Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation

and Applied Analysis 3 Let u 1 (t) = u 1 (t) + u 4 (t)u 14 , where u 14 = −u −1 44 u 14 , so we have u 1 (t) ∈ (T γ(t) W cu ) c ⋂(T γ(t) W s ) c, satisfying u 1 (−T) = (u 11 , u 12 , u 13 , 0) 󸀠 , u 1 (T) = (1, 0, 0, u 14 ) 󸀠 , (10) where u 1j = u 1j − u 4j u −1 44 u 14 , j = 1, 2, 3, obviously u 12 ̸ = 0. Noticing that the strong inclination property holds, it then follows that u 33 ̸ = 0, u 44 ̸ = 0. By diagA(t) → diag(0, −ρ 1 , λ 1 , −ρ 2 ) as |t| → ∞, one can easily know that |u 1j u −1 12 | ≪ 1 for j = 1, 3, |u 3j u −1 33 | ≪ 1 for j = 1, 2, 4, and |u 4j u −1 44 | ≪ 1 for j = 1, 2, 3. The proof is then finished. Let Φ(t) = (φ 1 , φ 2 , φ 3 , φ 4 ) = (U −1 (t)) 󸀠; from the matrix theory, we know thatΦ(t) is the fundamental solutionmatrix of (7). Introduce the following local moving frame coordinates: e (t) = γ (t) + U (t) L (t) , (11) where L(t) = (l 1 (t), 0, l 3 (t), l 4 (t)) 󸀠. Define the cross sections: S 0 = {w = e (T) = (x, y, u) : 󵄨󵄨󵄨󵄨li 󵄨󵄨󵄨󵄨 < δ} ⊂ U, S 1 = {w = e (−T) = (x, y, u) : 󵄨󵄨󵄨󵄨li 󵄨󵄨󵄨󵄨 < δ} ⊂ U, i = 1, 2. (12) Notice that if q 0 ∈ S 0 , q 1 ∈ S 1 , then q 0 = (x 0 , y 10 , u 0 , y 20 ) 󸀠 = γ (T) + U (T) L (T) , L (T) = (l 10 , 0, l 30 , l 40 ) 󸀠 , q 1 = (x 1 , y 11 , u 1 , y 21 ) 󸀠 = γ (−T) + U (−T) L (−T) , L (−T) = (l 11 , 0, l 31 , l 41 ) 󸀠 . (13) We may easily obtain the new coordinates for q 0 and q 1 as follows: l 10 = (u 12 ) −1 y 11 − u 42 (u 12 u 44 ) −1 y 21 , l 30 = x 1 − u 11 (u 12 ) −1 y 11 + (u 11 (u 12 ) −1 u 42 − u 41 ) (u 44 ) −1 y 21 , l 40 = (u 44 ) −1 y 21 ,


