# Bifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria

Authors: Fengjie Geng; Junfang Zhao
Journal: The Scientific World Journal
 (2014)

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## Abstract

The bifurcations of
heteroclinic loop with one nonhyperbolic equilibrium and one
hyperbolic saddle are considered, where the nonhyperbolic
equilibrium is supposed to undergo a transcritical bifurcation;
moreover, the heteroclinic loop has an orbit flip and an inclination
flip. When the nonhyperbolic equilibrium does not undergo a
transcritical bifurcation, we establish the coexistence and
noncoexistence of the periodic orbits and homoclinic orbits. While
the nonhyperbolic equilibrium undergoes the transcritical
bifurcation, we obtain the noncoexistence of the periodic orbits and
homoclinic orbits and the existence of two or three heteroclinic
orbits.


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## Body

## 1. Introduction

In recent years, a great deal of mathematical efforts has been devoted to the bifurcation problems of homoclinic and heteroclinic orbits with high codimension, for example, the bifurcations of homoclinic or heteroclinic loop with orbit flip, the bifurcations of homoclinic or heteroclinic loop with inclination flip, and so forth; see [1–5] and the references therein. However, most of these papers considered the bifurcation problems of orbits connecting hyperbolic equilibria, and limited work has been done in the corresponding problems with nonhyperbolic equilibria; see [6–8]. To fill this gap, we investigate the bifurcations of orbit and inclination flip heteroclinic orbits with one nonhyperbolic equilibrium and one hyperbolic saddle. The method is using the fundamental solution matrix of the linear variational system to obtain the Poincarémap, which is easier to get the bifurcation equations.Consider the followingCr (r≥5) system
(1)z˙=g(z,λ,μ)
and its unperturbed system
(2)z˙=f(z),
where z∈ℝ4, the vector fieldgdepends on the parameters (λ,μ), λ∈ℝ, μ∈ℝl, l≥2, 0≤λ, |μ|≪1, g(z,0,0)=f(z), g(p1,0,μ)=0, and g(p2,λ,μ)=0. Moreover, the parameterλgoverns bifurcation of the nonhyperbolic equilibrium, whileμcontrols bifurcations of the heteroclinic orbits.Assuming system (2) has a heteroclinic loop Γ connecting its two equilibria p1, p2, where Γ=Γ1⋃‍Γ2, Γi={z=ri(t):t∈ℝ}, ri(+∞)=ri+1(-∞)=pi+1, i=1,2, r3(t)=r1(t), and p3=p1. Furthermore, the linearizationDf(p1)has real eigenvalues0, λ11, -ρ11, and-ρ12satisfying-ρ12<-ρ11<0<λ11; Df(p2) has simple real eigenvalues λ21, λ22, -ρ21, and -ρ22 fulfilling -ρ22<-ρ21<0<λ21<λ22.The following conditions hold in the whole paper:(H1)
(3)ei±=lim⁡t→±∞r˙i(-t)|r˙i(-t)|,where e1+∈Tp1W1cu, e1-∈Tp2W2ss, e2+∈Tp2W2u, e2-∈Tp1W1s, and e1-∈Tp2W2ss mean that Γ1 is a heteroclinic orbit with orbit flip, W1cu is the center unstable manifold of p1, Wiu (resp., Wis) is the unstable (resp., stable) manifold of pi, and Wiuu (resp., Wiss) is the strong unstable (resp., stable) manifold of pi, i=1,2. Moreover,
(4)dim⁡(Tr1(t)W1c∩Tr1(t)W2s)=dim⁡(Tr1(t)W1cu∩Tr1(t)W2s)=1.(H2)
