# Bifurcations of Nontwisted Heteroclinic Loop with Resonant Eigenvalues

Authors: Yinlai Jin; Xiaowei Zhu; Zheng Guo; Han Xu; Liqun Zhang; Benyan Ding
Journal: The Scientific World Journal
 (2014)

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## Abstract

By using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits to establish the local
coordinate systems in the small tubular neighborhoods of the
heteroclinic orbits, we study the bifurcation problems of
nontwisted heteroclinic loop with resonant eigenvalues. The
existence, numbers, and existence regions of 1-heteroclinic loop,
1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two
1-periodic orbits are obtained. Meanwhile, we give the
corresponding bifurcation surfaces.


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## Body

## 1. Introduction and Hypotheses

With the development of nonlinear science and the deep study of chaotic phenomenon, many experts have been interested in the research on the bifurcation problems of homoclinic and heteroclinic loops for high-dimensional nonlinear dynamical systems (see [1–5] and references therein). Compared with the bifurcations of homoclinic and heteroclinic loops for planer systems, the obtained results have been not rich because of the complexity and the lack of methods. In 1990, Chow et al. studied the codimension 2 homoclinic bifurcation under nondegenerate condition at resonant eigenvalues in [6]. The traditional methods to construct Poincaré maps were adopted to later research. In 1998, Zhu studied the bifurcation problems of nondegenerated homoclinic loops for high-dimensional system z˙=f(z,α)+ϵg(z,μ,ϵ) in [7] and the bifurcations of nondegenerated heteroclinic loops for high-dimensional system z˙=f(z)+g(z,μ) in [8]. The methods used in [7, 8] were by generalizing the Floquet method to build the local coordinate systems and Poincaré map. The papers [7, 8] used the inherent characteristic values to describe the bifurcation surfaces and phenomena so that the results had the practicability and maneuverability.In [9], the authors studied the bifurcations of rough heteroclinic loops with two saddle point. In [10], Tian and Zhu studied the bifurcations of fine heteroclinic loops with two saddle points. The heteroclinic loops studied in [9, 10] were both nontwisted and with no resonant eigenvalues.In this paper, we study the bifurcation problems of heteroclinic loop with resonant eigenvalues for high-dimensional systemz˙=f(z)+g(z,μ). By simplifying the method in [7, 8], we use the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits to establish the suitable local coordinate systems in the small tubular neighborhoods of the heteroclinic orbits. Then, we get the Poincaré maps and bifurcation equations by means of the improved method.Consider theCr system
(1)z˙=f(z),
where r≥4, z∈Rm+n. We assume the following.(H1) Hyperbolic assumption:z=pi are hyperbolic critical points of (1), i=1,2. The stable manifold Wpis and the unstable manifold Wpiu of pi are m-dimensional and n-dimensional, respectively. Moreover, -ρi1 and λi1 are the simple real eigenvalues of Dzf(pi) such that any other eigenvalue σ of Dzf(pi) satisfies either Reσ<-ρi0<-ρi1<0 or 0<λi1<λi0<Reσ, where ρi0 and λi0 are some positive constants.(H2) Nondegeneration: system (1) has a heteroclinic loop Γ=Γ1∪Γ2, where Γi={z=ri(t):t∈R}, ri(+∞)=ri+1(-∞)=pi+1, r3(t)=r1(t), p3=p1. dim⁡(Tri(t)Wpiu∩Tri(t)Wpi+1s)=1.(H3) Strong inclination:lim⁡x→+∞(Tri(t)Wpiu+Tri(t)Wpi+1s)=Tpi+1Wpi+1uu⊕Tpi+1Wpi+1s, lim⁡x→-∞(Tri(t)Wpiu+Tri(t)Wpi+1s)=TpiWpiu⊕TpiWpiss, where Wpiuu and Wpiss are the strong unstable manifolds and the strong stable manifolds, respectively, TpiWpiuu is the generalized eigenspace corresponding to all the eigenvalues with larger real part than λi0, and TpiWpiss is the generalized eigenspace corresponding to all the eigenvalues with smaller real part than -ρi0. Denote ei±=lim⁡t→∓∞r˙i(t)/|r˙i(t)|, where ei+∈TpiWpiu and ei-∈Tpi+1Wpi+1s are unit eigenvectors corresponding to λi1 and -ρi+11, respectively. Furthermore, span⁡(TpiWpiuu,ei+)=TpiWpiu, span⁡(TpiWpiss,ei+1-)=TpiWpis.Now, we consider the followingCr system:
(2)z˙=f(z)+g(z,μ),
where μ∈Rl, l≥2, 0≤|μ|≪1, g(pi,μ)=g(z,0)=0, i=1,2.


