# Bifurcation of an Orbit Homoclinic to a Hyperbolic Saddle of a Vector Field inR4

Authors: Tiansi Zhang; Dianli Zhao
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field inR4. We give an expression of the gap between returning points in a transverse section by renormalizing system, through which we find the existence of homoclinic-doubling bifurcation in the case 1+α>β>ν. Meanwhile, after reparametrizing the parameter, a periodic-doubling bifurcation appears and may be close to a saddle-node bifurcation, if the parameter is varied. These scenarios correspond to the occurrence of chaos. Based on our analysis, bifurcation diagrams of these bifurcations are depicted.


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## Body

## 1. Introduction and Problems

Homoclinic orbits are crucial to know dynamics of differential systems in many application fields. For example, the famous FitzHugh-Nagumo equations, given by PDEs (see [1]), describe how action potentials in neurons occur and spread
(1)ut=uxx-fau-w,wt=εu-γw,
where fa(u)=u(u-a)(u-1). Through the variable transforming ζ=x+ct, system (1) is then in an ODE form:
(2)u˙=v,v˙=cv+fa(u)+w,w˙=εc(u-γw).
It has an orbit homoclinic to the equilibrium (u,v,w)=0 which corresponds to a solitary wave (u,w)(x,t)=(u,w)(ζ) of system (1). The authors detected how homoclinic branch converted a 1-homoclinic orbit to a N-homoclinic orbit.In [2], a reversible water wave model was studied:
(3)215viv-bv′′+av+32v2-12v′2+vv′′=0.
The system admits a flip orbit
(4)rt=3b+12sech2t342b+1,
for b>2, a>0, and shows the existence of the N-homoclinic orbit in some circumstances on two sides of the flip bifurcation.In fact, the homoclinic-doubling bifurcation, which switches a2n-1-homoclinic orbit to a 2n-homoclinic orbit, exists extensively in systems with flips; see [3–6] and the references therein. A simple and analytic model permitting these flips was initially given by Sandstede in a three-dimensional system in [7]. From then on, more and more excellent work has been done based on the model (see, e.g., [8, 9]). Now researchers even extend these flips phenomena to heterodimensional cycles and homoclinic bellows to study periodic orbits and homoclinic orbits; see [10–12]. But none of them aimed to investigate the homoclinic-doubling bifurcations. So in this paper we focus on the homoclinic-doubling problem for a kind of homoclinic flips.Throughout the paper, we consider the following ODE system:(5)x˙=f(x,ξ),(x,ξ)∈R4×Rl,
where f is sufficiently smooth and l≥4. Suppose that the system (5) has an orbit γ(t) of codimension-1 homoclinic to a saddle equilibrium p at ξ=0. Let T be a certain time, such that γ(T) and γ(-T) are in some small neighborhood U of p. Then we can take two sections vertical to Tγ(±T):
(6)S0:{x∣x=x(T)}⊂U;S1:{x∣x=x(-T)}⊂U.Generally, if the small parameterξ≠0, the homoclinic orbit γ(t) will not exist. But the system (5) must have solutions γ±(t,ξ) with the properties
(7)γ˙±=fγ±,ξ,γ+t,ξ∈Wsp,γ-t,ξ∈Wup,γ±t,0=γt,γ+T,ξ∈S0,γ--T,ξ∈S1,γ-(T,ξ)-γ+(T,ξ)≪1,
where Ws(p) and Wu(p) are the stable and unstable manifolds of the equilibrium p, and dim⁡(Tγ(t)Ws∩Tγ(t)Wu)=1. Notice that if the gap γ-(T,ξ)-γ+(T,ξ)=0 in the transverse section S0, it means that the homoclinic orbit is kept (see Figure 1(a)) but it may not be of codimension-1.The gapγ-T,ξ-γ+T,ξ≠0 in (a); there is no homoclinic orbit in general. The gap γiT+Ti+τi,ξ-γiT,ξ≠0 in (b); there is no periodic orbit in general.
(a)
(b)Moreover, the system (5) still has other solutions γi(t,ξ); i is a natural number. Set the time of the orbit γi(t,ξ) from S0 to S1 and from S1 to S0 to be τi and Ti, respectively; there are
(8)γ˙i=f(γi,ξ),γi(T+Ti+τi,ξ),γi(T,ξ)∈S0,γi(T+Ti+τi,ξ)-γi(T,ξ)≪1.
Actually γi(t,ξ) is a regular orbit and will be periodic if the gap ∥γi(T+Ti+τi,ξ)-γi(T,ξ)∥=0; namely, the orbit starting in S0 will return to S0 after the time Ti+τi; see Figure 1(b).From above, we see that the gap in the transverse sectionS0 of some orbits is crucial to study bifurcations of the system. So in the next section we try to quantitate the gap size.


