Bifurcation Analysis of the Multiple FlipsHomoclinic Orbit∗

Tiansi ZHANG Deming ZHU

1 Introduction and Hypothesis

Homoclinic bifurcation is one of the origins of chaotic behaviors and has important applications in many fields.Amongst them,the flips cases began to catch attention in the last decade.As is well-known,homoclinic orbits generically occur as a codimension-one phenomenon if the genericity conditions are all kept.Otherwise,a higher codimension instance may take place,such as the case of resonant eigenvalues,which was considered in[1];orbit flips,that is,the non-principal phenomenon,was treated in[2–3];and the case of inclination flips or critically twisted homoclinic orbits,for which one can refer to[4–5],etc,where the homoclinic-doubling bifurcation,a codimension-two transition from ann-homoclinic to a 2n-homoclinic orbit,is found to exist(see[6–10]).One may refer to the model for electro-chemical oscillators,or the FitzHugh-Nagumo nerve-axon equations(see[11]),a Shimitzu-Morioka equation for convection instabilities(see[12]),and a Hodgkin-Huxley model of thermally sensitive neurons(see[13])for some applications.

Recently,the flip of heterodimensional cycles or accompanied by transcritical bifurcation was studied(see[14–16]).The double and triple periodic orbit bifurcations were proved to exist,and also some coexistence conditions for the homoclinic orbit and the periodic orbit were given.But the research is not concerned with multiple flips or homoclinic-doubling bifurcations.

This paper produces mainly a study of the homoclinic bifurcation with multiple flips,concretely one orbit flip and two inclination flips,which takes place at least in a four-dimensional system.Compared with the above work mentioned,the subject of multiple flips,especially with resonance,is very challenging and difficult,because the codimension is higher and the relationship of eigenvalues is more subtle.All these lead to the stronger degeneracy of bifurcation equations.So it is extremely hard to solve the bifurcation equations.To well handle the problem,we choose the method initially established in[17],that is, first construct a local specifical active coordinate system which reflects sufficiently the geometric structure of the corresponding invariant manifolds in a small tubular neighborhood of the homoclinic orbit,then establish a returning map,the Poincar´e map,and get the associated successor function.By a delicate analysis of the bifurcation equation,we get the existence of 1-periodic orbit,1-homoclinic orbit,some double and triple periodic orbits,and particularly the 2n-homoclinic orbit and their corresponding bifurcation surfaces.

The system to be considered isCras

and its unperturbed system is

whereandg(0,μ)=g(z,0)=0.

Suppose that(1.2)has an orbithomoclinic to the hyperbolic equilibriumz=0,which has two negative and two positive eigenvaluessatisfyingSetWs(resp.Wss)andWu(resp.Wuu)to be the stable(resp.strong stable)manifold and unstable(resp.strong unstable)manifold of the equilibriumz=0,respectively.We further make three assumptions:

(H1)(Resonance)for,whereandρ1(0)=ρ1.

(H3)(Inclination Flips)Denote byande−the unit eigenvectors corresponding toλ2and−ρ1respectively,and let

Hypothesis(H3)means that in two directions when,the homoclinic orbit Γ comes through two inclination flips.Under(H1)–(H3),the genericity conditions are all broken,and therefore a codimension-4 homoclinic bifurcation occurs.

2 Normal Forms and Successor Functions

To well construct the Poincar´e map and get the associated successor functions,we first need to transform(1.1)into a normal form in some neighborhoodUof the originO.In fact,with hypotheses(H1)–(H3)and the normal form theory,there must be aCr−4system as follows after four successiveCrtoCr−3transformations inU(see[10]for the details),

with the assumption

Indeed we have

whereandk6+λ2(0)=λ2,ρ1(0)=ρ1andρ2(0)=ρ2;a(μ),b(μ),c(μ)andd(μ)are parameters depending onμ.Equivalently,inU,we have

where(resp.is the local weak unstable(resp.weak stable)manifold which is tangent toe+(resp.e−)atz=0 andv(y)satisfiesv(0)=v?(0)=0.Namely,we have straightened the corresponding invariant manifolds.It is possible to choose some momentT,such thatandγ(T)={0,0,0,δ},whereδis small enough and{(x,y,u,v):|x|,|y|,|u|,|v|<2δ}⊂U.