(5)lim⁡t→+∞Tr1(t)W1cu=span⁡{e1-,Tp2W2uu},lim⁡t→+∞Tr2(t)W2u=span⁡{e2-,Tp1W1u},lim⁡t→-∞Tr1(t)W2s=span⁡{e1+,Tp1W1ss},lim⁡t→-∞Tr2(t)W1s=span⁡{e2+,Tp2W2s},where the first three equations mean that the center unstable manifold W1cu of p1, the stable (resp., unstable) manifold W2s (resp., W2u) ofp2are fulfilling the strong inclination property. And the fourth equation implies that the stable manifold W1s is of inclination flip as t→-∞.It is worthy of noting that, for any integersm≥1 and n≥1, if we assume dim⁡(W1u)=dim⁡(W2uu)=m and dim⁡(W1ss)=dim⁡(W2ss)=n, then all the results achieved in this paper are still valid.Letλ∈ℝ be a parameter to control the transcritical bifurcation of system (1), let the x-axis be the tangent space of the center manifold at p1, and let θ(x,λ,μ) be the vector field defined on the center manifold; then by [9], we may assume(H3)
θ(xp1,λ,μ)=0, (∂θ/∂x)(xp1,0,0)=0, (∂2θ/∂x2)(xp1,0,0)>0, (∂2θ/∂x∂λ)(xp1,0,0)<0, (∂2θ/∂x∂μ)(xp1,0,μ)=0, where xp1 is the x component of p1.If(H3) is true, then system (1) exhibits the transcritical bifurcation, that is, when λ>0 (or λ<0; in this paper, we only consider the case λ>0; for the case λ<0, one may discuss it similarly); there are two hyperbolic saddles p10 and p11 bifurcated from p1. Denote by p10=p1=(0,0,0,0)* and p11=p1+(λp,0,0,0)*, where λp=θ0λ+O(λ2)+O(λμ) and θ0=-(∂2θ/∂x∂λ)(xp1,0,0)/(∂2θ/∂x2)(xp1,0,0). Moreover, dim⁡(Wp10s)=3,dim⁡(Wp10u)=1, and dim⁡(Wp11u)=dim⁡(Wp11s)=2.The present paper is built up as follows. In Section2, we devote it to deriving the successor functions by constructing a suitable Poincaré Map. The analysis to the bifurcations of system (2) is presented in Section 3, where we establish the existence of the heteroclinic loop, the homoclinic orbits, and the three or two heteroclinic orbits and the coexistence of a periodic orbit and a homoclinic loop, and the difference between the heteroclinic loop with hyperbolic equilibria and nonhyperbolic equilibria is revealed.


## 2. Normal Form and Poincaré Map

Let the neighborhoodUi of pi be small enough and straight the local manifolds of Wis, W2uu, Wiss, and i=1,2 in the neighborhood Ui. And then by virtue of the invariance of these manifolds and a scale transformation x→θxx-1(xp1,0,0)x and λ→-θxλ-1(xp1,0,0)λ, system (1) has the following expression inU1:
(6)x˙=-λpx+x2+O(u)[O(y)+O(v)]+O(x)[O(y)+O(u)+O(v)]+O(x)O(x2),y˙=[-ρ11(α)+⋯]y+O(v)[O(x)+O(u)],u˙=[λ11(α)+⋯]u+O(x)[O(y)+O(v)],v˙=[-ρ12(α)+⋯]v+O(y)[O(x)+O(y)+O(u)],
and in U2 it takes the following form:
(7)x˙=[λ21(α)+⋯]x+O(u)[O(y)+O(v)],y˙=[-ρ21(α)+⋯]y+O(v)[O(x)+O(u)],u˙=[λ22(α)+⋯]u+O(x)[O(x)+O(y)+O(v)],v˙=[-ρ22(α)+⋯]v+O(y)[O(x)+O(y)+O(u)],
where α=(λ,μ), λp=λ+O(λ2)+O(λμ), λ11(0)=λ11, ρij(0)=ρij, j=1,2,i=1,2, λ2j(0)=λ2j,j=1,2.From the normal form (6), (7), and the condition (H1), we may select -Ti and Ti such that
(8)r1(-T1)=(δ,0,0,0)*,r1(T1)=(0,0,0,δ)*,r1(-T2)=(δ,0,0,δu,0)*,r2(T2)=(0,δ,0,δv)*,
where δ>0 is small enough such that {(x,y,u,v):|x|,|y|,|u|,|v|<2δ}⊂Ui and |δu|=o(δ), |δv|=o(δ).Consider the linear variational system(9)iz˙=Df(ri(t))z
and its adjoint system
(10)iϕ˙=-(Df(ri(t)))*ϕ,i=1,2, where (Df(ri(t)))* is the transposed matrix of Df(ri(t)).SupposingZi(t)=(zi1(t),zi2(t),zi3(t),zi4(t)) is a fundamental solution matrix of  (9)i, then, we arrive at the following lemma.Lemma 1.