Introduction
It is well known that the nonhyperbolic equilibrium is unstable and always undergoes a saddle-node (resp., transcritical or pitchfork) bifurcation. So the bifurcation problems of homoclinic or heteroclinic orbits with nonhyperbolic equilibria are more difficult and challenging. And few of the papers take into account the homoclinic or heteroclinic orbits with nonhyperbolic equilibria. Zhu [1] gave the sufficient conditions for the existence of nongeneric heteroclinic orbits accompanied with saddle-node bifurcation by extending exponential trichotomy. Klaus and Knobloch [2] discussed the bifurcation of homoclinic orbit to a saddle-center in reversible system. Liu et al. [3] considered the bifurcations of homoclinic orbit with a nonhyperbolic equilibrium for a high dimensional system; they achieved the persistence of homoclinic orbit and the bifurcation of periodic orbit for the system accompanied by a pitchfork bifurcation. In 2012, we discussed the bifurcations of generic heteroclinic loop accompanied by pitchfork bifurcation [4]. For other works about bifurcations of the homoclinic or heteroclinic orbits with nonhyperbolic equilibria, the readers may see [5][6][7][8] and references therein.
Inspired by the above works, we deal with the nongeneric homoclinic bifurcation accompanied by a pitchfork bifurcation in a 4-dimensional system. By extending the method established in [7], we give the sufficient conditions for the existence of a generic (resp., a nongeneric homoclinic) orbit and a periodic orbit when pitchfork bifurcation does not happen, while the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we achieve the existence of homoclinic orbits connecting the bifurcated equilibrium and three heteroclinic orbits, where we may know the difference of bifurcations between the nongeneric homoclinic orbit and the generic one.
The rest of the paper is organized as follows. In Section 2, we present some hypotheses and give the normal form for the system considered in this paper. The Poincaré map and successor function are achieved in Section 3. Finally, the existence and nonexistence of homoclinic, heteroclinic, and periodic orbits are given in Section 4.
(H 3 ) Let -axis be the tangent space of the center manifold at , and let ( , , ) be the vector field defined on the center manifold and satisfies According to Wiggins [9], under the above assumption, the origin is a pitchfork bifurcation point, and is the parameter controlling the pitchfork bifurcation; that is to say, under small perturbation when > 0 the origin is perturbed into three hyperbolic saddles 0 , + , − (one may see Figure 1). Denote 0 = = (0, 0, 0, 0) , + = 0 + (√ , 0, 0, 0) , and In the whole paper, the sign " " denotes the transpose of the vector. It is easy to see that dim( According to the invariance of the manifolds, we may introduce a scale transformation and straighten the local manifolds of , , , ; then system (2) has the following expression in the small neighborhood of the origin: Due to the normal form (5) and (H 1 ), we may choose Take into account the linear variational system: and its adjoint systemΦ where = ( ( )) and is the transpose of . We introduce the following lemma; it is very significant in this paper.
Proof. According to the hypotheses (H 1 ) and (H 2 ), one may easily obtain the existence of the 2 ( ), 3 ( ), and 4 ( ) with the given expression at = ± . Based on the condition (H 2 ), we take 1 ( ) ∈ ( ( ) ) ⋂( ( ) ) , satisfying Noticing that the strong inclination property holds, it then follows that 33 The proof is then finished.
; from the matrix theory, we know that Φ( ) is the fundamental solution matrix of (7).

Poincaré Map and Successor Function
In this section, we establish the Poincaré map in the new coordinate system and then derive the successor function.
Let be the flying time from 0 to 1 , and set = − (where = min{ 1 ( ), 1 ( )}); utilizing the approximate solutions of system (5), it is easily to obtain the expression of 0 : 0 → 1 : where and the higher order terms are neglected.
(4) Establishment of the successor function.

Remark 4.
As we know that the homoclinic orbit is nongeneric, so the homoclinic orbit obtained in Theorem 3 (iii) comes from along the unstable manifold, while the orbit may come from the equilibrium along the weak unstable manifold (see Figure 2(b)).
Next, we consider the case > 0; the origin undergoes a pitchfork bifurcation in this case; namely, there are three equilibria + , 0 , and − bifurcated from the origin . And there always exist two straight segment orbits, one is heteroclinic to + and 0 and the other is heteroclinic to − and 0 ; their lengths are √ , and we denote the heteroclinic orbits by Γ * and Γ * , respectively. On the other hand, based on the definition of 0 , we will consider the bifurcations with three cases: for 0 ≥ √ , 0 ∈ [−√ , √ ) and 0 < −√ .
Next, we discuss the case 0 ∈ [−√ , √ ), as we know that the orbit will go into 0 (we denote = 0) in this case. While for 0 < −√ the orbit will keep away from − . However, the orbit that comes from the equilibrium will be decided by 1 . If 1 > √ , then the orbit comes from + ; if 0 < 1 < √ , then the orbit comes from 0 ; for −√ < 1 < 0, then the orbit comes from − (see Figure 1).
So we obtain the following result. such that system (1) has no other heteroclinic orbit and no homoclinic orbit for ( , ) ∈ Σ 4 ( , ).

Remark 7.
As we know from [3], the orbits in Figures 3(e)-3(h) cannot be bifurcated from the generic homoclinic orbit, which exactly reveals the difference between the bifurcation of generic homoclinic orbit and the nongeneric one.