## 2. Local Coordinate Systems

Suppose thatUi is a sufficiently small neighborhood of z=pi and (H1)~(H3) hold; then, for |μ| small enough, there exists a Cr transformation such that system (2) has the following form in Ui, respectively:
(3)x˙=[λi1(μ)+⋯]x+h.o.t.,y˙=[-ρi1(μ)+⋯]y+h.o.t.,u˙=[Bi1(μ)+⋯]u+h.o.t.,v˙=[-Bi2(μ)+⋯]v+h.o.t.,
where z=(x,y,u*,v*)*, x∈R1, y∈R1, u∈Rn-1, v∈Rm-1, * means transposition, λi1(0)=λi1, ρi1(0)=ρi1, Reσ(Bi1(0))>λi0, and Reσ(-Bi2(0))<-ρi0. Moreover, we suppose
(4)Wpiu={z:y=0,v=0},Wpis={z:x=0,u=0},Wpiuu={z:x=x(u),y=0,v=0},Wpiss={z:x=0,u=0,y=y(v)},Γ∩Wpiu={z:u=u(x),y=0,v=0},Γ∩Wpis={z:x=0,u=0,v=v(y)},
where x(0)=x˙(0)=0, y(0)=y˙(0)=0, u(0)=u˙(0)=0, v(0)=v˙(0)=0.Letri(t)=(rix(t),riy(t),(riu(t))*,(riv(t))*)*; we can select some times Ti0 and Ti1 such that ri(-Ti0)=(δ,0,δu*,0*)*, ri(Ti1)=(0,δ,0*,δv*)*, where δ is small enough such that {(x,y,u*,v*)*:|x|,|y|,|u|,|v|<2δ}⊂Ui, |δu|=O(δω), |δv|=O(δω), ω=min⁡{Reσ(Bi2(μ))/ρi1(μ),Reσ(Bi1(μ))/λi1(μ)}>1.Consider the linear variational system and its adjoint system(5)z˙=Df(ri(t))z,(6)ϕ˙=-(Df(ri(t)))*ϕ.By [8, 9, 11–13], system (5) and (6) have exponential dichotomies in R+ and R-. It follows from [4] that system (5) has a fundamental solution matrix Zi(t)=(zi1(t),zi2(t),zi3(t),zi4(t)), satisfying
(7)zi1(t)∈(Tri(t)Wpiu∪Tri(t)Wpi+1s)c,zi2(t)=-r˙i(t)|r˙iy(Ti1)|∈Tri(t)Wpiu∩Tri(t)Wpi+1s,zi3(t)=(zi3,1(t),…,zi3,n-1(t))∈Tri(t)Wpiu∩(Tri(t)Wpi+1s)c=Tri(t)Wpiuu,zi4(t)=(zi4,1(t),…,zi4,m-1(t))∈(Tri(t)Wpiu)c∩Tri(t)Wpi+1s=Tri(t)Wpi+1ss,Zi(-Ti0)=(wi11wi210wi41wi1200wi42wi13wi23Iwi43000wi44),Zi(Ti1)=(10wi31001wi32000wi330wi14wi24wi34I),
where wi21<0, wi12≠0, ∥wi33∥≠0, ∥wi44∥≠0, and |wi1j(wi12)-1|≪1, j≠2; |wi2j(wi21)-1|≪1, j=3,4; |wi3j(wi33)-1|≪1, j≠3; |wi4j(wi44)-1|≪1, j≠4 for δ small enough.DenoteΦi(t)=(ϕi1(t),ϕi2(t),ϕi3(t),ϕi4(t))=(Zi-1(t))*; then Φi(t) is a fundamental solution matrix of (6). It is well known that ϕi1(t) is bounded and tends to zero exponentially as t→±∞ [8, 9, 12].DenoteΔi=wi12/|wi12|. Γi is nontwisted if Δi=1 and twisted if Δi=-1. In this paper, we study the bifurcation problems under the nontwisted condition.Letzi1(t), zi2(t), zi3(t), zi4(t) be the local coordinate systems along Γi in the tubular neighborhood of Γi.