If conditions (H1)–(H3) are satisfied, then
(1) there exists a fundamental solution matrix of(9)1 satisfying(11)z11(t)∈(Tr1(t)W1cu)c∩(Tr1(t)W2s)c,z12(t)=-r˙1(t)|r˙1(T1)|∈Tr1(t)W1c∩Tr1(t)W2s,z13(t)∈Tr1(t)W1cu∩(Tr1(t)W2s)c,z14(t)∈(Tr1(t)W1cu)c∩Tr1(t)W2s
such that
(12)Z1(-T1)=(w111w1210w141w11200w142w113w1231w143000w144),Z1(T1)=(10w1310w~1120w132100w133001w1340);
(2)(9)2 has a fundamental solution matrix fulfilling
(13)z21(t)∈(Tr2(t)W2u)c∩(Tr2(t)W1s)c,z22(t)=-r˙2(t)|r˙2(T2)|∈Tr2(t)W2u∩Tr2(t)W1s,z23(t)∈Tr2(t)W2u∩(Tr2(t)W1s)c,z24(t)∈(Tr2(t)W2u)c∩Tr2(t)W1s
such that
(14)Z2(-T2)=(w211w2210w241000w242w213w2231w243w214000),Z2(T2)=(10w231001w232000w2330w~214w224w2341),
where wi21<0, w112wi33w214w242≠0, |(wi33)-1wi3j|≪1, j=1,2,4, i=1,2.Now, let(zi1(t),zi2(t),zi3(t),zi4(t)) be a new local active coordinate system along Γi. Given Φi(t)=(ϕi1(t),ϕi2(t),ϕi3(t),ϕi4(t))=(Zi-1(t))*, then Φi(t) is the fundamental solution matrix of (10)i, i=1,2.Letz=ri(t)+Zi(t)Ni(t)≜hi(t), where Ni(t)=(ni1,0,ni3,ni4)*, i=1,2. Defining the cross sections
(15)Si0={z=hi(-Ti):|x|,|y|,|u|,|v|<2δ},Si1={z=hi(Ti):|x|,|y|,|u|,|v|<2δ}
of  Γi at t=-Tiand t=Ti, respectively, i=1,2.Now that ifqi0∈Si0 and qi1∈Si1, then
(16)qi0=(xi0,yi0,ui0,vi0)*=ri(-Ti)+Z1(-Ti)Ni(-Ti),Ni(-Ti)=(ni0,1,0,ni0,3,ni0,4)*,qi1=(xi1,yi1,ui1,vi1)*=ri(Ti)+Zi(Ti)Ni(Ti),Ni(Ti)=(ni1,1,0,ni1,3,ni1,4)*.
Based on the expressions of Zi(-Ti) and Zi(Ti), we get their new coordinates of qi0(ni0,1,0,ni0,3,ni0,4)* and qi1(ni1,1,0,ni1,3,ni1,4)*; that is,
(17)n10,1=(w112)-1[y10-w142(w144)-1v10],n10,3=u10-w113(w112)-1y10+[w113w142(w112)-1-w143](w144)-1v10,n10,4=(w144)-1v10,x10=δ+w111n10,1+w141n10,4≈δ,n11,1=x11-w131(w133)-1u11,n11,3=(w133)-1u11,n11,4=y11-w~112x11+(w~112w131-w132)(w133)-1u11,v11≈δ,n20,1=(w214)-1v20,n20,3=u20-δ2u-w213(w214)-1v20-w243(w242)-1y20,n20,4=(w242)-1y20,x20≈δ,n21,1=x01-w231(w233)-1u01,n21,3=(w233)-1u01,n21,4=v01-δ2v-w~214x01+(w~214w231-w234)(w233)-1u01,y01≈δ.Next, we divide our establishment of the Poincarémap in the new coordinate system in three steps.First, consider the mapFi1:Si0↦Si1. Put z=hi(t) into (1); we have
(18)r˙i(t)+Z˙i(t)Ni(t)+Zi(t)N˙i(t)=g(ri(t)+Zi(t)Ni(t),λ,μ)=g(ri(t),0,0)+gz(ri(t),0,0)Zi(t)Ni(t)+gλ(ri(t),0,0)λ+gμ(ri(t),0,0)μ+h.o.t.=f(ri(t))+Df(ri(t))Zi(t)Ni(t)+gλ(ri(t),0,0)λ+gμ(ri(t),0,0)μ+h.o.t.
