# Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

Authors: Yinlai Jin; Feng Li; Han Xu; Jing Li; Liqun Zhang; Benyan Ding
Journal: Computational and Mathematical Methods in Medicine
 (2013)

---

## Abstract

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence ofk-homoclinic loop and k-periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.


---

## Body

## 1. Introduction

With the rapid development of nonlinear science, in the studies of many fields of research and application of medicine, life sciences and many other disciplines, there are a lot of variety high-dimensional nonlinear dynamical systems with complex dynamic behaviors. Homoclinic and heteroclinic orbits and the corresponding bifurcation phenomenons are the most important sources of complex dynamic behaviors, which occupy a very important position in the research of high-dimensional nonlinear systems. We know that in the study of high-dimensional dynamical systems of infectious diseases and population ecology we tend to ignore the stability switches and chaos when considered much more the nonlinear incidence rate, population momentum, strong nonlinear incidence rate, and so forth. The existence of transversal homoclinic orbits implies that chaos phenomenon occur; therefore, it is of very important significance to study the cross-sectional of homoclinic orbits and the preservation of homoclinic orbits for the system in small perturbation.In addition, in the study of infectious diseases and population ecology systems, we sometimes require the existence of periodic orbits. And, homoclinic and heteroclinic orbits bifurcate to periodic orbits in a small perturbation means that we can get the required periodic solution only by adding a small perturbation when using the similar system which exists homoclinic or heteroclinic orbits to represent the natural system. This also explains the importance of homoclinic and heteroclinic orbits bifurcating periodic orbits in real-world applications.Therefore, by using the research methods and theoretical results of qualitative and bifurcation problems of high-dimensional systems, especially the results of homoclinic and heteroclinic orbits and their bifurcations for the systems, to study the high-dimensional infectious disease dynamics and population ecology systems to reveal the complex dynamical behavior of the nonlinear dynamical systems and the corresponding reality systems is essential.About the study of bifurcation problems of homoclinic and heteroclinic loops for two-dimensional systems, a large number of papers were obtained and achieved many good results (for some results see [1–6]); but for higher-dimensional nonlinear systems, due to the complexity, the results we see today are not abundant. Chow S. N., Deng B., and Fiedler B. discussed the bifurcation of non-degenerated homoclinic loop [7], but the research method is abstract, and the results are focused on the theory. Some subsequent studies are mostly based on the traditional Poincaré map construction method. Zhu [8] discussed the non-degenerated bifurcation problems of homoclinic loop of the system z˙=f(z,α)+ϵg(z,μ,ϵ). Compared with [7], paper [8] described the bifurcation surface and bifurcation phenomenon by using the inherent eigenvalue, so that the results possess good operability.In this paper, the bifurcation and stability problems of non-degenerated homoclinic loops under non-resonant and resonant conditions for high dimensional systemz˙=f(z)+g(z,μ) are considered. The method to establish the local coordinates system in the tubular neighborhood of homoclinic loop used in [8] is simplified here. In [8], the author used the generalized Floquet method to establish local coordinate systems and Poincaré map. Here, we use the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local coordinates system of the perturbed system in the small neighborhood of the homoclinic orbit. The Poincaré Maps and bifurcation equations obtained by this method are more simple and convenient for analysis than [8]. Besides, this method does not only have important significance in theory, but it can also be operated well in applications.


## 2. Hypotheses

Suppose the followingCr system:
(1)z˙=f(z),
where r≥4, z∈Rm+n satisfies the following hypotheses.(H1)
(Hyperbolicity)z=0 is a hyperbolic equilibrium of system (1), the stable manifold W0s and the instable manifold W0u of z=0 are m-dimensional and n-dimensional, respectively. Moreover, Df(0) has simple eigenvalues λ1, -ρ1, such that, any remaining eigenvalue σ of Df(0) satisfies either Reσ<-ρ0<-ρ1<0 or Reσ>λ0>λ1>0, for some positive numbers λ0 and ρ0.(H2)
(Nondegeneration) System (1) has a homoclinic loop Γ={z=r(t):t∈R,r(±∞)=0}, for any P∈Γ, codim (TPWPu+TPWPs)=1.(H3)
(Strong inclination)(2)limt→+∞(Tr(t)Wr(t)s+Tr(t)Wr(t)u)=T0W0s⊕T0W0uu,limt→-∞(Tr(t)Wr(t)s+Tr(t)Wr(t)u)=T0W0ss⊕T0W0u,where,W0ss and W0uu are the strong stable manifold and the strong instable manifold of z=0, respectively, T0W0ss is the generalized eigenspace corresponding to those eigenvalues with smaller real part than -ρ0, and T0W0uu is the generalized eigenspace corresponding to those eigenvalues with larger real part than λ0. Let e±=limt→∓∞r˙(t)/|r˙(t)|, e+∈T0W0u and e-∈T0W0s be unit eigenvectors corresponding to λ1 and -ρ1, respectively. span(T0W0uu,e+)=T0W0u and span(T0W0ss,e-)=T0W0s.Now, we consider the bifurcation problems of the followingCr perturbation system:
(3)z˙=f(z)+g(z,μ),
where μ∈Rl, l≥2, 0≤|μ|≪1, and g(0,μ)=g(z,0)=0.


