# Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

Authors: Tiansi Zhang; Xiaoxin Huang; Deming Zhu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.


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## Body

## 1. Introduction and Hypotheses

The study of homoclinic flip bifurcations is comprehensively developed from the last two decades with the beginning work of Yanagida (1987) for homoclinic-doubling bifurcations. Generally there exist two kinds of homoclinic flips, namely the orbit flips and the inclination flips corresponding to nonprincipal homoclinic orbits or critically twisted homoclinic orbits, respectively. Kisaka et al. in [1, 2] and Naudot in [3] studied some cases of codimension two inclination flips; Morales and Pacifico in [4] and Naudot in [5] considered the orbit flips cases, while Homburg and Krauskopf in [6] proposed several unfoldings of the resonant homoclinic flip bifurcations around the central codimension-three point (the organizing centre) in parameter space to study the qualitative structure of bifurcation curves on a sphere and also that of Oldeman et al. in [7] by a numerical investigation with some software into these bifurcations in a specific three-dimensional vector field.Recently, Zhang et al. in [8–10] studied a kind of multiple flips homoclinic resonant bifurcation and got the existence of some saddle-node bifurcations and homoclinic-doubling bifurcations. Meanwhile Geng et al. in [11], Lu et al. in [12], and Liu in [13] discussed, respectively, a heterodimensional cycle flip or accompanied by transcritical bifurcation; they found the double and triple periodic orbit bifurcations and gave also some coexistence conditions for homoclinic orbits and periodic orbits.As mentioned in [6, 7], due to the break of three genericity conditions, there are many complicated homoclinic flips cases to study. In this paper, we confine our attention to a principal eigenvalue resonance of one orbit flip and one inclination flip homoclinic bifurcation. Compared with the above-mentioned work, our subject is very challenging and difficult because of the stronger degeneracy and the higher codimension. By constructing specifically a local active coordinate in a small tubular neighborhood of homoclinic orbit, we establish a regular map and then combine it with a singular map defined by the approximation solutions of system to build Poincaré return map (see also [14]). We obtain the existence of several 1-periodic orbit, 1-homoclinic orbit, and double 1-periodic orbits, as well as some bifurcation surfaces with the analysis of the bifurcation equation.We first consider aCr system
(1)z˙=f(z)+g(z,μ)
and its unperturbed system
(2)z˙=f(z),
where r≥3, z∈ℝ4, μ∈ℝl, l≥4, 0<|μ|≪1, f(0)=0, and g(0,μ)=g(z,0)=0. Suppose that the linearization Df(0) has four simple real eigenvalues λ1, λ2, -ρ1, and -ρ2 with λ2>λ1>0>-ρ1>-ρ2. Accordingly, the stable manifold Ws and the unstable manifold Wu are both two-dimensional. Let Wss and Wuu be the strong stable manifold and the strong unstable manifold of the saddle z=O, respectively. Assume further that system (2) has a homoclinic orbit Γ={z=r(t):t∈ℝ,r(±∞)=0}. Hereinafter, our arguments will spread based on the following three hypotheses.(H1)Resonance. λ1(μ)=ρ1(μ), |μ|≪1, where λ1(0)=λ1 and ρ1(0)=ρ1.(H2)Orbit Flip. Define e+=limt→-∞r˙(t)/|r˙(t)|; es-=limt→+∞r˙(t)/|r˙(t)|; then e+∈T0Wu and es-∈T0Wss are unit eigenvectors corresponding to λ1 and -ρ2, respectively, where T0Wu (resp., T0Wss) is the tangent space of the corresponding manifold Wu (resp., Wss) at the saddle z=O.(H3)Inclination Flip. Let eu+ and e- be the unit eigenvectors corresponding to λ2 and -ρ1, respectively, and
(3)limt→+∞{Tr(t)Wu,Tr(t)Ws,eu+}=ℝ4,limt→-∞{Tr(t)Wu,Tr(t)Ws,e-}=ℝ4.Remark 1.