According to the fact r˙i(t)=f(ri(t)) and Z˙i(t)=Df(ri(t))Zi(t), it then yields to that
(19)N˙i(t)=Zi-1(t)[gλ(ri(t),0,0)λ+gμ(ri(t),0,0)μ]+h.o.t.
Integrating the above equation from -Ti to Ti, we arrive at
(20)Ni(Ti)=Ni(-Ti)+∫-TiTi‍Zi-1(t)gλ(ri(t),0,0)λdt+∫-TiTi‍Zi-1(t)gμ(ri(t),0,0)μdt+h.o.t.
Noticing that Φi*(t)=Zi-1(t), then
(21)ini1,j=ni0,j+Miλjλ+Miμjμ+h.o.t.,j=1,3,4,
where
(22)iMiλj=∫-TiTi‍ϕij*gλ(ri(t),0,0)dt,Miμj=∫-TiTi‍ϕij*gμ(ri(t),0,0)dt,j=1,3,4.
Together with (17) and (21)i, (22)i then defines the map Fi1:Si0↦Si1, (ni0,1,0,ni0,3,ni0,4)↦(ni1,1,0,ni1,3,ni1,4).Next, to construct the mapFi0:Si-11↦Si0 (whereS01=S21). Let τi,i=1,2 be the flying time from qi-11(xi-11,yi-11,ui-11,vi-11)* to qi0(xi0,yi0,ui0,vi0)*; set s1=e-ρ11(α)τ1 and s2=e-ρ21(α)τ2. By virtue of the approximate solution of system (6) and (7), if we neglect the higher terms, then the expression of F10:S01↦S10 is(23)x01≈x10h(s1),y10≈s1y01,u01≈s1λ11(α)/ρ11(α)u10,v10≈s1ρ12(α)/ρ11(α)v01
and F20:S11↦S21 is
(24)x11≈s2λ21(α)/ρ21(α)x20,y20≈s2y11,u11≈s2λ22(α)/ρ21(α)u20,v20≈s2ρ22(α)/ρ21(α)v11,
where (si,ui0,vi-11), i=1,2 are called Shilnikov coordinates, and
(25)h(s)={(λp)-1[x10-(x10-λp)sλp/ρ11(α)],λp≠0,1-(ρ11(α))-1x10ln⁡s,λp=0.
Since the nonhyperbolic equilibrium p1 undergoes a transcritical bifurcation based on the structure of orbits in U1, we may see that the equation x01≈x10/h(s1) holds only for x01≥λp. While for x01∈[-β,λp)  (0<β≪1), the map F10 is well defined only if s1=0 (see Figure 1). So, we extend the domain of F10, defining
(26)x10=δ,s1=0,ifx01∈[-β,λp).Figure 1The final step is to compose the mapsFi0 and Fi1, and then F1=F11∘F10:S01↦S11 can be expressed as
(27)n11,1=(w112)-1δs1-(w112)-1w142(w144)-1s1ρ12(α)/ρ11(α)v01+M1λ1λ+M1μ1μ+h.o.t.,n11,3=u10-w113(w112)-1δs1+[w113w142(w112)-1-w143](w144)-1s1ρ12(α)/ρ11(α)v01+M1λ3λ+M1μ3μ+h.o.t.,n11,4=(w144)-1s1ρ12(α)/ρ11(α)v01+M1λ4λ+M1μ4μ+h.o.t.
and F2=F21∘F20:S11↦S21(=S01) as
(28)n21,1=(w214)-1δs2ρ22(α)/ρ21(α)+M2λ1λ+M2μ1μ+h.o.t.,n21,3=u20-δ2u-w213(w214)-1δs2ρ22(α)/ρ21(α)-w243(w242)-1s2y11+M2λ3λ+M2μ3μ+h.o.t.,n21,4=(w244)-1s2y11+M2λ4λ+M2μ4μ+h.o.t.SetGi=Fi(qi-11)-qi1, i=1,2. Combing (21)i, (23), (24), (27), and (28), we derive the successor functions Gij:
(29)G11=(w112)-1δs1-δs2λ21(α)/ρ21(α)+M1λ1λ+M1μ1μ+h.o.t.,G13=u10-w113(w112)-1δs1-(w133)-1s2λ22(α)/ρ21(α)u20+M1λ3λ+M1μ3μ+h.o.t.,G14=(w144)-1s1ρ12(α)/ρ11(α)v01-y11+w~112δs2λ21(α)/ρ21(α)+M1λ4λ+M1μ4μ+h.o.t.,G21=(w214)-1δs2ρ22(α)/ρ21(α)-δh(s1)+w231(w233)-1s1λ11(α)/ρ11(α)u10+M2λ1λ+M2μ1μ+h.o.t.,G23=u20-δ2u-w213(w114)-1δs2ρ22(α)/ρ21(α)-w243(w242)-1s2ρ21(α)/ρ21(α)y11-(w233)-1s1λ11(α)/ρ11(α)u10+M2λ3λ+M2μ3μ+h.o.t.,G24=(w244)-1s2y11-v01+δ2v+w~214δh(s1)-(w234-w~214w231)(w233)-1s1λ11(α)/ρ11(α)u10+M2λ4λ+M2μ4μ+h.o.t.