## 3. Local Coordinates

Suppose that the neighborhoodU of z=0 is small enough and (H1)~(H3) are established, then, for |μ| is small enough, we can introduce a Cr change such that system (3) has the following form in U:
(4)x˙=[λ1(μ)+⋯]x+h.o.t.,y˙=[-ρ1(μ)+⋯]y+h.o.t.,u˙=[B1(μ)+⋯]u+h.o.t.,v˙=[-B2(μ)+⋯]v+h.o.t.,
where λ1(0)=λ1, ρ1(0)=ρ1, Reσ(B1(0))>λ0, Reσ(-B2(0))<-ρ0, z=(x,y,u*,v*)*, x∈R1, y∈R1, u∈Rn-1, v∈Rm-1, and * means transposition. Moreover, in U, we suppose that the instable manifold, the stable manifold, the strong instable manifold, the strong stable manifold and the local homoclinic orbits have the following forms, respectively. 						(5)Wlocu={y=0,v=0},Wlocs={x=0,u=0},Wlocuu={x=x(u),y=0,v=0},Wlocss={x=0,u=0,y=y(v)},Γ∩Wlocu={y=0,v=0,u=u(x)},Γ∩Wlocs={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.Denoter(t)=(rx(t),ry(t),(ru(t))*,(rv(t))*)*. Taking a time transformation if necessary, we may assume r(-T)=(δ,0,δu*,0*)*, r(T)=(0,δ,0*, δv*)*, where δ is small enough such that {(x,y,u,v):|x|,|y|,|u|,|v|<2δ}⊂U, |δu|=O(δω), |δv|=O(δω), ω=min{Reσ(B2(μ))/ρ1(μ),Reσ(B1(μ))/λ1(μ)}>1.Consider the linear system(6)z˙=(Df(r(t)))z.Similar to [9–11], system (6) has a fundamental solution matrix Z(t)=(z1(t),z2(t),z3(t),z4(t)), satisfying
(7)z1(t)∈(Tr(t)Ws)c∩(Tr(t)Wu)c,z2(t)=-r˙(t)|r˙y(T)|∈(Tr(t)Ws)∩(Tr(t)Wu),z3(t)=(z31(t),…,z3n-1(t))∈(Tr(t)Ws)c∩(Tr(t)Wu)=Tr(t)Wuu,z4(t)=(z41(t),…,z4m-1(t))∈(Tr(t)Ws)∩(Tr(t)Wu)c=Tr(t)Wss,(8)z1(T)=(1,0,0*,w14*)*,z2(T)=(0,1,0*,w24*)*,z3(T)=(w31*,w32*,w33*,w34*)*,z4(T)=(0,0,0*,I)*,z1(-T)=(w11,w12,w13*,0*)*,z2(-T)=(w21,0,w23*,0*)*,z3(-T)=(0,0,I,0*)*,z4(-T)=(w41*,w42*,w43*,w44*)*,
where w21<0, w12≠0, detw33≠0, detw44≠0, and for δ small enough, |w1iw12-1|≪1, i≠2; |w2iw21-1|≪1, i=3,4; |w3iw33-1|≪1, i≠3; |w4iw44-1|≪1, i≠4.Thus, we may select(z1(t),z2(t),z3(t),z4(t)) as a local coordinate system along Γ.DenoteΦ(t)=(ϕ1(t),ϕ2(t), ϕ3(t),ϕ4(t))=(Z-1(t))*. So, Φ(t) is a fundamental solution matrix of the adjoint system ϕ˙=-(Df(r(t)))*ϕ of (6). And, ϕ1(t)∈(Tr(t)Ws)c∩(Tr(t)Wu)c is bounded and tends to zero exponentially as t→±∞ [8, 9, 12, 13].Denotew12=Δ|w12|,
(9)Δ={1,w12>0-1,w12<0.We say thatΓ is nontwisted as Δ=1, and twisted as Δ=-1. In this paper, we only consider the nontwisted case.


## 4. Poincaré Maps and Bifurcation Equations

DenoteN=(n1,0,n3*,n4*)*,  n3=(n31,…,n3n-1)*,  n4=(n41,…,n4m-1)*, h(t)=r(t)+Z(t)N=r(t)+z1(t)n1+z3(t)n3+z4(t)n4. Let S0={z=h(T):|x|,|y|,|u|,|v|<2δ}, S1={z=h(-T):|x|,|y|,|u|,|v|<2δ} be the cross sections of Γ at t=T and t=-T, respectively, where, δ is small enough such that S0, S1⊂U Figure 1.Figure 1Now, we set up Poincaré mapF=F1∘F0:S0↦S0, where, the map F0:S0↦S1 is definited by the approximate solution of (3) in the small neighborhood U of z=0 (similarly, we can set up the map by the flow of the linear system of (4) in the small neighborhood U of z=0), F1:S1↦S0 is set up by the solution manifold of (3) in the tubular neighborhood of Γ.