## 3. Poincaré Maps and Bifurcation Equations

Denotehi(t)=ri(t)+Zi(t)Ni, Ni=(ni1,0,(ni3)*,(ni4)*)*, ni3=(ni3,1,…,ni3,n-1)*, ni4=(ni4,1,…,ni4,m-1)*. Define the Poincaré sections of Γi at t=-Ti0 and t=Ti1 by
(8)Si0={z=hi(-Ti0):|x|,|y|,|u|,|v|<2δ},Si1={z=hi(Ti1):|x|,|y|,|u|,|v|<2δ},
respectively, where δ is small enough such that Si0⊂Ui, Si1⊂Ui+1, U3=U1 (Figure 1).Figure 1Now, we set up the Poincaré mapFi=Fi1∘Fi0: Si-11↦Si1, where Fi0: qi-11∈Si-11↦qi0∈Si0, Fi1: qi0∈Si0↦qi1∈Si1, S01=S21, and q01=q21. Here, Fi1 is constructed from the flow of (2) in the small tubular neighborhood of Γi; Fi0 is induced by the flow of the linear approximate system of (3) in the small neighborhood Ui of pi.Denote(9)qi0=(xi0,yi0,(ui0)*,(vi0)*)*=ri(-Ti0)+Zi(-Ti0)Ni0,qi1=(xi1,yi1,(ui1)*,(vi1)*)*=ri(Ti1)+Zi(Ti1)Ni1,
where Ni0=(ni0,1,0,(ni0,3)*,(ni0,4)*)*, Ni1=(ni1,1,0,(ni1,3)*,(ni1,4)*)*.By the expressions ofZi(-Ti0) and Zi(Ti1), we get that the relations of the two kinds coordinates of qij(j=0,1) are xi0≈δ, yi1≈δ, and
(10)ni0,1=(wi12)-1[yi0-wi42(wi44)-1vi0],ni0,3=ui0-δu+[wi11wi23(wi21)-1-wi13](wi12)-1yi0+{wi13(wi12)-1wi42-wi43-wi23(wi21)-1[wi11(wi12)-1]×[wi11(wi12)-1wi42-wi41]}(wi44)-1vi0,ni0,4=(wi44)-1vi0,ni1,1=xi1-wi31(wi33)-1ui1,ni1,3=(wi33)-1ui1,ni1,4=-wi14xi1+(wi14wi31+wi24wi32-wi34)(wi33)-1ui1+vi1-δv.First, we built the mapFi1:Si0↦Si1, qi0↦qi1. We make a coordinate transformation z=hi(t), t∈[-Ti0,Ti1] and substitute it into (2). By r˙i(t)=f(ri(t)), Z˙i(t)=Df(ri(t))Zi(t) and Φi*(t)Zi(t)=I; (2) becomes