## 2. Main Method

To well carry out our discussion, we give some hypotheses for the system (5) here.(A1)
The spectrumσ(Dxf(p,ξ))={λ1(ξ),λ2(ξ),-ρ1(ξ),-ρ2(ξ)}, and λ2(ξ)>λ1(ξ)>0>-ρ1(ξ)>-ρ2(ξ).(A2)
Ast→+∞, the homoclinic orbit γ(t)→p along the strong stable manifold Wss(p).(A3)
Vectors in the strong unstable (resp., stable) manifoldWuu (resp., Wss) return to the saddle p in the direction along Wu (resp., Ws).We know that the discontinuity of the functionsγ±(t,ξ) or γi(t,ξ) is confined in a special position in S0. Since dim⁡(Tγ(t)Ws∩Tγ(t)Wu)=1, the space
(9)TγtWs∩TγtWu⊥=span⁡φ1
is of one dimension, where φ1 can be taken as the solution of the linear variational system
(10)y˙=Dxfγt,0y,
and φ1(T)=(0,ω12,1,0), φ1(-T)=(ω11,0,ω13,ω14) based on the assumptions of (A2) and (A3); refer to [13] for the details.Beside this, by the theories of matrix, the other three solutions denoted byφ2, φ3, and φ4 of the system (10) can also be taken in the following ways:
(11)φ2=-γ˙(t)γ˙T∈Tγ(t)Wu∩Tγ(t)Ws,φ3∈Tγ(t)Wu,φ4∈Tγ(t)Ws,
satisfying
(12)φ2-T=ω21,0,0,0,φ2T=0,0,0,1,φ3-T=0,0,1,0,φ3T=ω31,ω32,0,ω34,φ4(-T)=(ω41,ω42,ω43,ω44),φ4(T)=(0,1,0,0).Now take a transformation(13)xt=γt+ΦtΞ=γt+φ1tχ1+φ3(t)χ3+φ4(t)χ4,
where Φ(t)=(φ1(t),φ2(t),φ3(t),φ4(t)) and Ξ=(χ1,0,χ3,χ4)*. Then system (5) becomes the following ODE in the new variable Ξ; namely,
(14)Ξ˙=Φ-1Dξf(γ(t),0)ξ+Φ-1Dxξ2f(γ(t),0)ΦΞξ+OΦΞ2+OΦ-1ξ2.
By (14),
(15)∫-TT‍Ξ˙dt=∫-TT‍Φ-1Dξfγt,0ξdt+h.o.t.
gives
(16)Ξ(T)=Ξ(-T)+Mξ+h.o.t.,