### 4.1. The Relationship between the Two Kinds of Coordinates

Denoteq2j∈S0, q2j+1∈S1, j=0,1,2,…, F0(q2j)=q2j+1, and F1(q2j+1)=q2j+2. Now, we can set up the relationship between the two kinds coordinates (xi,yi,ui*,vi*)* and (ni1,0,ni3*,ni4*)* of qi, i=0,1,2,…. Let
(10)q2j=r(T)+Z(T)N2j,q2j+1=r(-T)+Z(-T)N2j+1,
where N2j=(n2j,1,0,n2j,3*,n2j,4*)*, N2j+1=(n2j+1,1,0,n2j+1,3*,n2j+1,4*)*, j=0,1,2,….Using the expressions ofZ-1(T), Z-1(-T), and by simple calculation, we have y2j≈δ, x2j+1≈δ, and
(11)n2j,1=x2j-w31w33-1u2j,n2j,3=w33-1u2j,n2j,4=-w14x2j+(w14w31+w24w32-w34)w33-1u2j+v2j-δv,(12)n2j+1,1=w12-1(y2j+1-w42w44-1v2j+1),n2j+1,3=u2j+1-δu+(w11w23w21-1-w13)w12-1y2j+1+aw44-1v2j+1,n2j+1,4=w44-1v2j+1,
where a=-w43+w13w12-1w42-w23w21-1[-w41+w11w12-1w42].


### 4.2. Set UpF1

Substitutez=h(t) into (3), and use r˙(t)=f(r(t)), Z˙(t)=Df(r(t))Z(t), we have
(13)Z(t)(n˙1,0,n˙3*,n˙4*)*=gμ(r(t),0)μ+h.o.t..Multiplying both sides byΦ*(t), and using Φ*(t)Z(t)=I, we have (n˙1,0,n˙3*,n˙4*)*=Φ*(t)gμ(r(t),0)μ+h.o.t., that is,
(14)n˙i=ϕi*(t)gμ(r(t),0)μ+h.o.t.,i=1,3,4.Thus, from the flow of (14), we define the map F1:S1↦S0, N(-T)↦N(T) which has the following form
(15)ni(T)=ni(-T)+Miμ+h.o.t.,i=1,3,4,
where Mi=∫-∞+∞‍ϕi*(t)gu(r(t),0)dt, i=1,3,4, is said to the Melnikov vectors [8, 9, 14, 15].


### 4.3. Set UpF0

Now, we consider the mapF0:S0↦S1, q0↦q1 defined by the orbit of (4). For convenience, we assume that ρ1≥λ1. Obviously, for μ small enough, Reσ(B2(μ))>ρ0>λ1(μ).Letτ be the flying time from q0 to q1, and s=e-λ1(μ)τ be called Silnikov time. By (4), in U, we get
(16)x=eλ1(μ)(t-T-τ)x1+∫T+τt‍(h.o.t.)eλ1(μ)(t-ξ)dξ,y=e-ρ1(μ)(t-T)y0+∫Tt‍(h.o.t.)e-ρ(μ)(t-ξ)dξ,u=eB1(μ)(t-T-τ)u1+∫T+τt‍(h.o.t.)eB1(μ)(t-ξ)dξ,v=e-B2(μ)(t-T)v0+∫Tt‍(h.o.t.)e-B2(μ)(t-ξ)dξ.Neglecting the higher order terms, the Poincaré mapF0: q0↦q1 from S0 to S1 defined by (16) as follows:
(17)x0≈sδ,y1≈sρ1(μ)/λ1(μ)δ,u0≈sB1(μ)/λ1(μ)u1,v1≈sB2(μ)/λ1(μ)v0;(
s
,
u
1
,
v
0
) is called Silnikov coordinate.


### 4.4.F and Bifurcation Equation

LetF1(q1)=q2, by (15) we have
(18)n2,j=n1,j+Mjμ+h.o.t.,j=1,3,4.Substituting (11), (12), and (17) into (18), and neglecting the higher order terms, the Poincaré map F=F1∘F0:q0∈S0↦q2∈S0 is given by
(19)n2,1=w12-1δsρ1(μ)/λ1(μ)+M1μ+h.o.t.,n2,3=u1-δu+(w11w23w21-1-w13)w12-1δsρ1(μ)/λ1(μ)+M3μ+h.o.t.,n2,4=w44-1sB2(μ)/λ1(μ)v0+M4μ+h.o.t..So, from the above and (11), we have the successor function G(s,v0,u1)=(G1,G3,G4)=F(q0)-q0=(n2,1,0,n2,3*,n2,4*)*-(n0,1,0,n0,3*,n0,4*)* as follows:
(20)G1=δ(w12-1sρ1(μ)/λ1(μ)-s)+M1μ+h.o.t.,G3=u1-δu+(w11w23w21-1-w13)w12-1δsρ1(μ)/λ1(μ)-w33-1sB1(μ)/λ1(μ)u1+M3μ+h.o.t.,G4=-v0+δv+w14δs+w44-1sB2(μ)/λ1(μ)v0+M4μ+h.o.t..Thus, fors≥0, there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of (3) and the solution Q=(s,u1,v0) of the following equation:
(21)(G1,G3,G4)=0.Equation (21) is called bifurcation equation.