It is easy to see that what we need to do is considering the solutions of
(30)(G11,G13,G14,G21,G23,G24)=0
with s1≥0 and s2≥0. This is because the solution of (30) with s1=s2=0 (resp., s1>0, s2>0;s1=0, s2>0 or s1>0, s2=0) means that system (1) has a heteroclinic loop (resp., a periodic orbit; homoclinic loop).


## 3. Main Results

Based on the expressions of the successor functions and the implicit function theorem, we know that the equation(G13,G14,G23,G24)=0 has a unique solution (u10,u20,y11,v01). And putting it into (G11,G21)=0, then we obtain the following bifurcation equations:
(31)(w112)-1δs1-δs2λ21/ρ21+M1λ1λ+M1μ1μ+h.o.t.=0,(w214)-1δs2ρ22/ρ21-δh(s1)+M2λ1λ+M2μ1μ+w231(w233)-1s1λ11/ρ11[w113(w112)-1δs1+(w133)-1×s2λ22/ρ21(δ2u-M2λ3λ-M2μ3μ)-M1λ3λ-M1μ3μ]+h.o.t.=0.Firstly, we consider the caseλ=0, which means the transcritical bifurcation does not happen. By (23) and (25), (31) turns to
(32)(w112)-1δs1-δs2λ21/ρ21+M1μ1μ+h.o.t.=0,(w212)-1δs2ρ22/ρ21-δ1-(ρ11)-1δln⁡s1+M2μ1μ+w231(w233)-1w113(w112)-1δs1λ11/ρ11+1-w231(w233)-1s1λ11/ρ11M1μ3μ+h.o.t.=0.
Noticing that λ11/ρ11>0, which shows lim⁡s1→0s1λ11/ρ11(1-(ρ11)-1δln⁡s1)=0, it then follows that
(33)s1-w112s2λ21/ρ21+δ-1w112M1μ1μ+h.o.t.=0,(w212)-1s2ρ22/ρ21-11-(ρ11)-1δln⁡s1+δ-1M2μ1μ+h.o.t.=0.From the above bifurcation equations, we obtain the following results immediately.Theorem 2.
Let the conditions (H1)–(H3) be true and Miμ1≠0, i=1,2. Then, for λ=0 and 0<|μ|≪1, one has
(i) forrank⁡(M1μ1,M2μ1)=2, there exists a codimension 2 surface
(34)L12={μ:M1μ1μ+h.o.t.=M2μ1μ+h.o.t.=0}
such that system (1) has a unique heteroclinic loop near Γ if and only if μ∈L12, where the surface L12 has a normal plane span{M1μ1,M2μ1} at μ=0.
(ii) there exists an(l-1)-dimensional surface
(35)L12={μ:δ-1w212M2μ1μ+(δ-1M1μ1μ)ρ22/λ21+h.o.t.=0,M1μ1μ>0(δ-1M1μ1μ)ρ22/λ21}(36)(resp.,L21={μ:δ1-(ρ11)-1δln⁡⁡(-δ-1w112M1μ1μ)-M2μ1μ+h.o.t.=0,w112M1μ1μ<0δ(ρ11)-1δln⁡⁡(-δ-1w112M1μ1)})
such that system (1) has a unique homoclinic loop connecting p1 (resp., connecting p2) near Γ if and only if μ∈L12 (resp., μ∈L21).Proof.
The result (i) will be proved by puttings1=s2=0 into (33).
If we assumes1=0 and s2>0 in (33), then
(37)s2λ21/ρ21=δ-1M1μ1μ+h.o.t.,(w212)-1s2ρ22/ρ21+δ-1M2μ1μ+h.o.t.=0,
which means
(38)(w212)-1s2ρ22/λ21+δ-1M2μ1μ+h.o.t.=0.
