# Bifurcations of a Homoclinic Orbit to Saddle-Center in Reversible Systems

Authors: Zhiqin Qiao; Yancong Xu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The bifurcations near a primary homoclinic orbit to a saddle-center are investigated in a 4-dimensional reversible system. By establishing a new kind of local moving frame along the primary homoclinic orbit and using the Melnikov functions, the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding codimension 1 or
codimension 3 surfaces, are obtained.


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## Body

## 1. Introduction

In recent years, there are much interest in the phenomenon of homoclinics and heteroclinics in reversible dynamical systems because of their extensive applications in mechanics, fluids, and optics [1–10]. For example, Klaus and Knobloch [4] considered a homoclinic orbit to saddle-center with two-parameter families of non-Hamiltonian reversible vector fields by Lin's method. They derived the occurrence of 1-homoclinic orbits to the center manifold. Liu et al. [6] studied a singular perturbation system with action-angle variable and the unperturbed system was assumed to possess a saddle-center equilibrium in a general system without reversible or Hamiltonian structure. Mielke et al. [7] investigated bifurcations of homoclinic orbit to saddle-center in 4-dimensional reversible Hamiltonian systems. By using the Poincaré map and a special normal form, they detected the existence of N-homoclinic orbits to the equilibrium, N-periodic orbits, and chaotic behavior near the primary homoclinic orbit. As for purely Hamiltonian system in ℝ4, similar results are known from Koltsova and Lerman [2, 5]. In all of these papers the underlying Hamiltonian structure was heavily considered. Especially it was used to detect multiround orbits. Note that, the Hamiltonian can make the dynamics constrict in three-dimensional manifold in a zero level set. In this paper, we use the method originated by Zhu [9], by constructing a local moving coordinate system and Poincaré map near the primary homoclinic orbit, the existence of transversal homoclinic orbits and periodic orbits bifurcated from the primary homoclinic orbit are obtained in a 4-dimensional reversible system. It is worth to mention that a new kind of moving coordinates is introduced firstly in our paper in order to simplify and facilitate the reversible system.The remainder of this paper is organized as follows. Section2 contains the assumptions for the perturbed and unperturbed system. The local coordinate moving frame, cross sections, and Poincaré map are set up in Section 3. Finally, we obtain the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding surfaces with different condimensions in Sections 4 and 5.


## 2. The General Setup

Consider the following system(2.1)z˙=f(z)+ϵg(z;μ),z∈ℝ4,μ∈ℝl
and the corresponding unperturbed system
(2.2)z˙=f(z),
where 0<ϵ≪1, l≥1, f:ℝ4→ℝ4 and g:ℝ4×ℝ→ℝ4 are Cr(r≥4), f(0)=g(0;μ)=0 and μ is a parameter. Also, we need the following assumptions.(
A
1
) System (2.1) is reversible with respect to the linear involution R such that dim⁡(Fix⁡R)=2, f(Rz)+Rf(z)=g(Rz;μ)+Rg(z;μ)=0 for all z∈ℝ4 and μ∈Rl.Note that, throughout the paper, we will denoteR-symmetric orbit as symmetric orbit.(
A
2
) The origin O is a saddle-center equilibrium of (2.2), O∈Fix⁡R. More precisely, the Jacobian matrix A≜Df(0) has a pair of purely imaginary eigenvalues ±iω and two nonzero real eigenvalues, that is, σ(A)={±iω,±λ} with ω,λ>0.(
A
3
) System (2.2) has a symmetric homoclinic orbit Γ={z=r(t):t∈ℝ}, where r(±∞)=O.Note that,O is a symmetric equilibrium, and the eigenvalues of the Jacobian matrix A are symmetric with respect to the imaginary axis. Thus, assumption (A2) describes a scenario that is structurally stable. Furthermore, the saddle-center O has one-dimensional stable manifold Wϵ,μs(O) and one-dimensional unstable manifolds Wϵ,μu(O) (abbr. Ws and Wu as ϵ=0), and a two-dimensional center manifold Wϵ,μc(O) (abbr. Wc as ϵ=0) for ϵ close to 0. All of them are Cr. Confined to Wϵ,μc(O), O is a center. The reversibility implies that Wϵ,μu(O)=RWϵ,μs(O) and Wϵ,μc(O)=RWϵ,μc(O), and hence, the homoclinic orbit Γ is symmetric, that is, RΓ=Γ.