## 4.1. The Relationship between the Two Kinds of Coordinates

Denoteq2j∈S0, q2j+1∈S1, j=0,1,2,…, F0(q2j)=q2j+1, and F1(q2j+1)=q2j+2. Now, we can set up the relationship between the two kinds coordinates (xi,yi,ui*,vi*)* and (ni1,0,ni3*,ni4*)* of qi, i=0,1,2,…. Let
(10)q2j=r(T)+Z(T)N2j,q2j+1=r(-T)+Z(-T)N2j+1,
where N2j=(n2j,1,0,n2j,3*,n2j,4*)*, N2j+1=(n2j+1,1,0,n2j+1,3*,n2j+1,4*)*, j=0,1,2,….Using the expressions ofZ-1(T), Z-1(-T), and by simple calculation, we have y2j≈δ, x2j+1≈δ, and
(11)n2j,1=x2j-w31w33-1u2j,n2j,3=w33-1u2j,n2j,4=-w14x2j+(w14w31+w24w32-w34)w33-1u2j+v2j-δv,(12)n2j+1,1=w12-1(y2j+1-w42w44-1v2j+1),n2j+1,3=u2j+1-δu+(w11w23w21-1-w13)w12-1y2j+1+aw44-1v2j+1,n2j+1,4=w44-1v2j+1,
where a=-w43+w13w12-1w42-w23w21-1[-w41+w11w12-1w42].


## 4.2. Set UpF1

Substitutez=h(t) into (3), and use r˙(t)=f(r(t)), Z˙(t)=Df(r(t))Z(t), we have
(13)Z(t)(n˙1,0,n˙3*,n˙4*)*=gμ(r(t),0)μ+h.o.t..Multiplying both sides byΦ*(t), and using Φ*(t)Z(t)=I, we have (n˙1,0,n˙3*,n˙4*)*=Φ*(t)gμ(r(t),0)μ+h.o.t., that is,
(14)n˙i=ϕi*(t)gμ(r(t),0)μ+h.o.t.,i=1,3,4.Thus, from the flow of (14), we define the map F1:S1↦S0, N(-T)↦N(T) which has the following form
(15)ni(T)=ni(-T)+Miμ+h.o.t.,i=1,3,4,
where Mi=∫-∞+∞‍ϕi*(t)gu(r(t),0)dt, i=1,3,4, is said to the Melnikov vectors [8, 9, 14, 15].


## 4.3. Set UpF0

Now, we consider the mapF0:S0↦S1, q0↦q1 defined by the orbit of (4). For convenience, we assume that ρ1≥λ1. Obviously, for μ small enough, Reσ(B2(μ))>ρ0>λ1(μ).Letτ be the flying time from q0 to q1, and s=e-λ1(μ)τ be called Silnikov time. By (4), in U, we get
(16)x=eλ1(μ)(t-T-τ)x1+∫T+τt‍(h.o.t.)eλ1(μ)(t-ξ)dξ,y=e-ρ1(μ)(t-T)y0+∫Tt‍(h.o.t.)e-ρ(μ)(t-ξ)dξ,u=eB1(μ)(t-T-τ)u1+∫T+τt‍(h.o.t.)eB1(μ)(t-ξ)dξ,v=e-B2(μ)(t-T)v0+∫Tt‍(h.o.t.)e-B2(μ)(t-ξ)dξ.Neglecting the higher order terms, the Poincaré mapF0: q0↦q1 from S0 to S1 defined by (16) as follows:
(17)x0≈sδ,y1≈sρ1(μ)/λ1(μ)δ,u0≈sB1(μ)/λ1(μ)u1,v1≈sB2(μ)/λ1(μ)v0;(
s
,
u
1
,
v
0
) is called Silnikov coordinate.


## 4.4.F and Bifurcation Equation

LetF1(q1)=q2, by (15) we have
(18)n2,j=n1,j+Mjμ+h.o.t.,j=1,3,4.Substituting (11), (12), and (17) into (18), and neglecting the higher order terms, the Poincaré map F=F1∘F0:q0∈S0↦q2∈S0 is given by
(19)n2,1=w12-1δsρ1(μ)/λ1(μ)+M1μ+h.o.t.,n2,3=u1-δu+(w11w23w21-1-w13)w12-1δsρ1(μ)/λ1(μ)+M3μ+h.o.t.,n2,4=w44-1sB2(μ)/λ1(μ)v0+M4μ+h.o.t..So, from the above and (11), we have the successor function G(s,v0,u1)=(G1,G3,G4)=F(q0)-q0=(n2,1,0,n2,3*,n2,4*)*-(n0,1,0,n0,3*,n0,4*)* as follows:
(20)G1=δ(w12-1sρ1(μ)/λ1(μ)-s)+M1μ+h.o.t.,G3=u1-δu+(w11w23w21-1-w13)w12-1δsρ1(μ)/λ1(μ)-w33-1sB1(μ)/λ1(μ)u1+M3μ+h.o.t.,G4=-v0+δv+w14δs+w44-1sB2(μ)/λ1(μ)v0+M4μ+h.o.t..Thus, fors≥0, there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of (3) and the solution Q=(s,u1,v0) of the following equation:
(21)(G1,G3,G4)=0.Equation (21) is called bifurcation equation.