(11)n˙ij=ϕij*(t)gμ(ri(t),0)μ+h.o.t.,j=1,3,4.Thus, we defineFi1:Si0↦Si1, qi0↦qi1 as
(12)nij(Ti1)=nij(-T10)+Mijμ+h.o.t.,j=1,3,4.
That is,
(13)ni1,j=ni0,j+Mijμ+h.o.t.,j=1,3,4,
where Mij=∫-Ti0Ti1‍ϕij*(t)gμ(ri(t),0)dt, i=1,2, j=1,3,4, are called the Melnikov vectors.By [7–9, 14], Mij=∫-Ti0Ti1‍ϕij*(t)gμ(ri(t),0)dt=∫-∞+∞‍ϕij*(t)gμ(ri(t),0)dt, i=1,2, j=1,3,4.Second, we built the mapFi0:Si-11↦Si0, qi-11↦qi0, S01=S21, q01=q21. Without loss of generality, we assume ρi1≥λi1. Let τi be the flying time from qi-11 to qi0; we say si=e-λi1(μ)τi isSilnikov time. By (3) and neglecting the higher order terms, we have
(14)xi-11≈sixi0,yi0≈siρi1(μ)/λi1(μ)yi-11,ui-11≈siBi1(μ)/λi1(μ)ui0,vi0≈siBi2(μ)/λi1(μ)vi-11.
Here, (si,ui0,vi-11) is called the Silnikov coordinate.Last, it follows from (10)~(14) that we get the expression of Poincaré map Fi=Fi1∘Fi0 as follows:
(15)ni1,1=(wi12)-1δsiρi1(μ)/λi1(μ)+Mi1μ+h.o.t.,ni1,3=ui0-δu+[wi11wi23(wi21)-1-wi13](wi12)-1δsiρi1(μ)/λi1(μ)+Mi3μ+h.o.t.,ni1,4=(wi44)-1siBi2(μ)/λi1(μ)vi-11+Mi4μ+h.o.t.DenoteGi(qi-11)=(Gi1,Gi3,Gi4)=Fi(qi-11)-qi1. By (10) and (15), we get the successor function Gi as follows:
(16)Gi1=δ[(wi12)-1siρi1(μ)/λi1(μ)-si+1]+Mi1μ+h.o.t.,Gi3=ui0-δu+[wi11wi23(wi21)-1-wi13](wi12)-1δsiρi1(μ)/λi1(μ)-(wi33)-1si+1Bi+11(μ)/λi+11(μ)ui+10+Mi3μ+h.o.t.,Gi4=-vi1+δv+wi14δsi+1+(wi44)-1siBi2(μ)/λi1(μ)vi-11+Mi4μ+h.o.t.Let(17)(G11,G13,G14,G21,G23,G24)=0.Equation (17) is called the bifurcation equation.