Now we turn to consider the linear variational system of(1.2)and its adjoint system

From the matrix theory,we immediately have the following lemma.

Lemma 2.1There exists a fundamental solution matrixof(2.2)satisfying

whereand

ProofNotice that the tangent subspaceis invariant andis straightened to beuaxis.It is possible to choosesince.While for,it is becauseandpoints to thexaxis.

Remark 2.1The matrixis a fundamental solution matrix of(2.3),denoted byThenis bounded and tends to zero exponentially as

Take a new coordinateand define

Substitute it into(1.1),

A simple calculation gives the system in a new coordinates,

Now we want to construct a Poincar´e map.Firstly choose two cross sections of Γ (see Figure 1),

Then integrating both sides from−TtoTof the above equation,we further achieve

which means a regular map is well defined(see Figure 1(1)):

Figure 1 Transition Maps

Set pointsandThen taket=−TandTrespectively in(2.4),and it is easy to get the following relationship between two coordinate systems:

and

Next,we set up a singular map(see Figure 1(2))induced by the solutions of system(2.1)in the neighborhoodU.Firstly there is

whereτis the time going fromtoDenote the Silnikov timeThen there is.By the same approach,one can also get the valuesy1,u0andv1as below

It is exactly the definition of the mapF0.

With all the equations from(2.5)–(2.9),the Poincar´e mapis well defined as

Then we can get the associated successor function

3 Bifurcation Analysis

Notice that if the flying timeτof an orbit starting at a point inS0toS1is finite,then a periodic orbit of(1.1)exists,and accordingly the Silnikov timeifτis in finite,then a homoclinic orbit exists,and accordinglys=0.So from the establishment of the associated successor function,it is enough to look for the nonnegative solutionssof(2.10)to study the bifurcations.

First of all,G3=0 andG4=0 reveal that

Putting them intoG1=0,we get immediately the bifurcation equation

Here for concision,we have omitted the parameterμinλi(μ)andρi(μ),and replaced the exponentby one owing to(H1).Settingand,we find that,when

Therefore the implicit function theorem reveals thatG=0 has a unique solution

satisfyings(0)=0,u1(0)=0 andy0(0)=0.So(1.1)has a unique periodic orbit ass>0 or a unique homoclinic orbit ass=0,and they do not coexist.Furthermore,F(s,μ)=0 has explicitly a sufficiently small positive solutionOn the other hand,it has a solutions=0 when.So we have the following theorem.

Theorem 3.1Suppose that M1?=0and w33?=0hold.Then(1.1)has at most one1-periodic orbit or one1-homoclinic orbit in the neighborhood ofΓ.Moreover,a1-periodic orbit exists(resp.does not exist)as μis in the region defined by w31w33M1μ<0(resp.>0)and a1-homoclinic orbit exists as μ∈H1,but they do not coexist(see Figure2(1–2)).

From Theorem 3.1,we know that(1.1)may have a codimension-1 orbit near Γ homoclinic to the equilibriumOalongxandyaxes whenμ∈H1,while a codimension-2 orbit flip homoclinic orbit could exist ify0=M4μ+h.o.t.=0,wherey0is given byG4=0(see Figure 2(3)).So the following corollary is true.

Corollary 3.1Assume that the hypotheses of Theorem3.1are still valid.Then(1.1)has exactly a codimension-2orbit flip homoclinic orbit near04

To well develop our study,we de fi ne two functions

whereis a line andis a curve according to the variables.Clearly

Figure 2 Existence of the 1-periodic orbit(1-P)and the 1-homoclinic orbit(1-H)

Theorem 3.2Suppose thatRankand w33=0are valid.Then

(1)(1.1)has a unique(resp.not any)1-periodic orbit forand and

(2)(1.1)has a unique(resp.not any)1-periodic orbit for andand

Figure 3 Existence of 1-periodic orbit

Now ifwe can verify thatwhereforthat is,the termhas a lower order insois a positive solution ofH2(s,μ)=0.With the similar study of the solution ofH1(s,μ)=H2(s,μ),it is easy to get the result(2).This completes the proof.