## 3. Local Moving Frame and Poincaré Map

Suppose the neighborhoodU0 of O is small enough, we can firstly straighten the center-stable manifold, the center-unstable manifold, subsequently, then the stable manifold and the unstable manifold in U0 by using the method introduced in Zhu [9]. According to the invariance and symmetry of these manifolds, we can deduce that system (2.1) has the following form in U0:
(3.1)x˙=x(λ(ϵ,μ)+O(1)),y˙=y(-λ(ϵ,μ)+O(1)),u˙=v(ω(ϵ,μ)+O(1))+u(O(x)+O(y)+O(u))+O(xy),v˙=u(-ω(ϵ,μ)+O(1))-v(O(x)+O(y)+O(v))-O(xy),
where λ(0,μ)=λ, ω(0,μ)=ω, O(1)=O(x)+O(y)+O(u)+O(v), the system is Cr-2, and the corresponding involution acts as R(x,y,u,v)=(y,x,v,u).In fact, by a linear transformation, system (2.1) takes the form in a small neighborhood of U0 as follows:
(3.2)x˙=λ(ϵ,μ)x+O(2),y˙=-λ(ϵ,μ)y+O(2),u˙=ω(ϵ,μ)v+O(2),v˙=-ω(ϵ,μ)u+O(2),
and R(x,y,u,v)=(y,x,v,u). By the invariant manifold theorem, we know that there exist a local Cr center-stable manifold Wϵ,μcs={z=(x,y,u,v):x=xϵ,μcs(y,u,v),xϵ,μcs(0,0,0)=0,Dxϵ,μcs(0,0,0)=0,z∈U0}, a local Cr center-unstable manifold Wϵ,μcu={z=(x,y,u,v):y=yϵ,μcu(x,u,v),yϵ,μcu(0,0,0)=0,Dyϵ,μcu(0,0,0)=0,z∈U0} and RWϵ,μcs=Wϵ,μcu.By the straightening coordinate transformation which is similar to that of [1, 6, 9], now we straighten the local manifolds Wϵ,μcs and Wϵ,μcu, such that Wϵ,μcs={z∈U0:x=0},Wϵ,μcu={z∈U0:y=0}.Notice that the invariance ofWϵ,μcs and Wϵ,μcu implies the local invariance of {z∈U0:x=0} and {z∈U0:y=0}, respectively, which produces that, in U0,
(3.3)x˙=x(λ(ϵ,μ)+O(1)),y˙=y(-λ(ϵ,μ)+O(1)).Now the system isCr-1 and still reversible. By using a similar procedure to straighten the local Cr-1 stable manifold Wϵ,μs and unstable manifold Wϵ,μu, and the invariance and symmetry of these two local manifolds (that means the transformation is also symmetric), we get system (3.1). Clearly, corresponding to system (3.1), the center manifold Wϵ,μc is locally in the u-v plane, and the stable manifold Wϵ,μs (resp., unstable manifold Wϵ,μu) is locally the y-axis (resp., x-axis) when they are confined in U0.Definer(-T)=(δ,0,0,0)*, r(T)=(0,δ,0,0)* for T≫1, where δ>0 is small enough such that {(x,y,u,v)*:|x|,|y|,|u2+v2|1/2<2δ}⊂U0.Consider the linear variational system(3.4)z˙=Df(r(t))z,
and its adjoint system