## 4. The Preservations of Heteroclinic Orbits and Bifurcations of Homoclinic Loops

In this section, we discuss the nontwisted heteroclinic loop with resonant eigenvalues. Without loss of generality, we assume the following.(H4) Resonant condition:β1=ρ11/λ11>1, β2=ρ21/λ21=1.Letβi(μ)=ρi1(μ)/λi1(μ), βi=βi(0), ρ21(μ)=(1+α(μ))λ21(μ). By (H4) and the continuity of function, we get β1(μ)>1, β2(μ)=1+α(μ), |α(μ)|≪1, |α(0)|=0, and β1(μ)β2(μ)>1, for |μ|=1.Consider the solution of (17). Clearly, by the implicit function theorem, the equation (G13,G14,G23,G24)=0 has a unique solution ui0=ui0(s1,s2,μ), vi1=vi1(s1,s2,μ), i=1,2. Substituting it into (G11,G21)=0, we have
(18)δ[(w112)-1s1β1(μ)-s2]+M11μ+h.o.t.=0,δ[(w212)-1s21+α(μ)-s1]+M21μ+h.o.t.=0.
Equation (18) is equivalent to
(19)s1=(w212)-1s21+α(μ)+δ-1M21μ+h.o.t.,s2=(w112)-1s1β1(μ)+δ-1M11μ+h.o.t.Now, we study the existence of heteroclinic orbits and the1-homoclinic bifurcations. Denote
(20)R12={μ:M11μ>0,w212M21μ<0,|μ|≪1},R21={μ:M21μ>0,w112M11μ<0,|μ|≪1}.Theorem 1.
Suppose that hypotheses (H1)~(H4) are valid, and rank (M11,M21)=2; then one has the following.
(i) There exists a(l-1)-dimensional surface Li with a normal vector Mi1 at μ=0 such that (2) has a heteroclinic orbit joining p1 and p2 near Γi if and only if μ∈Li, |μ|≪1, where i=1,2. Moreover, (2) has a 1-heteroclinic loop near Γ if and only if μ∈L12=L1∩L2 and |μ|≪1, where L12 is (l-2)-dimensional surface and 0∈L12. That is, heteroclinic loop Γ is persistent.
(ii) There exists a(l-1)-dimensional surface L12⊂R12 which is tangent to L2(L1) at μ=0 as α(μ)>0(α(μ)<0), such that (2) has a unique 1-homoclinic loop Γ12 homoclinic to p1 near Γ as μ∈R12∩L12 and |μ|≪1. Meanwhile, there also exists a (l-1)-dimensional surface L21⊂R21 which is tangent to L1 at μ=0 such that (2) has a unique 1-homoclinic loop Γ21 homoclinic to p2 near Γ as μ∈R21∩L21 and |μ|≪1.Proof.
(i) If (19) has a solution s1=s2=0, then, we have
(21)Mi1μ+h.o.t.=0,i=1,2.
IfMi1≠0, then, there exists a (l-1)-dimensional surface Li defined by (21) with a normal vector Mi1 at μ=0 such that the ith equation of (19) has a solution s1=s2=0 as μ∈Li and |μ|≪1. That is, Γi is persistent.
If rank(M11,M21)=2, then there exists a (l-2)-dimensional surface L12=L1∩L2 such that (19) has a solution s1=s2=0 as μ∈L12 and |μ|≪1. That is, Γ is persistent.
(ii) If (19) has a solution s1=0, s2>0, then, we have
(22)s2=δ-1M11μ+h.o.t.,(δ-1M11μ+h.o.t.)1+α(μ)=-δ-1w212M21μ+h.o.t.
IfM11μ>0, w212M21μ<0, then, (22) defined a (l-1)-dimensional surface L12 with a normal vector M21(M11) at μ=0 as α(μ)>0(α(μ)<0), such that (19) has a solution s1=0, s2>0 as μ∈L12 and |μ|≪1. That is, the system (2) has a 1-homoclinic orbit to p1 near Γ for μ∈L12 and |μ|≪1.
(iii) If (19) has a solution s1>0, s2=0, then, we have
(23)s1=δ-1M21μ+h.o.t.,(δ-1M21μ+h.o.t.)β1(μ)=-δ-1w112M11μ+h.o.t.
IfM21μ>0, w112M11μ<0, then, (23) defined a (l-1)-dimensional surface L21 with a normal vector M11 at μ=0 such that (19) has a solution s1>0, s2=0 as μ∈L21 and |μ|≪1. That is, the system (2) has a 1-homoclinic orbit to p2 near Γ for μ∈L21 and |μ|≪1.