## 5. Nonresonant Bifurcations

(H4)
(Nonresonant condition)λ1<ρ1.Obviously, forμ small enough, we may assume ρ1(μ)>λ1(μ).Theorem 1.
Suppose that hypotheses (H1)~(H4) are valid, then if|μ| is small enough, the system (3) exists no more than one 1-homoclinic loop or one 1-periodic orbit in the neighbourhood of Γ. Moreover, the 1-homoclinic loop and the 1-periodic orbit cannot coexist.Proof.
Consider the solution of (21). Let G~=∂(G1,G3,G4)/∂Q. From (20), we have G~|Q=0,μ=0=diag(-δ,1,-δ)+(gij), where elements of (gij) are all zero except g31=w14δ. Therefore, detG~≠0. According to the implicit function theorem, in the neighbourhood of (Q,μ)=(0,0), (21) exists a unique solution
(22)s=s(μ),u1=u1(μ),v0=v0(μ),
which satisfies s(0)=0, u1(0)=0, v0(0)=0.
Ifs=0, then the solution (22) corresponds to a 1-homoclinic loop of the system (3), that is, the homoclinic loop Γ is persistent.
Ifs>0, then the solution (22) corresponds to a 1-periodic orbit of the system (3), that is, the homoclinic orbit Γ bifurcates to a periodic orbit.
The proof is complete.Theorem 2.
IfM1≠0, then there exists a (l-1)-dimensional surface L⊂Rl in the small neighbourhood of μ=0, such that when μ∈L and |μ|≪1, the system (3) exists a unique homoclinic loop near Γ. If M1μ>0, then the system (3) exists a unique periodic orbit near Γ. If M1μ<0, then the system (3) has no any homoclinic loop and periodic orbit near Γ. L is called bifurcation surface, its analytical expression is M1μ+h.o.t.=0.Proof.
From (20), for s≥0 and |μ| small enough, equations G3=0 and G4=0 always have a unique solution u1=u1(s,μ), v0=v0(s,μ). Substituting into G1=0, we get
(23)δ(w12-1sρ1(μ)/λ1(μ)-s)+M1μ+h.o.t.=0.
IfM1≠0, then, according to the implicit function theorem, in the neighbourhood of μ=0, the equation M1μ+h.o.t.=0 defines a unique (l-1)-dimensional surface L⊂Rl, such that if μ∈L and |μ|≪1, (23) has the solution s=0, the uniqueness can be obtained by the Theorem 1.
IfM1μ>0, then (23) has the small positive solution s=δ-1M1μ+h.o.t..
IfM1μ<0, then (23) has nonzero negative solution s=δ-1M1μ+h.o.t.. At present, the system (3) has neither homoclinic loop no periodic orbit near the neighbourhood of Γ.
The proof is complete.Now, we consider the nonexistence ofk-homoclinic loop and k-periodic orbit, where k>1. Firstly, we consider the case of k=2.We rewrite the time fromq0 and q1 as τ1, s1=e-λ1(μ)τ1. Suppose F0(q2)=q3, F1(q3)=q4=q0, and let τ2 be the time from q2 to q3, s2=e-λ1(μ)τ2.Similar to the previous discussion, we can get its associated successor functionG2=(G11,G31,G41,G12,G32,G42) as follows:
(24)G11=δ(w12-1s1ρ1(μ)/λ1(μ)-s2)+M1μ+h.o.t.,G31=u1-δu+(w11w23w21-1-w13)w12-1δs1ρ1(μ)/λ1(μ)-w33-1s2B1(μ)/λ1(μ)u3+M3μ+h.o.t.,G41=-v2+δv+w14δs2+w44-1s1B2(μ)/λ1(μ)v0+M4μ+h.o.t.,G12=δ(w12-1s2ρ1(μ)/λ1(μ)-s1)+M1μ+h.o.t.,G32=u3-δu+(w11w23w21-1-w13)w12-1δs2ρ1(μ)/λ1(μ)-w33-1s1B1(μ)/λ1(μ)u1+M3μ+h.o.t.,G42=-v0+δv+w14δs1+w44-1s2B2(μ)/λ1(μ)v2+M4μ+h.o.t..DenoteQ2=(s1,s2,u1,v0,u3,v2), G~2=∂(G12,G11,G31,G42,G32,G41)/∂Q2, then G~2|Q2=0,μ=0=diag(-δ,-δ,1,-1,1,-1)+(gij)n×n, where the elements of (gij) are all zero except g41=w14δ, g62=w14δ.Hence,detG~2|Q2=0,μ=0≠0. According to the implicit function theorem, near (Q2,μ)=(0,0), the bifurcation equation
(25)(G11,G31,G41,G12,G32,G42)=0
has a unique solution
(26)s1=s1(μ),u1=u1(μ),v0=v0(μ),s2=s2(μ),u3=u3(μ),v2=v2(μ),
which satisfies s1(0)=0, u1(0)=0, v0(0)=0, s2(0)=0, and u3(0)=0, v2(0)=0.Ifs1=s2=0, then the homoclinic loop of the system (3) which the solution (26) corresponds to is the 1-homoclinic loop.Because 1-periodic orbit obviously,corresponds to the solutions1=s2>0 of (26) then, by the uniqueness of solution, the system (3) has no 2-periodic orbit.Ifs1>0, s2=0, or s1=0, s2>0, then G11=0 and G12=0 will get the contradiction.And, so, for anyk>1, we have the following.Theorem 3.
Suppose that (H1)~(H4) are fulfilled, k>1, then the system (3) does not have any k-homoclinic loop and k-periodic orbit for |μ| sufficiently small.