Hypothesis (H2) is called an orbit flip because homoclinic orbit trends from the weak unstable manifold toward the strong stable manifold. Hypothesis (H3) means an inclination flip for its equivalence to(4)Tr(t)Wu⟶span{es-,e+},Tr(t)Ws⟶span{es-,e+}.


## 2. Poincaré Return Map

This section treats mainly the establishment of Poincaré return map with two steps. To begin we first need to transform system (1) into a normal form in some neighborhood of the origin O.It is well known that there are always twoCr and Cr-1 transformations successively, also by the stable (or unstable) manifold theorem in [15], to straighten the local manifolds Wlocs and Wlocu as Wlocs={z∈U,x=u=0} and Wlocu={z∈U,y=v=0}, respectively, Wlocss={z∈U,x=y=u=0} (resp., Wlocuu={z∈U,x=y=v=0}); see [8–10]. Notice that now Γ∩Wlocu={z∈U,u=u(x),y=v=0} and Γ∩Wlocs=Wlocss, where z=(x,y,u,v)∈ℝ4 and u(0)=u′(0)=0. Thus, system (1) can be changed to a Cr-2 form in the neighborhood U as follows:
(5)x˙=[λ1(μ)+a(μ)xv+o(|xv|)]x+O(u)[O(x2v)+O(y)],y˙=[-ρ1(μ)+b(μ)xv+o(|xv|)]y+O(v)[O(xyv)+O(u)],u˙=[λ2(μ)+c(μ)xv+o(|xv|)]u+x2H1(x,y,v),v˙=[-ρ2(μ)+d(μ)xv+o(|xv|)]v+y2H2(x,y,u),
where H1(x,0,0)=0, H2(0,y,0)=0, λ1(0)=λ1, λ2(0)=λ2, ρ1(0)=ρ1, and ρ2(0)=ρ2. a(μ), b(μ), c(μ), and d(μ) are parameters depending on μ.Owing to the above straightness of the invariant manifolds, it is easy to find some momentT, such that r(-T)={δ,0,δu,0} and r(T)={0,0,0,δ} for some sufficiently small δ and |δu|=O(δ2) with {(x,y,u,v):|x|,|y|,|u|,|v|<2δ}⊂U. Wherefore one can choose
(6)S0={z=z(T):|x|,|y|,|u|,|v|<2δ}⊂U,S1={z=z(-T):|x|,|y|,|u|,|v|<2δ}⊂U,
as the cross-sections of Γ at t=T and t=-T, respectively. Let τ be the time going from q0(x0,y0,u0,v0)∈S0 to q1(x1,y1,u1,v1)∈S1 and the Silnikov time s=e-λ1(μ)τ, with the help of the linear approximation solutions of system (5) (see [8–10]); we have thereby
(7)x0=x(T)=sx1+O(x12y0s2lns),y1=y(T+τ)=sρ1(μ)/λ1(μ)y0+O(x1y02s2lns),u0=u(T)=sλ2(μ)/λ1(μ)u1+O(x1y0u1s(λ2(μ)/λ1(μ))+1lns),v1=v(T+τ)=sρ2(μ)/λ1(μ)v0+O(x1y0v0s(ρ2(μ)/λ1(μ))+1lns),
which give explicitly the definition of the local transition map F0:S0→S1:q0↦q1; see Figure 1(a).Transition maps.
(a)
F
0: S0→S1
(b)
F
1: S1→S0In the following part we construct the mapF1:S1→S0. Firstly consider the linear variational system
(8)z˙=Df(r(t))z
and its adjoint system
(9)z˙=-(Df(r(t)))*z.Lemma 2.
There exists a fundamental solution matrixZ(t)=(z1(t), z2(t), z3(t), z4(t)) of system (8) satisfying
(10)Z(-T)=(w11w210w41w1200w42w13w231w43000w44),Z(T)=(00w310w140w32110w33001w340),
where z1(t)∈(Tr(t)Wu)c∩(Tr(t)Ws)c, z2(t)=-r˙(t)/|r˙y(T)|∈Tr(t)Wu∩Tr(t)Ws, z3(t)∈Tr(t)Wu, z4(t)∈Tr(t)Ws and w12w21w31w44≠0, w23<0. Moreover, |w1iw12-1|≪1, i≠2, |w3iw31-1|≪1,i≠1, |w4iw44-1|≪1, i≠4 for T→+∞.Proof.