## 5. The Periodic Orbits Bifurcations

Next, we consider the bifurcation problems of1-periodic orbits near Γ. In other words, we study the solutions of (19) satisfying s1>0, s2>0. We assume the following.(H5) Nontwisted condition:Δ1=Δ2=1.Obviously, if (H5) holds, we have(24)R12={μ:M11μ>0,M21μ<0,|μ|≪1},R21={μ:M11μ<0,M21μ>0,|μ|≪1}.At first, we consider the bifurcations inR12. From (19), we get
(25)(s1β1(μ)+δ-1w112M11μ+h.o.t.)1+α(μ)=(w112)1+α(μ)w212(s1-δ-1M21μ+h.o.t.).DenoteV1(s1) and N1(s1) are the left and right hands of (25), respectively; then, we have the following.Lemma 2.
Suppose that hypotheses (H1)~(H5) are valid; then V1(s1) is tangent to N1(s1) at some point s1 satisfying 0<s1, |μ|≪1 if and only if μ∈R12, α(μ)>0, |M11μ|≪|M21μ|, and
(26)δ-1M11μ=(-M11μw212M21μ)-1/α(μ)-(w112)-1×(-w112M11μ(1+α(μ))β1(μ)M21μ)β1(μ)/(β1(μ)-1)+h.o.t.Proof.
V
1
(
s
1
) is tangent to N1(s1) at some point s1 if and only if V1(s1)=N1(s1) and V˙1(s1)=N˙1(s1); that is,
(27)(s1β1(μ)+δ-1w112M11μ+h.o.t.)1+α(μ)=(w112)1+α(μ)w212(s1-δ-1M21μ+h.o.t.),(1+α(μ))β1(μ)s1β1(μ)-1(s1β1(μ)+δ-1w112M11μ+h.o.t.)α(μ)=(w112)1+α(μ)w212.
So
(28)s1=(-w112M11μ(1+α(μ))β1(μ)M21μ)1/(β1(μ)-1)+h.o.t.
Substituting (28) into the second expression of (27), we have that (26) holds.
Equation (28) means |M11μ∥≪∥M21μ∥, and (26) means μ∈R12 and α(μ)>0.IfM11 and M21 are linearly independent, then there exists a (l-1)-dimensional surface L~12 defined by (26) in the small neighborhood of μ=0. It is easy to know that L~12∈R12 is tangent to L2 for α(μ)>0. By (21), (22), and (26), we get
(29)δ-1M11μ|L12=(-M11μw212M21μ)-(1/α(μ))+h.o.t.>(-M11μw212M21μ)-(1/α(μ))-(w112)-1×(-w112M11μ(1+α(μ))β1(μ)M21μ)β1(μ)/(β1(μ)-1)+h.o.t.=δ-1M11μ|L~12>0=δ-1M11μ|L1.
That is to say, L~12 is located in the open region between L1 and L12. Thus, we have the following.(
1
) For α(μ)>0, we define the following three open sections in R12: (R12)1 is bounded by L2 and L12, (R12)2 is bounded by L12 and L~12, and (R12)0 is bounded by L~12 and L1, and they have nonempty intersection with R12.(
2
) For α(μ)<0, we define the following two open sections in R12: (R12)1 is bounded by L2 and L12 and (R12)0 is bounded by L12 and L1, and they have nonempty intersection with R12.Now, we consider the nonnegative solutions ofV1(s1)=N1(s1) defined by (25). By V1(s1)=(s1β1(μ)+δ-1w112M11μ+h.o.t.)1+α(μ), N1(s1)=(w112)1+α(μ)w212(s1-δ-1M21μ+h.o.t.), we get V1(0)=(δ-1w112M11μ+h.o.t.)1+α(μ), N1(0)=(w112)1+α(μ)w212(-δ-1M21μ+h.o.t.). From (22), V1(0)=N1(0) holds for μ∈L12, |μ|≪1.By Lemma2, it is easy to get the following conclusions.(1)
Forα(μ)>0, we have the following (see Figure 2).Ifμ∈(R12)1, then V1(s1)=N1(s1) has one small positive solution.Ifμ∈L12, then V1(s1)=N1(s1) has one small positive solution and one zero solution.Ifμ∈(R12)2, then V1(s1)=N1(s1) has two small different positive solutions.Ifμ∈L~12, then V1(s1)=N1(s1) has one small twofold positive solution.Ifμ∈(R12)0, then V1(s1)=N1(s1) does not have any small nonnegative solutions.(2)