(3.5)ϕ˙=-(Df(r(t)))*ϕ.Based on the invariance and symmetry of manifoldsWcs and Wcu, it is easy to know that system (3.4) has a fundamental solution matrix Z(t)=(z1(t),z2(t),z3(t),z4(t)) satisfying z1(t)∈(Tr(t)Wcu)c, z2(t)=-r˙(t)/|r˙(T)|∈Tr(t)(Ws∩Wu), z3(t),z4(t)∈Tr(t)Wcu and
(3.6)Z(-T)=(0-100100000100001),Z(T)=(10000100000100-10).Actually, in the resulting coordinates,Ws∩U0 and Wu∩U0 are y-axis and x-axis, respectively, combining with the symmetry, it follows that
(3.7)z2(T)=(0,1,0,0)*,z2(-T)=(-1,0,0,0)*.On the other hand, in a small tubular neighborhood of the homoclinic loopΓ, the center-unstable manifold Wcu (resp., center-stable manifold Wcs) can be foliated into a family of leaves, each is a 2-dimensional surface and asymptotic to Wc as the base point z=r(t)∈Γ tends to O as t→-∞ (resp., +∞). Notice that the limit of the linearization (3.4) of system (2.2) with respect to Γ as t→±∞ is
(3.8)x˙=λx,y˙=-λy,u˙=ωv,v˙=-ωu.
When confined on Wc, it becomes the following subsystem
(3.9)u˙=ωv,v˙=-ωu,
which is reversible with the involution (u,v)→(v,u). Obviously, for any a>0 and t0∈R, u=u(t)=asinω(t-t0), v=v(t)=acos⁡ω(t-t0) is a solution of (3.9), which defines a closed orbit Γ^ on Wc. More precisely, its tangent vectors u1(t)=u˙(t)=aωcos⁡ω(t-t0), v1(t)=v˙(t)=-aωsinω(t-t0) and its normal vectors u2(t)=-v˙(t)=aωsinω(t-t0), v2(t)=u˙(t)=aωcos⁡ω(t-t0) are the solutions of (3.9). Choose some appropriate a and t0 such that aω=1,ω(-T-t0)=2kπ for some k∈Z, then (u(-T),v(-T))=(0,a)=(0,w-1),z-3(-T)=(u1(-T),v1(-T))=(1,0),z-4(-T)=(u2(-T),v2(-T))=(0,1). Based on the reversibility, we have (u(T),v(T))=(a,0)=(w-1,0),z-3(T)=(u1(T),v1(T))=(0,-1),z-4(T)=(u2(T),v2(T))=(1,0) (see Figure 1 for details). Thus, if we take solutions z3(t) and z4(t) in Tr(t)Wcu satisfying z3(-T)=(0,0,1,0)*,z4(-T)=(0,0,0,1)*, then, restricted to the u-v plane, z3(-T) is the unit tangent direction of the closed orbit Γ^ at (u,v)=(0,a)=(0,ω-1). In addition, the restriction of z4(-T) is its unit exterior normal direction at the same point. By the reversibility, it is easy to obtain z3(T)=(0,0,0,-1)*, z4(T)=(0,0,1,0)*.Figure 1
The geometry of vectors on the center manifold.Finally, if we choose a solutionz1(t)∈(Tr(t)Wcu)c with z1(-T)=(0,1,0,0)*, then the symmetry says that z1(T)=(1,0,0,0)*.Therefore, we have demonstrated the existence of the fundamental matrixZ(t) with the specified properties.Remark 3.1.