Notice that the tangent subspaceTγ(-T)Wu is invariant and Wlocuu∩U is straightened to be u axis; it is possible to choose z3(-T)=(0,0,1,0) since z3(t)∈Tγ(t)Wu. While for w31≠0, it is because limt→+∞Tγ(t)Wu=span{e+,es-} and z3(T)∈Tγ(T)Wu corresponding to x axis.
As tozi(-T) or zi(T), i=1,2,4, one may refer to [8, 9] for the similar proof, but we omit the details here.Remark 3.
The matrix(Z-1(t))* is a fundamental solution matrix of system (9). We denote it as Φ(t)=(ϕ1(t),ϕ2(t),ϕ3(t),ϕ4(t))=(Z-1(t))*; then ϕ1(t)∈(Tr(t)Wu)c∩(Tr(t)Ws)c is bounded and tends to zero exponentially as |t|→+∞ due to <ϕi(t),zi(t)≥1 and zi(t) tends exponentially to infinity.In fact(z1(t),z2(t),z3(t),z4(t)) can be regarded as a new local coordinate system along Γ. So we may make a coordinate change as
(11)s(t)=r(t)+Z(t)N=r(t)+z1(t)n1+z3(t)n3+z4(t)n4,
where N=N(t)=(n1(t),0,n3(t),n4(t))*. Note that the new s(t) should satisfy system (1); that is,
(12)s˙(t)=f(s(t))+g(s(t),μ)=f(r(t)+Z(t)N)+g(r(t)+Z(t)N,μ).
An asymptotic expansion with respect to r(t) shows that
(13)n˙i=ϕi*(t)gμ(r(t),0)μ+O(|μ|2)+O(|N|2)+O(|μ||N|),i=1,3,4.
Via integrating both sides from -T to T of this equation, one can finally obtain
(14)ni(T)=ni(-T)+Miμ+h.o.t.,i=1,3,4,
where Mi=∫-TT‍ϕi*(t)gμ(r(t),0)dt,i=1,3,4 are the Melnikov vectors. And further M1=∫-TT‍ϕ1*(t)gμ(r(t),0)dt=∫-∞+∞‍ϕ1*(t)gμ(r(t),0)dt.Equation (14) defines exactly the map F1:S1→S0; N(-T)↦N(T) under the new coordinate system; see Figure 1(b).In order to combineF0 and F1 into the Poincaré return map, we still need to establish a relationship between the original and the new coordinate systems. In doing so, recall that z(t)=r(t)+Z(t)N(t); then by taking time t=T and t=-T, respectively, together with z(T)=q2j(x2j,y2j,u2j,v2j)∈S0, z(-T)=q2j+1(x2j+1,y2j+1,u2j+1,v2j+1)∈S1 and N2j(T)=(n2j,1,0,n2j,3,n2j,4), N2j+1(-T)=(n2j+1,1,0,n2j+1,3, n2j+1,4), j=0,1,2,…, we obtain immediately the following formulas:
(15)n2j,1=u2j-w33w31-1x2j,n2j,3=w31-1x2j,n2j,4=y2j-w14u2j+(w14w33-w32)w31-1x2j,n2j+1,1=w12-1y2j+1-w42w12-1w44-1v2j+1,n2j+1,3=u2j+1-δu-w13w12-1y2j+1+(w13w42w12-1-w43)w44-1v2j+1,n2j+1,4=w44-1v2j+1,x2j+1≈δ,v2j≈δ.With all of the above, the Poincaré return map is given asF=F1∘F0. Therefore, the associated successor function G(s, u1, y0)=(G1, G3, G4)=F(q0)-q0 is
(16)G1=w12-1sρ1(μ)/λ1(μ)y0-u1sλ2(μ)/λ1(μ)+w33w31-1δs-w42w12-1w44-1δsρ2(μ)/λ1(μ)+M1μ+h.o.t.,G3=u1-δu-w13w12-1sy0-w31-1δs+(w13w42w12-1-w43)w44-1δsρ2(μ)/λ1(μ)+M3μ+h.o.t.,G4=w44-1δsρ2(μ)/λ1(μ)-y0+w14sλ2(μ)/λ1(μ)u1+(w32-w14w33)w31-1δs+M4μ+h.o.t..