## 6. Resonance Bifurcation

We say that the homoclinic loop is Resonance ifλ1=ρ1. For convenience, we assume the resonant condition has the following form.(H5)
(Resonant condition)λ1=ρ1=λ, λ1(μ)=λ, ρ1(μ)=λ+α(μ)λ, where α(μ)∈R1, |α(μ)|≪1, and α(0)=0.At first, we discuss the bifurcations of 1-homoclinic loop and 1-periodic orbit. Now, the bifurcation equation has the following form:(27)G1=δ(w12-1s(1+α(μ))-s)+M1μ+h.o.t.=0,G3=u1-δu+(w11w23w21-1-w13)w12-1δs(1+α(μ))-w33-1sB1(μ)/λu1+M3μ+h.o.t.=0,G4=-v0+δv+w14δs+w44-1sB2(μ)/λv0+M4μ+h.o.t.=0.
Similarly for s≥0, there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of (3) and the solution Q=(s,u1,v0) of the bifurcation equation (27). It is Easy to see that, for the sufficiently small s≥0 and |μ|, equations G3=0, G4=0 of (27) always have a unique solution u1=u1(s,μ), v0=v0(s,μ). Substituting it into G1=0, we get
(28)s1+α(μ)=w12(s-δ-1M1μ)+h.o.t..DenoteN(s)=s1+α(μ), V(s)=w12(s-δ-1M1μ)+h.o.t., we have the following conclusion.Lemma 4.
Suppose (H1)~(H3) and (H5) are fulfilled, then, for 0<s0,|μ|≪1, the necessary condition of N(s) and V(s) are tangent at s0 is that α(μ)M1μ>0. Meanwhile, if Δ=1 and 0<s0≪1, N(s), and V(s) are tangent at s0 if and only if α(μ)(1-w12)>0, and
(29)M1μ=β1(μ)∶=δα(μ)(1+α(μ))-1-(1/α(μ))(w12)1/α(μ)+h.o.t..Proof.
N
(
s
) and V(s) are tangent at s=s0 if and only if N(s0)=V(s0), N˙(s0)=V˙(s0), that is,
(30)s01+α(μ)=w12(s0-δ-1M1μ)+h.o.t.,(1+α(μ))s0α(μ)=w12.
Solving the above equations, we haves0=((1+α(μ))/α(μ))δ-1M1μ+h.o.t.. Substituting it into (30), (29) is fulfilled.
The proof is complete.SupposeΣ1 is the surface defined by (29), Σ¯2(s) is the surface defined by (28), Σ2=Σ¯2(0). Besides, if μ∈Σ2, (28) is turned to M1μ=β2(μ).Theorem 5.
Suppose (H1)~(H3) and (H5) are fulfilled, α(μ)>0, 0<w12<1. If M1≠0, then, in the neighborhood of μ=0, there exists two (l-1)-dimensional surfaces Σ1 and Σ2, such that, for sufficiently small |μ|, we have the following.(i)
The system (3) has a unique 2-fold 1-periodic orbit near Γ if and only if μ∈Σ1.(ii)
The system (3) has no 1-homoclinic orbit and 1-periodic orbit near Γ if and only if μ satisfies M1μ>β1(μ).(iii)
The system (3) has exactly two 1-periodic orbits near Γ if and only if μ satisfies β2(μ)<M1μ<β1(μ).(iv)
The system (3) has exactly a 1-homoclinic orbit and a 1-periodic orbit near Γ if and only if μ∈Σ2.(v)
The system (3) has exactly a unique 1-periodic orbit near Γ if and only if μ satisfies M1μ<β2(μ).Proof.
Through taking a proper scale transformation, such ass=|μ|2, we can treat s as a small parameter in (28). According to M1≠0 and the implicit equation theorem, near μ=0, (28) can define a (l-1)-dimensional surface Σ¯2(s), such that if μ∈Σ2∶=Σ¯2(0), (28) has exactly two negative small solutions s1=0 and s2=(w12)1/α(μ)+h.o.t.>0.
If-1≪M1μ<β2(μ), then (28) has exactly a unique negative small solution s1>0.
IfM1μ=β1(μ), that is μ∈Σ1, then (28) has exactly two negative small solutions s1=s2>0.
IfM1μ>β1(μ), then (28) has no negative small solution.
Ifβ2(μ)<M1μ<β1(μ), then (28) has exactly two negative small solutions s1>0 and s2>0, moreover, s1≠s2.
The proof is complete.Similarly, we can define the correspondingβ1(μ) and β2(μ), and the corresponding Σ1 and Σ2, to obtain the following theorem.Theorem 6.