Forα(μ)<0, we have the following.Ifμ∈(R12)1, then V1(s1)=N1(s1) has one small positive solution.Ifμ∈L12, then V1(s1)=N1(s1) has one zero solution.Ifμ∈(R12)0, then V1(s1)=N1(s1) has not any small nonnegative solutions.(a)
(b)
(c)
(d)
(e)Thus, we have shown the following conclusions.Theorem 3.
Suppose that hypotheses (H1)~(H5) are valid, μ∈R12, α(μ)>0; then the following conclusions are true.(i)
System (2) has one simple 1-periodic orbit near Γ as μ∈(R12)1.(ii)
System (2) has one simple 1-periodic orbit and one 1-homoclinic loop homoclinic to p1 near Γ as μ∈L12.(iii)
System (2) has two simple 1-periodic orbits near Γ as μ∈(R12)2.(iv)
System (2) has a unique twofold 1-periodic orbit near Γ as μ∈L~12.(v)
System (2) has not any 1-periodic orbits and 1-homoclinic loops near Γ as μ∈(R12)0.Theorem 4.
Suppose that hypotheses (H1)~(H5) are valid, μ∈R12, α(μ)<0; then the following conclusions are true.(i)
System (2) has one simple 1-periodic orbit near Γ as μ∈(R12)1.(ii)
System (2) has one 1-homoclinic loop homoclinic to p1 near Γ as μ∈L12.(iii)
System (2) does not have any 1-periodic orbits and 1-homoclinic loops near Γ as μ∈(R12)0.Next, we consider the bifurcations inR21. Similarly, from (19), we get
(30)(s21+α(μ)+δ-1w212M21μ+h.o.t.)β1(μ)=w112(w212)β1(μ)(s2-δ-1M11μ+h.o.t.).
Denote V2(s2) and N2(s2) are the left and right hands of (30), respectively; then one has the following.Lemma 5.
Suppose that hypotheses (H1)~(H5) are valid, then V2(s2) is tangent to N2(s2) at some point s2 satisfying 0<s2, |μ|≪1 if and only if α(μ)<0, ∥M11μ∥≪∥M21μ∥, μ∈R21 and
(31)δ-1M21μ=(-w112M11μM21μ)1/(β1(μ)-1)-(w212)-1×(-(1+α(μ))β1(μ)M11μw212M21μ)-(1+α(μ))/α(μ)+h.o.t.Proof.
V
2
(
s
2
) is tangent to N2(s2) at some point s2 if and only if V2(s2)=N2(s2) and V˙2(s2)=N˙2(s2); that is,
(32)(s21+α(μ)+δ-1w212M21μ+h.o.t.)β1(μ)=w112(w212)β1(μ)(s2-δ-1M11μ+h.o.t.),(1+α(μ))β1(μ)s2α(μ)(s21+α(μ)+δ-1w212M21μ+h.o.t.)β1(μ)-1=w112(w212)β1(μ)+h.o.t.
The solution of (32) is
(33)s2=(-w212M21μ(1+α(μ))β1(μ)M11μ)1/α(μ)+h.o.t.
Substituting (33) into the second expression of (32), we obtain (31).
Equation (33) holds means |M11μ∥≪∥M21μ∥, α(μ)<0, or |M21μ∥≪∥M11μ∥, α(μ)>0. But, if |M21μ∥≪∥M11μ∥, α(μ)>0, then, by (31), we get that δ-1M21μ will be tented to ∞. Last, μ∈R21 is obvious.IfM11 and M21 are linearly independent, then there exists a (l-1)-dimensional surface L~21 defined by (31) in the small neighborhood of μ=0. It is easy to know that L~21 is tangent to L1 at μ=0. By (21), (23), and (31), we get
(34)δ-1M21μ|L21=(-w112M11μM21μ)1/(β1(μ)-1)+h.o.t.>(-w112M11μM21μ)1/(β1(μ)-1)-(w212)-1×(-(1+α(μ))β1(μ)M11μw212M21μ)-(1+α(μ))/α(μ)+h.o.t.=δ-1M21μ|L~21>0=δ-1M21μ|L2.
That is, L~21 is located in the open region between L2 and L21. Thus, we have the following.(
1
) For α(μ)>0, we define the following two open sections: (R21)1 is bounded by L1 and L21 and (R21)0 is bounded by L21 and L2, and they have nonempty intersection with R21.(