## 3. Bifurcation Results

From the definition of theSilnikov time s=e-λ1(μ)τ, we know that a solution with s>0 of (16) or equivalently τ>0 corresponds to a periodic orbit near Γ and a solution with s=0 of (16) or equivalently τ=+∞ corresponds to a homoclinic orbit near Γ. It is enough to study the solutions of the successor function G(s,u1,y0)=0 for bifurcation analysis. Consider for concision that we omit the dependence on parameter μ from now on in the exponent notation.FromG3=0 and G4=0, there are
(17)u1=δu+w31-1δs-M3μ+O(sy0)+O(sρ2/λ1),y0=(w32-w33w14)w31-1δs+M4μ+O(sρ2/λ1)+O(sλ2/λ1).Putu1 and y0 into G1=0; we have
(18)F(s,μ)≡M4μs-w42w44-1δsρ2/λ1+w12w33w31-1δs-δuw12sλ2/λ1+w12M3μsλ2/λ1+(w32-w33w14)w31-1δs2+w44-1δs(ρ1+ρ2)/λ1-w12w31-1δs1+(λ2/λ1)+w12M1μ+w14δus(ρ1+λ2)/λ1+w13M4μs(λ1+λ2)/λ1+h.o.t.=0,
which is the bifurcation equation. Furthermore, for w33≠0,
(19)detG~|μ=0Q=0ml=|w33w31-1δ00-δw31-110(w32-w33w14)w31-1δ0-1|≠0,
where G~=∂(G1,G3,G4)/∂Q(s,u1,y0). Implicit function theorem reveals that (16) has a unique solution as
(20)s=s(μ),u1=u1(μ),y0=y0(μ),
with s(0)=0, u1(0)=0, and y0(0)=0. It means that system (1) has a unique periodic orbit as s>0 or a unique homoclinic orbit as s=0, and they cannot coexist.Theorem 4.
Suppose thatM1≠0 and w33≠0 are true; then system (1) has a unique 1-periodic orbit near Γ for w31w33M1μ<0 or has a unique 1-homoclinic orbit Γμ near Γ as μ∈H1=Δ{μ:M1μ+h.o.t.=0}, and they do not coexist.Proof.
ClearlyF(s,μ)=0 has a small positive solution s=-δ-1w33-1w31M1μ+h.o.t., for w31w33M1μ<0, and has a zero solution s=0 for μ∈H1 which is a codimension-one hypersurface.In the following part we restrict our attention on the casew33=0 for 2λ1>λ2>ρ2. Define
(21)Rijkl={μ:(-1)k(w12)lMiμMjμ>0},Ejkl={μ:(-1)k(w12)lw42w44Mjμ>0},
where k,l=0,1,i=1,3,j=1,3,4.In order to well solve (18), we rewrite it into two parts, namely, a line W=P(s,μ) and a curve W=Q(s,μ) with respect to s:
(22)P(s,μ)=M4μs+w12M1μ+h.o.t.,Q(s,μ)=w42w44-1δsρ2/λ1+δuw12sλ2/λ1-w12M3μsλ2/λ1-w32w31-1δs2-w44-1δs(ρ1+ρ2)/λ1+w12w31-1δs(λ1+λ2)/λ1+h.o.t..Then there are firstly the following conclusions based on an analysis of the relative position of the lineW=P(s,μ) and the curve W=Q(s,μ).Lemma 5.
Suppose that2λ1>λ2>ρ2, w33=0, and w42≠0 hold; then F(s,μ)=0 has a unique small positive solution s~∈(0,s*) for μ∈E101, where s*=[w44w42-1δ(2w12M1μ+M4μ)]λ1/ρ2.Proof.