where M=(M1,0,M3,M4)*=∫-TT‍Φ-1Dξf(γ(t),0)dt.Notice that in (13), Ξ represents in some meaning the deviation in the normal direction of the manifolds Tγ(t)Ws∩Tγ(t)Wu, so Ξ(-T)∈S1 and Ξ(T)∈S0; in other words, (16) maps a point in S1 to a point in S0.On the other hand, from assumption (A1), system (5) admits a local linearization
(17)Dxfp,ξ=λ1(ξ)x1∂∂x1-ρ1(ξ)x2∂∂x2+λ2(ξ)x3∂∂x3-ρ2(ξ)x4∂∂x4,
where x=(x1,x2,x3,x4).SupposeΞ(T)=Ξ0=(x10,x20,x30,x40) and Ξ(-T)=Ξ1=(x11,x21,x31,x41). Rescale the time s=e-λ1(ξ)τ and write α=ρ1(ξ)/λ1(ξ), β=ρ2(ξ)/λ1(ξ), and ν=λ2(ξ)/λ1(ξ). Without loss of generality, we assume α≥1 for sufficiently small ξ. Otherwise, we can set s=e-ρ1(ξ)τ and α=λ1(ξ)/ρ1(ξ). Clearly, β>α≥1.Then by the linear approximation solutions of system (17), we have thereby
(18)x=eλ1(ξ)τx10∂x∂x1+e-ρ1(ξ)τx20∂x∂x2+eλ2(ξ)τx30∂x∂x3+e-ρ2(ξ)τx40∂x∂x4=s-1x10∂x∂x1+sαx20∂x∂x2+s-νx30∂x∂x3+sβx40∂x∂x4.
Formula (18) indeed maps a point in S0 to a point in S1 in some subset of U if we substitute x by Ξ1.Now takeΞ0∈S0 as the initial point. System (5) must have an orbit γ(t,ξ) starting at Ξ0, passing through S1 with an intersection Ξ1, and finally returning to S0 at some point Ξ2; see Figure 1. From (13), (16), and (18), we can derive Ξ2-Ξ0=(χ12,0,χ32,χ42)-(χ10,0,χ30,χ40)≜(ε1(s,ξ),0,ε3(s,ξ),ε4(s,ξ)), where
(19)ω14ε1s,ξ=δsβ-ω14x31sν-ω44ω42-1sαx20+ω14M1ξ+h.o.t.,ε3s,ξ=x31-ω31-1δs+ω13ω44ω14-1-ω43ω42-1sαx20+M3ξ+h.o.t.,ε4s,ξ=ω42-1sαx20-x20+ω12sνx31+ω32ω31-1δs+M4ξ+h.o.t.
Denote a=ω44w42-1, b=ω32w31-1, and ω=ω14. Then the gap between the points Ξ2 and Ξ0 can be represented by (refer to [13–15])
(20)γ2T+τ,ξ-γT,ξ=εγs,ξ=-aM4ξsα+δsβ+ωM3ξsν-abδs1+α+ωM1ξ+Os1+ν.
Obviously, εγ(s,ξ)=0 means that system (5) has closed orbits.