2
) For α(μ)<0, we define the following three open sections: (R21)1 is bounded by L1 and L21, (R21)2 is bounded by L21 and L~21, and (R21)0 is bounded by L~21 and L2, and they have nonempty intersection with R21.Now, we consider the nonnegative solutions ofV2(s2)=N2(s2) defined by (30). By V2(s2)=(s21+α(μ)+δ-1w212M21μ+h.o.t.)β1(μ), N2(s2)=w112(w212)β1(μ)(s2-δ-1M11μ+h.o.t.), we get V2(0)=(δ-1w212M21μ+h.o.t.)β1(μ), N2(0)=w112(w212)β1(μ)(-δ-1M11μ+h.o.t.). From (23), V2(0)=N2(0) holds for μ∈L21, |μ|≪1.By Lemma5, it is easy to get the following conclusions.(1)
Forα(μ)>0, we have the following.
Ifμ∈(R21)1, then V2(s2)=N2(s2) has one small positive solution.Ifμ∈L21, then V2(s2)=N2(s2) has one zero solution.Ifμ∈(R21)0, then V2(s2)=N2(s2) does not have any small nonnegative solutions.(2)
Forα(μ)<0, we have the following.
Ifμ∈(R21)1, then V2(s2)=N2(s2) has one small positive solution.Ifμ∈L21, then V2(s2)=N2(s2) has one small positive solution and one zero solution.Ifμ∈(R21)2, then V2(s2)=N2(s2) has two different positive solutions.Ifμ∈L~21, then V2(s2)=N2(s2) has one twofold positive solution.Ifμ∈(R21)0, then V2(s2)=N2(s2) does not have any small nonnegative solutions.Thus, we have shown the following conclusions.Theorem 6.
Suppose that hypotheses (H1)~(H5) are valid, μ∈R21, α(μ)>0; then the following conclusions are true.(i)
System (2) has one simple 1-periodic orbit near Γ as μ∈(R21)1.(ii)
System (2) has one 1-homoclinic loop homoclinic to p2 near Γ as μ∈L21.(iii)
System (2) does not have any 1-periodic orbits and 1-homoclinic loops near Γ as μ∈(R21)0.Theorem 7.
Suppose that hypotheses (H1)~(H5) are valid, μ∈R21, α(μ)<0; then the following conclusions are true.(i)
System (2) has one simple 1-periodic orbit near Γ as μ∈(R21)1.(ii)
System (2) has one simple 1-periodic orbit and one 1-homoclinic loop homoclinic to p2 near Γ as μ∈L21.(iii)
System (2) has two simple 1-periodic orbits near Γ as μ∈(R21)2.(iv)
System (2) has a unique twofold 1-periodic orbit near Γ as μ∈L~21.(v)
System (2) does not have any 1-periodic orbits and 1-homoclinic loops near Γ as μ∈(R21)0.LetD12 be an open region that is bounded by L1 and L2; meanwhile, D12∩{μ:M11μ>0,M21μ>0,|μ|≪1}≠∅. Let D21 be an open region that is bounded by L2 and L1; meanwhile, D21∩{μ:M11μ<0,M21μ<0,|μ|≪1}≠∅.By (19), it is easy to see that (19) has an unique solution s1>0, s2>0 for μ∈D12 and does not have any nonnegative solutions for μ∈D21. So we get the following conclusions.Theorem 8.
Supposing that hypotheses (H1)~(H5) are valid, then(i)
system (2) has one simple 1-periodic orbit near Γ as μ∈D12,(ii)
system (2) has not any 1-periodic orbits, 1-homoclinic loops, and heteroclinic orbits near Γ as μ∈D21.Combining Theorems1~8, we get the bifurcation figures (see Figures 3 and 4).Figure 3Figure 4



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