In the following, we will regard(z1(t),z2(t),z3(t),z4(t)) as a moving coordinate in a small tubular neighborhood of Γ. This new kind of moving frame is firstly introduced for the homoclinic orbit to a saddle-center, which is the extension of the corresponding coordinates built in [1, 6, 9] for the homoclinic orbit to a saddle. The explicit advantage is that, these coordinate vectors inherit and exhibit the geometrical and dynamical properties of those invariant manifolds. As mentioned above, they will greatly simplify the original reversible system.LetΣu={(x,y,u,v)∣x=δ,|y|,|u2+v2|1/2≤δ/2}⊂U0 and Σs={(x,y,u,v)∣y=δ,|x|,|u2+v2|1/2≤δ/2}⊂U0 be the cross sections of Γ at t=-T and t=T, respectively.Now we turn to seek the new coordinates ofq0∈Σu and q1∈Σs (see Figure 2 for details) under the transformation z(t)=r(t)+Z(t)·N, where N=(n1,0,n3,n4)*. Take
(3.10)q0≜(x0,y0,u0,v0)=r(-T)+Z(-T)·(n10,0,n30,n40)*,q1≜(x1,y1,u1,v1)=r(T)+Z(T)·(n11,0,n31,n41)*,
which are solved by
(3.11)x0=δ,n10=y0,n30=u0,n40=v0;y1=δ,n11=x1,n31=-v1,n41=u1.Figure 2
Setup of cross sections and Poincaré map.Putting the transformationz(t)=r(t)+Z(t)·N into (2.1), we get
(3.12)r˙(t)+Z˙(t)·N+Z(t)·N˙=f(r(t)+Z(t)·N)+ϵg(r(t)+Z(t)·N,μ)=f(r(t))+Df(r(t))Z(t)·N+ϵg(r(t),μ)+O((ϵ+N)2).
That implies Z(t)·N˙=ϵg(r(t),μ)+O((ϵ+N)2), which is Cr-3. Then multiplying the resulting equation by Z-1(t)=(ϕ1*,ϕ2*,ϕ3*,ϕ4*)* and integrating it on both sides from -T to T, we have the regular map ΠR:Σu→Σs:
(3.13)ni(T)=ni(-T)+ϵMi(T,μ)+O((ϵ+N(-T))2),i=1,3,4,
where Mi(T,μ)=∫-TT‍ϕi*gi(r(t),μ)dt are the Melnikov functions.For conciseness, we will denoteλ(ϵ,μ)≜λ,ω(ϵ,μ)≜ω. Now we consider the local map ΠS induced by the flow (3.1) in U0, where
(3.14)ΠS:Σs⟶Σu,q1=(n11,0,n31,n41)⟶q0=(n10,0,n30,n40).
Let τ be the flying time from q1 to q0, by variation of constants formula, we can get the following expression:
(3.15)x1=x(T)=x0e-λτ(1+ρ1),y0=y(T+τ)=y1e-λτ(1+ρ2),u0=u(T+τ)=[u1cos⁡(ωτ)+v1sin(ωτ)](1+ρ3)+ρ5,v0=v(T+τ)=[-u1sin(ωτ)+v1cos⁡(ωτ)](1+ρ4)+ρ6,
where ρi=ρi(x0,y1,u1,v1,τ,ϵ,μ), ρi∈C0, i=1,…,6, and ρi=O(δ) for i=1,2,3,4, ρj=O(δ2e-λτ) for j=5,6.Denotes=e-λτ, then we get ΠS:Σs→Σu defined by
(3.16)n10=δs(1+ρ2),n30=[n41cos⁡(ωλlog⁡⁡s)+n31sin(ωλlog⁡⁡s)](1+ρ3)+ρ5,n40=[n41sin(ωλlog⁡⁡s)-n31cos⁡(ωλlog⁡⁡s)](1+ρ4)+ρ6.Combining the maps (3.13) and (3.16), we obtain the Poincaré map Π=ΠR∘ΠS:Σs→Σs defined as
(3.17)n1(T)=δs(1+ρ2)+ϵM1(T,μ)+h.o.t.,n3(T)=[n41cos⁡(ωλlog⁡⁡s)+n31sin(ωλlog⁡⁡s)](1+ρ3)+ϵM3(T,μ)+ρ5+h.o.t.,n4(T)=[n41sin(ωλlog⁡⁡s)-n31cos⁡(ωλlog⁡⁡s)](1+ρ4)+ϵM4(T,μ)+ρ6+h.o.t.