It follows that there exists an(l-1)-dimensional surface L12 given by (35) such that (33) has a unique solution s1=0, s2=s2(μ)>0 asμ∈L12 and 0<|μ|≪1. This implies system (1) has a homoclinic loop connecting p1. The existence of L21 can be obtained similarly.
This completes the proof.Remark 3.
There is no difficulty to see thatL12 has a normal vector M2μ1 at μ=0 as ρ22>λ21, while for ρ22<λ21 (resp., ρ22>λ21) it has a normal vector M1μ1 (resp., M1μ1+w212M2μ1) at μ=0.Theorem 4.
Assume the conditions(H1)–(H3) hold and Miμ1≠0, i=1,2. Then for λ=0, μ∈L12, and 0<|μ|≪1, the periodic orbit and homoclinic loop with p1 of system (1) cannot coexist.Proof.
Theorem2 shows that if μ∈L12 and 0<|μ|≪1, then system (1) has a homoclinic loop withp1. Setting s1≥0, s2λ21/ρ21=(w112)-1s1+δ-1M1μ1μ+h.o.t.>0, and μ∈L12, then (33) is reduced to
(39)V1(s1)≜[(w112)-1s1+δ-1M1μ1μ]ρ22/λ21+δ-1w212M2μ1μ+h.o.t.=w2121-(ρ11)-1δln⁡s1≜N1(s1).
Notice that V1(0)=N1(0) and
(40)V1′(s1)=ρ22λ21(w112)-1[(w112)-1s1+δ-1M1μ1μ]ρ22/λ21-1,N1′(s1)=w212(ρ11)-1δ(1-(ρ11)-1δln⁡s1)2s1.
Ifw112w212<0, then V1′(s1)N1′(s1)<0; it is obvious that V1(s1)=N1(s1) has no sufficiently small positive solutions.
Whileρ22>λ21, then |V1′(s1)|≪1 and |N1′(s1)|≫1 hold for 0<s1≪1, which shows that V1(s1)=N1(s1) has no sufficiently small positive solution.
Next, we only consider the caseρ22≤λ21 and w112w212>0. As μ∈L12, we have M1μ1μ>0, and then, for wi12>0, i=1,2 we see that
(41)V1′(s1)≤(w112)-ρ22/λ21s1ρ22/λ21-1<N1′(s1)for0<s1≪1.
In fact, ρ22<λ21 yields that lim⁡s1→0+s1ρ22/λ21-1=+∞, lim⁡s1→0+N1′(s1)=+∞, and lim⁡s1→0s1ρ22/λ21-1/N1′(s1)=0, which shows V1(s1)=N1(s1) has no sufficiently small positive solutions. Obviously, the conclusion is correct asρ22=λ21.
Similarly, forρ22<λ21,wi12<0, i=1,2, there does not exist a small positive solution for V1(s1)=N1(s1).
The proof is then completed.Theorem 5.
Assume that the conditions (H1)–(H3) hold and Miμ1≠0, i=1,2. Let λ=0, μ∈L21, and 0<|μ|≪1; then (i)
the periodic orbit and the homoclinic loop connectingp2 of system (1) cannot coexist as ρ22≥λ21 or w112w212<0;(ii)
at least one periodic orbit and the homoclinic loop connectingp2 of system (1) coexist as ρ22<λ21, w112>0, and w212>0;(iii)
a unique periodic orbit and the homoclinic loop connectingp2 of system (1) coexist as ρ22<λ21, w112<0, and w212<0.Proof.
By Theorem2, the condition μ∈L21for 0<|μ|≪1 implies that system (1) has a homoclinic loop connecting p2.
(i) Lets2=e-ρ21τ2 and eliminatings1in (33), we derive
(42)V2(s2)≜s2+δ-1w212M2μ1μ+h.o.t.=w2121-(ρ11)-1δln⁡(w112(s2λ21/ρ22-δ-1M1μ1μ))≜N2(s2).
Note that V2(0)=N2(0) as μ∈L21. Moreover,
(43)V2′(s2)=1,N2′(s2)=w212(ρ11)-1δ[1-(ρ11)-1δln⁡(w112(s2λ21/ρ22-δ-1M1μ1μ))]2·(λ21/ρ22)s2(λ21-ρ22)/ρ22(s2λ21/ρ22-δ-1M1μ1μ).