## 3. Saddle-Node Bifurcations

From this section, we analyze bifurcation construction of system (5). Firstly, set r=sα. Then εγ(r,ξ)=0 equals
(21)aM4ξr-δrβ/α-ωM3ξrν/α+abδr1+1/α-ωM1ξ+Or1+ν/α=0
or
(22)r=δaM4ξrβ/α+ωM3ξaM4ξrν/α-abδaM4ξr1+1/α+ωM1ξaM4ξ+1aM4ξOr1+ν/α.DefineS(ξ)=ωM1ξ/aM4ξ. When S(ξ)=O(∥M4ξ∥α/β-α), aM4ξr is the leading term in (21), so (21) has a small solution r=S(ξ)+h.o.t.>0; but as ∥Sξ∥≫∥M4ξ∥α/β-α, r=(-δ-1ωM1ξ)α/β+h.o.t.>0. No matter which case, a periodic orbit of system (5) exists.Theorem 1.
Under(A1)–(A3) and for 1+α>β, system (5) has a 1-periodic orbit.To look for saddle-node bifurcations of 1-periodic orbits, it is enough to differentiate (22) with respect to r. Consider
(23)1=βδαaM4ξrβ/α-1+νωM3ξαaM4ξrν/α-1+1aM4ξO(r1/α).
Solving (23) for r, there is
(24)r=αaM4ξβδα/β-α+OM3ξα/β-αM4ξαν-α/β-α2.
Then substituting (24) into (22), an asymptotic expression for a saddle-node bifurcation is given by
(25)Sξ=β-αβαaM4ξβδα/β-α+O(M3ξM4ξν-β+α/β-α).Furthermore, if we continue to differentiate (23), there is
(26)0=β-αβδα2aM4ξrβ/α-2+ν-ανωM3ξα2aM4ξrν/α-2+1aM4ξO(r1-α/α).
Equation (26) is solvable for β>ν with
(27)r=-(ν-α)νωM3ξ(β-α)δβα/β-ν+O(M3ξα(1-α)/β-ν2).
This is a triple solution of (22). It means that a saddle-node bifurcation of a triple 1-periodic orbit exists. The asymptotic expression can be derived from (22) and (23):
(28)S(ξ)=-(ν-α)νωM3ξ(β-α)δβα/β-ν+O(M3ξα/β-ν),
where
(29)ωM3ξ=-β-αδβν-αναν-αaM4ξδβν-ββ-ν/β-α+o(M4ξβ-ν/β-α).Theorem 2.
Under(A1)–(A3) and for 1+α>β, system (5) has a saddle-node bifurcation SN of a double 1-periodic orbit given by (25) in the parameter space; moreover, for β>ν, system (5) has a saddle-node bifurcation SN2 of a triple 1-periodic orbit given by (28).Remark 3.
For the caseβ<ν, (26) has no sufficiently small positive solution, so there does not exist n-multiple 1-periodic orbit bifurcation for n≥3.Now we define a surface in the parameter space ofξ:
(30)H1(ξ)=ξ:M1ξ+o1=0.
On the surface H1, (21) equals
(31)raM4ξ-δrβ/α-1-ωM3ξrν/α-1+abδr1/αk+Or1+ν/α-1=0.
Clearly, it has a zero solution r1=0. If we differentiate the part in the parentheses in (31) for r, we get
(32)-β-ααδrβ/α-2-ν-ααωM3ξrν/α-2+O(r1/α-1)=0.
It has a solution for β>ν:
(33)r2=-ν-α(β-α)δωM3ξα/β-ν+O(M3ξα(1+α-ν)/β-ν2).
Then we obtain another saddle-node bifurcation similarly:
(34)Rξ=β-νβ-α-ν-αβ-αδωM3ξν-α/β-ν+O(M3ξ1+ν-β/β-ν),
where R(ξ)=aM4ξ/ωM3ξ.Notice that (32) has no solution for β<ν. But from (31), a small positive solution in the form r2′=(δ-1aM4ξ)α/β-α+O(∥M3ξ∥α/β-α∥M4ξ∥α(ν-α)/(β-α)2) exists.So we can conclude the following.Theorem 4.
Under(A1)–(A3) and for 1+α>β>ν, system (5) has a homoclinic-saddle-node bifurcation HSN of a 1-homoclinic orbit and a double 1-periodic orbit confined on H1∩R(ξ) while, for β<ν, system (5) has only a 1-homoclinic orbit and a 1-periodic orbit in the parameter space and the 1-homoclinic orbit is of codimension-1.Remark 5.
In Theorem4, the 1-homoclinic orbit may be nongeneral, that is, may be a flip orbit, because the orbit can connect the saddle along the weak unstable and strong stable directions if M4ξ=0.