## 4. Existence of 1-Homoclinic Orbit and 1-Periodic Orbit

LetG(q1)≜Π(q1)-q1 be the displacement function. Based on (3.11), (3.17), and the new coordinate of q1=(n11,0,n31,n41), we see the small zero point (s,u1,v1) of G will satisfy the following equations:
(4.1)δs(ρ2-ρ1)+ϵM1(T,μ)+O(|s,ϵ|2)=0,v1+[u1cos⁡(ωλlog⁡⁡s)-v1sin(ωλlog⁡⁡s)](1+ρ3)+ϵM3(T,μ)+ρ5+h.o.t.=0,u1-[u1sin(ωλlog⁡⁡s)+v1cos⁡(ωλlog⁡⁡s)](1+ρ4)-ϵM4(T,μ)-ρ6+h.o.t.=0.Due to the coordinate transformations introduced inU0 at the beginning of Section 3, the unstable manifold and the stable manifold are locally x-axis and y-axis, respectively, so it is evident that, near Γ, system (2.1) has a symmetric 1-homoclinic orbit to O if and only if (4.1) have a solution (s,u1,v1) with s=u1=v1=0, and system (2.1) has an 1-homoclinic orbit to a periodic orbit on the center manifold and an 1-periodic orbit if and only if (4.1) have a solution (s,u1,v1) with s=0, u12+v12>0 and s>0, u12+v12≥0, respectively. Clearly, system (4.1) is C1 in (s,u1,v1) as 0≤s≪1, u12+v12≪1, and C3 in (ϵ,μ), and has a solution s=u1=v1=0 as ϵ=0, thus we can rescale s=ϵ2s-, u1=ϵ2u, v1=ϵ2v, such that system (4.1) is reformulated as
(4.2)M1(T,μ)+O(ϵ)=0,M3(T,μ)+O(ϵ)=0,M4(T,μ)+O(ϵ)=0,
where O(ϵ) depends on ϵ,s-,u,v and μ.Now the following results are verified directly by the implicit function theorem.Theorem 4.1.
Assume (A1)–(A3) are satisfied, M1(T,μ0)=M3(T,μ0)=M4(T,μ0)=0 and Rank(∂(M1,M3,M4)/∂μ)(T,μ0)=3 for l≥3 and μ0∈ℝl, then, for any given r0>0, s-0>0 and ϵ0>0 small enough, there exist a single-parameter family of codimension 3 surface Σϵ and a four-parameter family of codimension 3 surface Σϵuvs- near μ0 such that Σ0=Σ0uvs-={μ|μ=μ0}, lims-→0Σϵuvs-=Σϵuv0 for 0<ϵ<ϵ0, 0≤u2+v2<r02, 0<s-<s-0, and system (2.1) has a symmetric 1-homoclinic orbit to O as μ∈Σϵ, a 1-periodic orbit with approximate period 2T-λ-1log⁡ϵ2s- as μ∈Σϵuvs-, and a 1-homoclinic orbit to a periodic orbit on the center manifold with u12(t)+v12(t)=ϵ4(u02+v02) as μ∈Σϵu0v00, Σϵu0v00-{μ0}≠∅, 