Suppose (H1)~(H3) and (H5) are fulfilled,α(μ)<0, w12>1. If M1≠0, then, in the neighborhood of μ=0, there exists two (l-1)-dimensional surfaces Σ1 and Σ2 such that for sufficiently small |μ|, the following conclusions hold.(i)
The system (3) has a unique 2-fold 1-periodic orbit near Γ if and only if μ∈Σ1.(ii)
The system (3) has no 1-homoclinic loop and 1-periodic orbit near Γ if and only if μ satisfies M1μ<β1(μ).(iii)
The system (3) has exactly two 1-periodic orbits near Γ if and only if μ satisfies β2(μ)>M1μ>β1(μ).(iv)
The system (3) has exactly a unique 1-homoclinic loop and 1-periodic orbit near Γ if and only if μ∈Σ2.(v)
The system (3) has exactly a 1-periodic orbit near Γ if and only if μ satisfies M1μ>β2(μ).Σ
1 is called 2-fold 1-periodic orbit bifurcation surface, Σ2 is called 1-homoclinic loop bifurcation surface Figure 2.Figure 2Now, we consider the nonexistence ofk-homoclinic loop and k-periodic orbit, where k>1. We may assume that k=2.We rewrite the time fromq0 and q1 as τ1, s1=e-λτ1. Suppose F0(q2)=q3, F1(q3)=q4=q0, let τ2 be the time from q2 to q3, s2=e-λτ2. Similar to the previous discussion, we can get its associated successor function G2=(G11,G31,G41,G12,G32,G42) as follows:
(31)G11=δ(w12-1s1(1+α(μ))-s2)+M1μ+h.o.t.,G31=u1-δu+(w11w23w21-1-w13)w12-1δs1(1+α(μ))-w33-1s2B1(μ)/λu3+M3μ+h.o.t.,G41=-v2+δv+w14δs2+w44-1s1B2(μ)/λv0+M4μ+h.o.t.,G12=δ(w12-1s2(1+α(μ))-s1)+M1μ+h.o.t.,G32=u3-δu+(w11w23w21-1-w13)w12-1δs2(1+α(μ))-w33-1s1B1(μ)/λu1+M3μ+h.o.t.,G42=-v0+δv+w14δs1+w44-1s2B2(μ)/λv2+M4μ+h.o.t..DenoteQ2=(s1,s2,u1,v0,u3,v2). The equation (G31,G41,G32,G42)=0 always has a unique solution u1=u1(s1,s2,μ),  v0=v0(s1,s2,μ),  u3=u3(s1,s2,μ),  v2=v2(s1,s2,μ), substituting it into (G11,G12)=0, we get
(32)δ[(w12)-1s1(1+α(μ))-s2]+M1μ+h.o.t.=0,δ[(w12)-1s2(1+α(μ))-s1]+M1μ+h.o.t.=0.
If (32) has solution s1=s2=0, then (32) is turned to
(33)M1μ+h.o.t.=0.IfM1≠0, the above formula defines a (l-1)-dimensional surface L, now the 2-homoclinic loop is the 1-homoclinic loop.If (32) has the solution s1>0, s2=0, then (32) is turned to
(34)s1=δ-1M1μ+h.o.t.,s11+α(μ)=-δ-1w12M1μ+h.o.t..Easily, the necessary condition that (34) has solutions s1>0, s2=0 is M1>0, Δ=-1, that is, Γ is twisted. (32) is turned to
(35)(δ-1M1μ+h.o.t.)1+α(μ)=-δ-1w12M1μ+h.o.t..The above formula defines(l-1)-dimensional surface L1, such that when |μ|≪1 and μ∈L1, the system (3) has a unique 2-homoclinic loop.If (32) has the solution s1>0, s2>0, then (32) turns to
(36)(w12-1s21+α(μ)+δ-1M1μ)1+α(μ)=(w12)(s2-δ-1M1μ)+h.o.t..LetN(s2) and V(s2) be the left and right of the above formula, respectively, then N(s2) and V(s2) are tangent at some point if and only if (36) and the following formulas are fulfilled. 						(37)(1+α(μ))2(s21+α(μ)+δ-1w12M1μ)α(μ)s2α(μ)=(w12)2+α(μ)+h.o.t..Thus, we can get(38)s2=[-w12(1+α(μ))2]1/α(μ)+h.o.t.,Obviously, the necessary condition that (38) has solutions is Δ=-1 and (1+w12)α(μ)>0, that is, Γ is twisted. At present, s2 satisfies 0<s2<δ-1M1μ≪1. Substituting (38) into (37), we get
(39)δ-1M1μ=β*(μ):=(-w12)1/α(μ)[1+(1+α(μ))-2-(2/α(μ))]+h.o.t..Suppose the surface defined by (35) is δ-1M1μ=β*(μ). We notice that every 1-periodic orbit also corresponds to a solution s1=s2>0 of (32). Thus, we have the following theorem.Theorem 7.
Suppose (H1)~(H3) and (H5) are fulfilled, ifΔ=1, k>1, then system (3) has no any k-homoclinic loop and k-periodic orbit in the small neighborhood of  Γ for |μ| sufficiently small.