## 4. Homoclinic-Doubling and Periodic-Doubling Bifurcations

Now we focus on 2-homoclinic orbits and 2-periodic orbits. Correspondingly, the gap functions are(35)r1-δaM4ξr1β/α-ωM3ξaM4ξr2ν/α+abδaM4ξr11+1/α-S(ξ)+1aM4ξO(r11/αr2ν/α)=0,r2-δaM4ξr2β/α-ωM3ξaM4ξr1ν/α+abδaM4ξr21+1/α-S(ξ)+1aM4ξO(r21/αr1ν/α)=0.To find a 2-homoclinic orbit, the above two equations must have a kind of solutions withr1=0 and r2>0. That is,
(36)r2ν/α+ωM1ξωM3ξ+1ωM3ξO(r2ν+β/α)=0,r2-δaM4ξr2β/α+abδaM4ξr21+1/α-Sξ+1aM4ξOr21+β/α=0.
Then we get
(37)Sξ=-M1ξM3ξα/ν-δaM4ξ-M1ξM3ξβ/ν+1aM4ξOM1ξM3ξ1+α/ν,
in the region defined by ∥M1ξ∥≪∥M3ξ∥β/β-ν.To find a 2-periodic orbit, the gap functions will have two positive solutionsr1 and r2. We suppose that r2=(1+ϵ)r1 after the reparametrization ξ=(ξ1,ϵ). Then there are
(38)r1-δaM4ξr1β/α-ωM3ξaM4ξ1+ϵν/αr1ν/α-S(ξ)+1aM4ξOr11+α/α=0,1+ϵr1-δaM4ξ1+ϵβ/αr1β/α-ωM3ξaM4ξr1ν/α-Sξ+1aM4ξO1+ϵ1+α/αr11+α/α=0.
Subtracting the two equations, there is
(39)r1=δ-1ϵ1+ϵβ/α-1α/β-αaM4ξα/β-α+OM3ξα/β-αϵr1ν-α/β-α.
Finally, we get the 2-periodic orbit bifurcation:
(40)Sξ=1-ϵ1+ϵβ/α-1δ-1ϵ1+ϵβ/α-1α/β-α×aM4ξα/β-α+OM3ξM4ξα+ν-β/β-α.Remark 6.
Obviously, ifr1=r2, the 2-periodic orbit is close to the double 1-periodic orbit perturbed from the saddle-node bifurcation. This is true by taking limit ϵ→0 in (40), and one may get the similar approximate expression given in (25).If we continue the computation, we can finally get an asymptotic expression of the homoclinic-doubling bifurcation of2n-homoclinic orbit and the periodic-doubling bifurcation of 2n-periodic orbit with the same leading terms as in (37) and (40), respectively. For example, for a 4-homoclinic orbit or a 4-periodic orbit, the gap functions are
(41)r1-δaM4ξr1β/α-ωM3ξaM4ξr2ν/α+abδaM4ξr11+1/α-S(ξ)+1aM4ξo(1)=0,r2-δaM4ξr2β/α-ωM3ξaM4ξr3ν/α+abδaM4ξr21+1/α-S(ξ)+1aM4ξo(1)=0,r3-δaM4ξr3β/α-ωM3ξaM4ξr4ν/α+abδaM4ξr31+1/α-S(ξ)+1aM4ξo(1)=0,r4-δaM4ξr4β/α-ωM3ξaM4ξr1ν/α+abδaM4ξr41+1/α-S(ξ)+1aM4ξo(1)=0.
We need only to consider solutions r1=0 and ri>0, i=2,3,4, for the 4-homoclinic orbit or all the positive solutions for 4-periodic orbit. For concision, we omit the details here.Now we can claim our last theorem.Theorem 7.
Under(A1)–(A3) and for 1+α>β>ν, system (5) has a homoclinic-doubling bifurcation HD2n of 2n-homoclinic orbit and a periodic-doubling bifurcation PD2n of 2n-periodic orbit defined by (37) and (40), respectively, in the parameter region ∥M1ξ∥≪∥M3ξ∥β/β-ν.From the above analysis, one may see that all of these bifurcation surfaces have the same orderS(ξ)=O(∥M4ξ∥α/β-α) except HSN and are tangent to H1. To be clear, we illustrate these bifurcation surfaces in the parameter plane in Figure 2.Bifurcation surfaces for1+α>β, ν>β in (a) and for 1+α>β>ν in (b). 0 means no periodic orbits and n means n periodic orbits. Chaos occurs in the region bounded by HD2n and PD2n.
(a)
(b)



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