It is clear that(23)P(0,μ)=w12M1μ+h.o.t.,Q(0,μ)=0,P′(s,μ)=M4μ+h.o.t.,Q′(s,μ)=ρ2λ1-1w42w44-1δs(ρ2/λ1)-1+h.o.t..
Therefore, the line W=P(s,μ) intersects the curve W=Q(s,μ) at a unique point s~ for μ∈E101. Notice that Q(s*,μ)=2w12M1μ+M4μ+h.o.t.>w12M1μ+M4μs*+h.o.t.=P(s*,μ), so s~∈(0,s*).Theorem 6.
Suppose that2λ1>λ2>ρ2, w33=0, and w42≠0 hold; then system (1) has exactly a unique (resp., not any) 1-periodic orbit for μ∈E101 (resp., μ∈E111∩R1401).Proof.
From Lemma5, we know that F(s,μ)=0 has exactly a unique small positive solution for μ∈E101 which corresponds exactly to a 1-periodic orbit of system (1). Moreover, F(s,μ)=0 does not have any small positive solutions for μ∈E111∩R1401.Theorem 7.
Suppose that2λ1>λ2>ρ2, w33=0, and w42≠0 hold; then, for μ∈E111∩R1411 and Rank (M1,M4)=2, system (1) has a unique double 1-periodic orbit near Γ on the bifurcation surface
(24)SN1:w12M1μ+ρ2-λ1ρ2(λ1w44ρ2w42δ)λ1/(ρ2-λ1)×(M4μ)ρ2/(ρ2-λ1)+h.o.t.=0,
which has a normal vector M1 at μ=0. The corresponding double positive zero point is
(25)s*=(λ1w44M4μρ2w42δ)λ1/(ρ2-λ1)+h.o.t.
as μ∈E400 (see Figure 2(a)).Rank(M1,M4)=2 and μ∈E111∩R1411.
(a)
Double 1-periodic orbit
(b)
Two 1-periodic orbits
(c)
No 1-periodic orbitProof.
We know that the existence of a double 1-periodic orbit corresponds to the equationsP(s,μ)=Q(s,μ), P′(s,μ)=Q′(s,μ), and P′′(s,μ)≠Q′′(s,μ), that is,
(26)M4μs+w12M1μ=w42w44-1δsρ2/λ1+δuw12sλ2/λ1-w12M3μsλ2/λ1-w32w31-1δs2-w44-1δs(ρ1+ρ2)/λ1+w12w31-1δs(λ1+λ2)/λ1+h.o.t.,M4μ=ρ2λ1w42w44-1δs(ρ2/λ1)-1+λ2λ1δuw12s(λ2/λ1)-1-λ2λ1w12M3μs(λ2/λ1)-1-2w32w31-1δs-ρ1+ρ2λ1w44-1δsρ2/λ1+λ1+λ2λ1w12w31-1δsλ2/λ1+h.o.t.,0≠ρ2(ρ2-λ1)w42w44-1δs(ρ2/λ1)-2+λ2(λ2-λ1)δuw12s(λ2/λ1)-2-λ2(λ2-λ1)w12M3μs(λ2/λ1)-2-2λ12w32w31-1δ+ρ2(ρ1+ρ2)w44-1δs(ρ2/λ1)-1+λ2(λ1+λ2)w12w31-1δs(λ2/λ1)-1+h.o.t.,
having solutions. Indeed, the second equation of (26) permits the double positive zero point s* as μ∈E400. Putting it into the first equation of (26), there is
(27)M4μ(λ1w44M4μρ2w42δ)λ1/(ρ2-λ1)+w12M1μ=w42w44-1δ(λ1w44M4μρ2w42δ)ρ2/(ρ2-λ1)+h.o.t..
Then SN1 exists for μ∈E111∩R1411.From the above proof, we see that, whenM4μ>0 and w12M1μ<0, the line W=P(s,μ) has a positive slope lying under the curve W=Q(s,μ) when w42w44>0, so if w12M1μ increases, the line must intersect the curve at two sufficiently small positive points, which can be equal to the existence of two 1-periodic orbits of system (1). For M4μ<0, w42w44<0, and w12M1μ>0, the arguments are similar. So we have immediately a complement of Theorem 7.Corollary 8.