Forρ22≥λ21 and N2′(s2)≪1=V2′(s2), this means V2(s2)=N2(s2) has no sufficiently small positive solutions.
Now we turn to the casew112w212<0, since we are interested in sufficiently small positive solutions of (33), it suffices to consider the sufficiently small positive solutions of V2(s2)=N2(s2) satisfying w112(s2λ21/ρ22-δ-1M1μ1μ)>0, which implies that s2λ21/ρ22-δ-1M1μ1μ<0 (resp., s2λ21/ρ22-δ-1M1μ1μ>0) for w112<0 (resp., w112>0). It is easy to see that V2(s2)=N2(s2) has no sufficiently small positive solutions as w112w212<0.
(ii) Forρ22<λ21, we have V2′(0)=1>0=N2′(0), which implies that there exists an 0<s~2≪1 such that V2(s2)>N2(s2) for 0<s2<s~2.
Choosings^2=δ-1w212M2μ1μ>0, then
(44)V2(s^2)=2δ-1w212M2μ1μ+h.o.t.,N2(s^2)=w2121-(ρ11)-1δln⁡(w112(s^2λ21/ρ22-δ-1M1μ1μ)).
In view of ln⁡(w112(s^2λ21/ρ22-δ-1M1μ1μ))>ln⁡(w112s^2λ21/ρ22)=ln⁡(w112(δ-1w212M2μ1μ)λ21/ρ22) for w112>0, so
(45)N2(s^2)>w2121-(ρ11)-1δln⁡(w112(w112(δ-1w212M2μ1μ)λ21/ρ22))≫2w112(δ-1w212M2μ1μ)=V2(s^2)
when w212>0. As a result, N2(s2)=V2(s2) has at least one solution s-2 satisfying 0<s~2<s-2<s^2≪1.
(iii)s2 must fulfill 0<s2<(δ-1M1μ1μ)ρ22/λ21 as w112<0; with similar arguments in proof of (ii), we can prove that there exists a 0<s2*≪1 such that V2(s2*)=N2(s2*) for 0<s2*<(δ-1M1μ1μ)ρ22/λ21≪1. It is easy to compute that N2′′(s2)>0 for w212<0, 0<s2<(δ-1M1μ1μ)ρ22/λ21, and μ∈L21. Combining with the fact V2(0)=N2(0), N2′(s2)>0, and V2′(s2)=1, we immediately know that s2* is unique.
This completes the proof.Now, we turn to discussing the bifurcations of the heteroclinic loop forλ>0, when p1 undergoes a transcritical bifurcation. From Figure 1, we know that when λ>0, after the creation of the equilibria p10 and p11, there always exists a straight segment orbit heteroclinic to p11 and p10, its length is λp, and we denote this heteroclinic orbit by Γ*. Moreover, x01=λp is a critical position.Firstly, we take into account the casex01≥λp. In this case, (31) becomes
(46)(w112)-1δs1-δs2λ21/ρ21+M1λ1λ+M1μ1μ+h.o.t.=0,(w212)-1δs2ρ22/ρ21-δλp[δ-(δ-λp)s1λp/ρ11]-1+M2λ1λ+M2μ1μ+w231(w233)-1s1λ11/ρ11[w113(w112)-1δs1+(w133)-1δs2λ22/λ21-M1λ3λ-M1μ3μ(w112)-1]+h.o.t.=0.
Let s=s1λp/ρ11 (s=0 means s1=0 and vice versa); by virtue of Taylor’s development for δλp/(δ-(δ-λp)s1λp/ρ11), we have
(47)(w112)-1sρ11/λp-s2λ21/ρ21+δ-1M1λ1λ+δ-1M1μ1μ+h.o.t.=0,(w212)-1δs2ρ22/ρ21-λp-λp(δ-λp)δs+M2λ1λ+M2μ1μ+h.o.t.=0.With similar arguments toλ=0, we may easily obtain the following results.Theorem 6.
