteroclinic orbit γ i and establish two-side exponential dichotomies. We introduce some notations and present the main result. In Section 3, we provide the proof of the main result. First, we study the existence of periodic solution near the heteroclinic loop Γ in Section 3.1. In Section 3.2, we construct the homoclinic solution near the heteroclinic loop Γ . Hence, under some conditions, periodic or homoclinic solution can bifurcated from heteroclinic loop Γ under periodic perturbation.

2 Preliminaries and main result

2.3 Statement of the main result

Before presenting the main result, we introduce some notations. Let u k i denote the k th column of U i . Renumbering if necessary, we can assume that

P u u i = I d i + ( − 1 ) i s 0 0 0 0 d i 0 0 0 0 , P s s i = 0 d i + ( − 1 ) i s 0 0 0 I d i 0 0 0 0 ,

where I d i + ( − 1 ) i s and 0 d i + ( − 1 ) i s are ( d i + ( − 1 ) i s ) × ( d i + ( − 1 ) i s ) identity and zero matrixes, respectively, for i = 1 , 2. Then, we have

P u u i U i = ( u 1 i , u 2 i , … , u d i + ( − 1 ) i s i , 0 , … , 0 ) , P s s i U i = ( 0 , … , 0 , u d i + ( − 1 ) i s + 1 , u d i + ( − 1 ) i s + 2 , … , u d i + ( − 1 ) i s + d i , 0 , … , 0 ) ,

for i = 1 , 2.

Let U i − 1 denote the inverse of U i . Then, we have U i − 1 U i = I . Differentiating with respect to t , we obtain U ˙ i − 1 U i + U i − 1 U ˙ i = 0 and hence, U ˙ i − 1 = − U i − 1 U ˙ i U i − 1 = − U i − 1 D f ( γ i ) . Then, ( U ˙ i − 1 ) T = − D f ( γ i ) T ( U i − 1 ) T , where T denotes the transpose of a matrix. Hence, ( U i − 1 ) T is a fundamental matrix solution of the adjoint equation (2.4). Let ( u j i ) ⊥ denote the j th row of U i − 1 . ( u j i ) ⊥ can be obtained from the fundamental matrix solution of the adjoint equation. Clearly, ⟨ ( u p i ) ⊥ , u q i ⟩ = δ p q , the Kronecker delta. By the definition of P u u i , we have

P u u i U i − 1 = ( ( u 1 i ⊥ ) T , ( u 2 i ⊥ ) T , … , ( u d i + ( − 1 ) i s i ⊥ ) T , 0 , … , 0 ) T ,

for i = 1 , 2.

Let

a j , k i ( α i ) = ∫ − ∞ + ∞ ⟨ ( u j i ) ⊥ ( s ) , g k ( γ i ( s ) , 0 , s + α i ) ⟩ d s b j , p q i = ∫ − ∞ + ∞ ⟨ ( u j i ) ⊥ ( s ) , D 11 f ( γ i ( s ) ) u p i ( s ) u q i ( s ) ⟩ d s ,

i = 1 , 2, j = 1 , … , d i + ( − 1 ) i s , p , q = d i + ( − 1 ) i s + 1 , … , d i + ( − 1 ) i s + d i − 1 . Using those notations, we let

M j i ( β , μ , α i ) = ∑ k = 1 2 a j , k i ( α i ) μ k + 1 2 ∑ p = 1 d i − 1 ∑ q = 1 d i − 1 b j , p q i β p i β q i ,

for i = 1 , 2,  j = 1 , … , d i + ( − 1 ) i s , p , q = d i + ( − 1 ) i s + 1 , … , d i + ( − 1 ) i s + d i − 1 . And define M i ( β , μ , α i ) : R d 1 + d 2 − 2 × R 2 × R → R d i + ( − 1 ) i s by

M i ( β , μ , α i ) = ( M 1 i ( β , μ , α i ) , … , M d i + ( − 1 ) i s i ( β , μ , α i ) ) .

Further, we let M : R d 1 + d 2 − 2 × R 2 × R 2 → R d 1 − s × R d 2 + s be given by

(2.6) M ( β , μ , α ) = ( M 1 ( β , μ , α 1 ) , M 2 ( β , μ , α 2 ) ) ,

where α = ( α 1 , α 2 ) .

Our main result can be stated as follows.

Theorem 1

Assume that (H1)–(H5) hold. Let M ( β , μ , α ) be as in (2.6). If there are some points ( β 0 , μ 0 , α 0 ) ∈ R d 1 + d 2 − 2 × R 2 × R 2 , such that

M ( β 0 , μ 0 , α 0 ) = 0

and

D ( β , μ ) M ( β 0 , μ 0 , α 0 )

is a nonsingular ( d 1 + d 2 ) × ( d 1 + d 2 ) matrix, then there exists constants T 0 > 0 , r 0 > 0 and a differentiable function ψ ˆ * : B 2 ( α 0 , r 0 ) × ( T 0 , + ∞ ) → B ¯ 1 ( r 0 ) , where B 2 ( α 0 , r 0 ) ⊂ R 2 , B ¯ 1 ( r 0 ) ⊂ R d 1 + d 2 , such that the perturbed equation (1.2) has periodic or homoclinic solution γ ( t ) near the unperturbed heteroclinic loop Γ when μ = μ 0 + ψ ˆ * .

The proof establishing the existence of a periodic solution bifurcating from the heteroclinic loop Γ is detailed in Section 3.1. Figure 2 illustrates the associated bifurcation phenomenon. The proof of the homoclinic solution bifurcated from heteroclinic loop Γ is given in Section 3.2. And, the Figure 3 illustrates the associated bifurcation phenomenon.

Figure 2

The periodic solution bifurcated from heteroclinic loop Γ .

Figure 3

The homoclinic solution bifurcated from heteroclinic loop Γ . (a) Homoclinic solution γ ( t ) asymptotic to hyperbolic equilibrium P − near the unperturbed heteroclinic loop Γ and (b) homoclinic solution γ ( t ) asymptotic to hyperbolic equilibrium P + near the unperturbed heteroclinic loop Γ .

3 Proof of Theorem 1

For the proof of the conclusion of Theorem 1, we apply Lin’s method for constructing Lin orbits x i , i = 1 , 2 that are composed of piecewise continuous orbits x i , 1 , x i , 2 , i = 1 , 2 near the heteroclinic loop, which is characterized by a gap ζ i on Σ P ± → P ± in a distinguished direction Z . The orbit x 1 = { x 1,1 , x 2,2 } starts in Σ P − → P + , follows Σ P − → P + until it reaches a neighborhood of P + follows then Σ P + → P − , and x 2 = { x 1,2 , x 2,1 } starts in Σ P + → P − , stays further close to Σ P + → P − until it reaches a neighborhood of P − , follows then Σ P − → P + again, and terminates finally in Σ P − → P + . Because the periodic and homoclinic solutions are constructed in different ways, we divide the proof of Theorem 1 into two parts, which prove that the periodic solution can be bifurcated from heteroclinic loop Γ in Section 3.1 and the homoclinic solution bifurcated from heteroclinic loop Γ in Section 3.2, respectively. First, we seek the periodic solution near the unperturbed heteroclinic loop Γ .