Assume that the hypotheses of Theorem7 are valid, system (1) then has two (resp., not any) 1-periodic orbits near Γ when μ lies on the side of SN1 which points to the direction (sgnw12w42w44)M1 (resp., in the opposite direction) (see Figures 2(b) and 2(c)).AsMelnikov functions generally play an important role in bifurcation study, the following theorem shows also the existence of some double 1-periodic orbits relying on the investigation of Mi=0 for i=1,3,4.Theorem 9.
Suppose2λ1>λ2>ρ2, w33=0, and w42≠0 are valid; then the following applies.(1)
ForM1=0 or M12+M32=0, system (1) has exactly one 1-homoclinic orbit and one (resp., not any) 1-periodic orbit near Γ and they (resp., do not) coexist as μ∈E400 (resp., μ∈E410).(2)
ForM3=0, system (1) has exactly one (resp., not any) 1-periodic orbit near Γ as μ∈E101 (resp., μ∈E111∩R1401).system (1) has a unique double 1-periodic orbit near Γ as μ∈E111∩R1411 and Rank(M1,M4)=2. Accordingly, the double 1-periodic orbit bifurcation surface is SN1:w12M1μ+((ρ2-λ1)w42δ/λ1w44)(λ1w44M4μ/ρ2δw42)ρ2/(ρ2-λ1)+h.o.t.=0 with a normal vector M1 at μ=0, and it may generate two 1-periodic orbits when μ lies in the direction (sgnw42w44w12)M1 of SN1 and no such a 1-periodic orbit in the opposite direction.(3)
ForM4=0 or M32+M42=0, system (1) has only one (resp. not any) 1-periodic orbit near Γ as μ∈E101 (resp., μ∈E111).(4)
ForM12+M42=0, system (1) does not have any 1-periodic orbit near Γ.(5)
ForM12+M32+M42=0, system (1) has only one 1-homoclinic orbit near Γ.Proof.
WhenM1=0 or M1=M3=0, F(s,μ)=s(M4μ-w42w44-1δs(ρ2-λ1)/λ1-δuw12s(λ2-λ1)/λ1+w12M3μs(λ2-λ1)/λ1+w32w31-1δs+w44-1δsρ2/λ1+h.o.t.)=0 has two solutions s1=0 and s2=(w44M4μ/w42δ)λ1/(ρ2-λ1)+h.o.t. for μ∈E400 which correspond to a 1-periodic orbit, and a 1-homoclinic orbit respectively. Thus, (1) is true.
The result of the cases(2) is exactly the same as that of Theorem 7.
IfM4=0 or M3=M4=0, F(s,μ)=0 has only a positive solution s3=(w44w12M1μ/δw42)λ1/ρ2+h.o.t. for μ∈E101, which means system (1) has a 1-periodic orbit. Then (3) is valid.
In case ofM12+M42=0, it is clear that F(s,μ)=0 does not have any small nonnegative solutions, so system (1) does not have, any 1-homoclinic orbits or 1-periodic orbits.
The last conclusion is obvious. Thereby, the proof is complete.Remark 10.
Notice that, in Theorem9  (1) and (5), F(s,μ)=0 has a solution s=0, which means that system (1) has a codimension-1 1-homoclinic orbit (see Figure 3(a)), so the existing homoclinic orbit is no longer orbit flip for y0=M4μ+h.o.t.≠0. But if y0=0, an orbit flip homoclinic orbit could still exist, where y0 is given by G4=0; see Figure 3(b).1-homoclinic orbit (1-H) and (1-OH).
(a)
F
(
0
,
μ
)
=
0,
y0=M4μ+h.o.t.≠0
(b)
F
(
0
,
μ
)
=
0, y0=M4μ+h.o.t.=0Remark 11.
There still exist some double 1-periodic orbits or triple 1-periodic orbits for the casew42=0 and δu≠0; one may pursue the similar process to discuss, so we leave it here.



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