## 7. Stability

Now, we consider the stability of homoclinic loopΓ.According toF(q0)=F(n0,1,0,n0,3*,n0,4*)*=q2=(n2,1,0,n2,3*,n2,4*)*, if μ=0, and ρ1≠λ1, we can get the following by (19)
(40)n2,1=w12-1δsρ1/λ1+h.o.t.,n2,3=u1-δu+(w11w23w21-1-w13)w12-1δsρ1/λ1+h.o.t.,n2,4=w44-1sB2(0)/λ1v0+h.o.t..By (17), we get s≈δ-1x0, u1≈s-B1(0)/λ1u0, substituting into (40), we get
(41)n2,1=w12-1(δ-1x0)ρ1/λ1-1x0+h.o.t.,n2,3=s-B1(0)/λ1u0-δu+(w11w23w21-1-w13)w12-1δsρ1/λ1+h.o.t.,n2,4=w44-1sB2(0)/λ1v0+h.o.t..According to (11), we know x0≈n0,1, u0=w33n0,3, v0=O(n0,4), x2≈n2,1, u2=w33n2,3, v2=O(n2,4). And by Reσ(-B1(0))<0, ReσB2(0)>0, s≪1, we can get u2=O(s-B1(0)/λ1u0)≫u0, v2=O(sB2(0)/λ1v0)≪v0. Meanwhile, we make Poincaré map F restrict at the half transversal section S0+={(x,y,u,v)∈S0,0≤x<δ1<δ}, F makes the transversal line Lx={0≤x<δ1,y=δ,u=0,v=0} maps to the segment Lx′={0≤x<γδ1,y=δ,u=0,v=0} approximately, where γ is called the shrinkage (expansion) rate. So, when ρ1/λ1>1(<1), we can get γ=w12-1(δ-1x0)ρ1/λ1-1<1(>1) for δ≪1, x0≪1. Hence, we have the following.Theorem 8.
Ifρ1/λ1>1, homoclinic loop Γ is weak stability, and Γ has a (m+1)-dimensional stable manifold and a n-dimensional instable manifold, If ρ1/λ1<1, homoclinic loop Γ is weak instability, and Γ has a m-dimensional stable manifold and a (n+1)-dimensional instable manifold.
Ifμ=0 and ρ1=λ1, similar to above, we can get γ=w12-1 not difficulty. Thus, we have the following.Theorem 9.
Ifw12-1<1, homoclinic loop Γ is weak stable, and Γ has a (m+1)-dimensional stable manifold and a n-dimensional instable manifold. If w12-1>1, homoclinic loop Γ is weak instable, and Γ has a m-dimensional stable manifold and a (n+1)-dimensional instable manifold.Besides, by the above discussion, we can get the following.Theorem 10.
The homoclinic loop or the periodic orbit of the perturbed system have the same stability with the homoclinic loop of the unperturbed system.


## 8. Example

Now, suppose theCr system (3) is 2-dimensional, z=(x,y)∈R2, r≥5. We consider the stability of the homoclinic orbit Γ under the resonance case. Denote f(z)=(f1(z),f2(z))*, σ=exp{∫‍-∞+∞(∂f1/∂x+∂f2/∂y)(r(t))dt}, σ-=exp{∫‍-TT(∂f1/∂x+∂f2/∂y)(r(t))dt}, We have the following two lemmas.Lemma 11.
σ is convergent, and σ=σ-.Proof.
In fact, from [1–3], there always exist a Cr transformation coordinates, such that the system (3) has the following form in a small neighborhood of the origin:
(42)x˙=λx+x2yH1(x,y),y˙=-λy+xy2H2(x,y),
where H1(x,y), H2(x,y) is Cr-3. So, in U, we have Γ∩W0s={(x,y):x=0},Γ∩W0u={(x,y):y=0}. Thus, in U, if (x,y)∈Γ, (∂f1/∂x)+(∂f2/∂y)=xyH(x,y)=0.
The proof is complete.Remark 12.
Because the divergence integration is the invariant under theC2 transformation (refer [4, 16]), so the function f(·) and r(t) of the divergence integration σ can be thought of the original forms of (3).Lemma 13.
w
21
=
-
1
,
w
12
=
1
/
σ.Proof.
According tor˙(T)=(0,-λδ), r˙(-T)=(λδ,0), we get z2(-T)=-r˙(-T)/|r˙y(T)|=(-1,0), so, w21=-1. And by the Liouville formula, we have |1001|=|w11w21w120|σ, therefore, -w12w21σ=1, and by w21=-1, we get w12=1/σ.
The proof is complete.Combined with the Theorem9, we have the following.Theorem 14.
Ifσ<1, the homoclinic orbit Γ is stable; If σ>1, the homoclinic orbit Γ is instable.



---

*Source: 582820-2013-11-12.xml*