0<u02+v02<r02. Moreover, these codimension 3 surfaces are either coincident, or tangent to each other at μ=μ0, and they have a common 3-dimensional normal space spanned by M1μ(T,μ0),M3μ(T,μ0), and M4μ(T,μ0) as l>3.In order to study the existence of the1-periodic orbit bifurcation surface with lower codimension, we reconsider (4.1). Let s=e-(λ/ω)(2kπ-α), α∈[-(π/2),(3π/2)) and H be the vector defined by the left hand side of the last two equations of (4.1), then, by using
(4.3)W≜det⁡[∂H∂(u1,v1)|u1=v1=ϵ=0k=∞]=-[2+ρ3+ρ4+ρ3ρ4-(2+ρ3+ρ4)sinα]|u1=v1=ϵ=0k=∞,
it follows that, for δ>0 so small that |ρ3+ρ4|<0.1, |ρ3+ρ4+ρ3ρ4|<0.1, we have W≠0 for α∈[-(π/4),(5π/4)]. Therefore, there is a unique pair (u1,v1) which solves the last two equations as k>-(ω/λπ)log⁡ϵ and ϵ>0 is small enough.Substituting this solution(u1,v1) into the first equation of (4.1), and using s=O(ϵ2) for k>-(ω/λπ)log⁡ϵ, it produces
(4.4)M1(T,μ)+O(ϵ)=0,
where the term O(ϵ) depends on ϵ,k,α and μ.At this stage, we have actually demonstrated the following existence theorem.Theorem 4.2.
Suppose (A1)–(A3) are satisfied, M1(T,μ0)=0, (∂M1/∂μ)(T,μ0)≠0 as l≥1 and μ0∈ℝl, then for ϵ0>0, δ>0 small enough, 0<ϵ<ϵ0, k>-(ω/λπ)log⁡ϵ and α∈[-(π/4),(5π/4)], there exists a 3-parameter (resp., 2-parameter) family of codimension 1 surface Σϵkα (resp., Σϵα=Σϵ∞α) near μ0 satisfying Σ0∞α={μ0}, such that system (2.1) has one 1-periodic orbit L(ϵ,k,α) (resp., 1-homoclinic orbit L(ϵ,∞,α)) near Γ as μ∈Σϵkα(resp., μ∈Σϵα), and L(ϵ,k,α) has an approximate period 2T+((2kπ-α)/ω). In addition, if M1(T,μ)≠0 for any μ∈ℝl, then system (2.1) has not any 1-periodic orbit near Γ.Remark 4.3.
When confined in a neighborhood of the local center-manifold, the (u,v)-coordinates of 1-periodic or 1-homoclinic orbit guaranteed by Theorem 4.1 as μ∈Σϵuvs- or μ∈Σϵu0v00 satisfying u12+v12=O(ϵ4), whereas, these coordinates of the corresponding orbits given in Theorem 4.2 as μ∈Σϵkα or μ∈Σϵα have the scale O(ϵ2) in case M42(T,μ0)+M32(T,μ0)≠0 and k>-(ω/λπ)log⁡ϵ.