Suppose the conditions (H1)–(H3) hold, 0<λ≪1; then
(i) ifrank⁡(M1μ1,M2μ1)=2, there exists an(l-2)-dimensional surface
(48)L12λ={μ(λ):M1μ1μ+M1λ1λ+h.o.t.=M2μ1μ+M2λ1λ-λ+h.o.t.=0}
such that system (1) has a unique heteroclinic loop if and only if μ∈L12λ and 0<|μ|≪1;
(ii) there exists an(l-1)-dimensional surface
(49)L1λ2={μ(λ):W12(λ,μ)=(w212)-1[δ-1(M1μ1μ+M1λ1λ)]β2+δ-1(M2μ1μ+M2λ1λ)-δ-1λp+h.o.t.=0,M1μ1μ+M1λ1λ>0[δ-1(M1μ1μ+M1λ1λ)]β2}([-δ-1w112(M1λ1λ+M1μ1μ)]λp/(ρ11)resp.,L2λ1={[-δ-1w112(M1λ1λ+M1μ1μ)]λp/(ρ11)μ(λ):W21(λ,μ)=δλp+λp(δ-λp)×[-δ-1w112(M1λ1λ+M1μ1μ)]λp/ρ11-δM2λ1λ-δM2μ1μ+h.o.t.=0,w112(M1λ1λ+M1μ1μ)<0[-δ-1w112(M1λ1λ+M1μ1μ)]λp/(ρ11)})
such that system (1) has one homoclinic loop connecting p11 (resp., connecting p2) if and only if μ∈L1λ2 and 0<|μ|≪1.Theorem 7.
Suppose hypotheses (H1)–(H3) hold, Miμ1≠0, i=1,2, 0<λ, |μ|≪1, and w112w212<0. Then, except the homoclinic loop connecting p11 (resp., p2), system (1) has no periodic orbits as μ∈L1λ2 (resp., μ∈L2λ1).Remark 8.
It is easy to see that homoclinic loop connectingp10 and heteroclinic loop joining p10, p2 cannot be bifurcated from Γ, which is exactly determined by the generic condition (H1).Finally, we consider the case-β≤x01<λp. Due to Figure 1 and (25), it follows from (31) that
(50)s2=[δ-1(M1λ1λ+M1μ1μ)]ρ21/λ21+h.o.t.,x01=(w212)-1δs2ρ22/ρ21+M2λ1λ+M2μ1μ+h.o.t.Theorem 9.
Assume the conditions (H1)–(H3) are true, rank⁡(M1λ1,M1μ1)>0 and rank⁡(M2λ1,M2μ1)>0. Then, (i)
there exists a surface(51)Σ1(μ,λ)={μ(λ):[δ-1(M1λ1λ+M1μ1μ)]ρ21/λ21+h.o.t.=0,-β≤M2λ1λ+M2μ1μ+h.o.t.<λp,0<|μ|,λ≪1[(M1λ1λ+M1μ1μ)]ρ21/λ21},
such that system (1) has two orbits heteroclinic to p11, p2, p10 as μ∈Σ1(μ,λ);(ii)
there exists a region in the(λ,μ) space
(52)Δ={(λ,μ):-β≤(w212)-1δ(λ21-ρ22)/λ21×(M1λ1λ+M1μ1μ)ρ22/λ21+M2λ1λ+M2μ1μ+h.o.t.<λp,M1λ1λ+M1μ1μ>0,0<|μ|,λ≪1δ(λ21-ρ22)/λ21},
such that system (1) has a heteroclinic orbit connecting p11 and p10 for (λ,μ)∈Δ.Proof.
(i) Ifs2=0 in (50), then
(53)0=[δ-1(M1λ1λ+M1μ1μ)]ρ21/λ21+h.o.t.,x01=M2λ1λ+M2μ1μ+h.o.t.
which shows that there exists a surface Σ1(μ,λ) such that (50) has a solution s2=0 and -β≤x01<λp for μ∈Σ1(μ,λ), then system (1) has two heteroclinic orbits, one is heteroclinic top11andp2 and the other is heteroclinic to p2 andp10.
(ii) Ifs2>0 in (50), one attains M1λ1λ+M1μ1μ>0. Eliminating s2 in (50), we achieve
(54)x01=(w212)-1δ(λ21-ρ22)/λ21(M1λ1λ+M1μ1μ)ρ22/λ21+M2λ1λ+M2μ1μ+h.o.t.,
which shows that there exists a region Δ such that when (λ,μ)∈Δ, system (1) has one heteroclinic orbit heteroclinic to p11 and p10.Remark 10.
All the heteroclinic orbits joiningp10 will go into p10 in different ways according to different fields of x01; see Figure 2.(a)
(b)
(c)



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*Source: 585609-2014-03-23.xml*