## 5. Existence of Symmetric 1-Periodic Orbit

In this section, we turn to seek the existence of symmetric1-periodic orbit. Note that, a 1-periodic orbit L near Γ is symmetric if and only if R(L∩Σs)=L∩Σu, equivalently L∩Σs=(x1,δ,u1,v1) and L∩Σu=(δ,y0,u0,v0) should satisfy
(5.1)x1=y0,u1=v0,v1=u0.
Due to (3.15), it is equivalent to
(5.2)ρ1-ρ2=0,u1-[u1sin(ωλlog⁡⁡s)+v1cos⁡(ωλlog⁡⁡s)](1+ρ4)-ρ6=0,v1-[u1cos⁡⁡(ωλlog⁡⁡s)-v1sin⁡(ωλlog⁡⁡s)](1+ρ3)-ρ5=0.
Using (5.2) and rescaling u1=ϵu, v1=ϵv, s=e-(λ/ω)(4kπ+2θ)=O(ϵ2) for k>-(ω/2λπ)log⁡⁡ϵ, θ∈[0,2π), system (4.1) reads as
(5.3)M1(T,μ)+O(ϵ)=0,2v+M3(T,μ)+O(ϵ)=0,M4(T,μ)+O(ϵ)=0.Solving the second equation of (5.3), we have
(5.4)v=-12M3(T,μ)+O(ϵ).On the other hand,L∩Σs and L∩Σu are symmetric if and only if their middle point (x(T+(1/2)τ),y(T+(1/2)τ),u(T+(1/2)τ),v(T+(1/2)τ)) lies on the plane Fix⁡(R), that is, (5.1) is equivalent to
(5.5)x(T+12τ)=y(T+12τ),u(T+12τ)=v(T+12τ),
where
(5.6)x(T+12τ)=δs(1+ρ11),y(T+12τ)=δs(1+ρ21),u(T+12τ)=[u1cos⁡⁡(ωτ2)+v1sin⁡(ωτ2)](1+ρ31)+ρ51,v(T+12τ)=[-u1sin⁡(ωτ2)+v1cos⁡⁡(ωτ2)](1+ρ41)+ρ61,
and ρi1=ρi1(x1,y1,u1,v1,τ,ϵ,μ) for i=1,…,6, ρi1=O(x1/s)+Oi(y1)+O(u12+v12)=O(δ), Oi(y1)=O(y1) for i=1,…,4, ρj1=O(x1y1)=O(δ2s) for j=5,6.Explicitly, (5.5) can be formulated as
(5.7)ρ11-ρ21=0,u1[cos⁡⁡(ωτ2)+sin⁡(ωτ2)]-v1[cos⁡⁡(ωτ2)-sin⁡(ωτ2)]+O(δ)+O(δ2s)=0.
Due to (5.4) and (1/2)ωτ=2kπ+θ, θ∈[0,2π), we can uniquely solve the second equation in (5.7) by
(5.8)u1=-ϵ(cos⁡⁡θ-sin⁡θ)2(cos⁡⁡θ+sin⁡θ)M3(T,μ)(1+O(δ))+O(ϵ2)
as θ≠(3/4)π, (7/4)π, and δ, ϵ small enough.Up to now, there are three equations(5.9)ρ11-ρ21=0,M1(T,μ)+O(ϵ)=0,M4(T,μ)+O(ϵ)=0
left which should be fulfilled, where v1=ϵv and u1 are given by (5.4) and (5.8).Applying the implicit function theorem to (5.9), where O(ϵ) depends on ϵ,θ,k and μ, we derive the following result.Theorem 5.1.
Suppose that (A1)–(A3) hold, M1(T,μ0)=0, M4(T,μ0)=0 and
(5.10)Rank{∂(ρ11-ρ21)∂μ|ϵ=0,μ=μ0k=∞,∂(M1,M4)∂μ(T,μ0)}=3
as μ0∈ℝl(l≥3), there exist ϵ0>0, θ0>0 small enough, and a 3-parameter family of codimension 3 surface Σϵθk near μ0, such that, system (2.1) has a symmetric 1-periodic orbit L near Γ with approximate period 2T+(1/ω)(4kπ+2θ) as μ∈Σϵθk for ϵ,θ,k satisfying 0<ϵ<ϵ0, θ∈[0,(3/4)π-θ0)∪((3/4)π+θ0, (7/4)π-θ0)∪((7/4)π+θ0,2π), k>-(ω/2λπ)log⁡⁡ϵ. Moreover, if there is no μ0∈ℝl satisfying
(5.11)M12(T,μ0)+M42(T,μ0)+(ρ11-ρ21)2|ϵ=0,μ=μ0k=∞=0,
then system (2.1) has no symmetric 1-periodic orbit in the small neighborhood of Γ.Remark 5.2.
Similarly, if we takes≜e-λτ=u1=v1=0 in (3.15), then x1=y0=u0=v0=u1=v1=0 (i.e., condition (5.1)), it means that there is a symmetric 1-homoclinic orbit to O if and only if s=u1=v1=0 in Section 4.Remark 5.3.
In Theorem4.2 (resp., Theorem 5.1), the geometric meaning of k is that, confined in U0, the u-v component of the above 1-periodic orbit bifurcated from Γ makes circle k (resp., 2k) times around the saddle-center.Remark 5.4.
Fromx1/s=δs(1+O(δ)), s=O(ϵ2) and the constitution of ρ11 and ρ21, it is easy to know that the necessary condition for ρ11=ρ21 is O1(y1)=O2(y1).



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*Source: 678252-2012-11-04.xml*