Next we begin to look for the double 1-periodic orbit bifurcation.Redefine two functions

We know that a double 1-periodic orbit bifurcation surface exists if and only ifP(s,μ)=hold,that is,

In the region,the second equation of(3.2)yields a solution

asandorand.Combining with the value ofs∗,the first equation of(3.2)gives a tangency condition,which corresponds to the existence of the double periodic orbit bifurcation surface

forandand

Notice that,when the tangency takes place,the lineW=P(s,μ)lies under the curveW=Q(s,μ).So if−w14M1μincreases(resp.decreases),the line must intersect the curve at two(resp.no)sufficiently small positive points.Namely,F(s,μ)=0 has two small positive solutions,or equivalently,two 1-periodic orbits appear on the side ofwhich points to(sgnM3μ)M1.

asAnother double 1-periodic orbit bifurcation surfaceexists with the expression

con fi ned toandwhich has the normal vectorM1atμ=0.

Furthermore,whenis solvable with the solution

forw14M3μ<0,which is a triple small positive solution ofP(s,μ)=Q(s,μ).Multiplying both sides of the second equation of(3.2)bys∗and substituting it into the first one,we get easily the triple 1-periodic orbit bifurcation surface

forand

Obviously,the hypersurface has a normal plane span{M1,M4}atμ=0.To sum up,there are the following results.

Theorem 3.3Suppose thatRank(M1,M3,M4)=3,2λ1>ρ2>λ2and w33=0hold.Then in the small neighborhood of the orgin of μ space,

(1)For w14M1μ>0,w14M3μ>0and w42w44M4μ>0(or w14M1μ<0,w14M3μ<0andthere exists a double1-periodic orbit bifurcation surface as|M4μ|?or a double1-periodic orbit bifurcation surface

Moreover,they both have the normal vector M1at μ=0and bifurcate two1-periodic orbits on their side pointing to the direction(sgnM3μ)M1or(sgnw14)M1respectively,and have no1-periodic orbits on the other side(see Figure4).

(2)For and there exists a triple1-periodic orbit bifurcation surface SN2with the normal planespan{M1,M4}at μ=0,such that(1.1)has exactly a triple1-periodic orbit when μ∈SN2.

Figure 4

Remark 3.1There does not exist ann-multiple 1-periodic orbit bifurcation surface in any case forn>3.

In the case ofMiμ=0 fori=1,3,4,some fruitful bifurcation results are obtained.

Theorem 3.4Suppose thatand w33=0hold.Then there are the following results:

(1)If M1=0,(1.1)has exactly a1-homoclinic orbit,and moreover,

(a)(1.1)has a unique(resp.not any)1-periodic orbit for w42w44M4μ>0(resp.w42w44M4μ<0and w14M3μ>0);

(b)forRank(M3,M4)=2,w14M3μ<0and w42w44M4μ<0,there exists a double1-periodic orbit bifurcation surface

with a normal vector M4at μ=0,which undergoes two1-periodic orbits when μlies on the side ofpointing to the direction(sgnw42w44)M4and no undergoes1-periodic orbit in the opposite direction.

(2)If M3=0,

(a)(1.1)has exactly a(resp.not any)1-periodic orbit for w14M1μ<0(resp.w14M1μ>0and w42w44M4μ<0);

(b)forRank(M1,M4)=2,w14M1μ>0and w42w44M4μ>0,there exists a double1-periodic orbit bifurcation surface

with a normal vector M1at μ=0,which bifurcates two1-periodic orbits when μlocates on the side ofdirected by−(sgnw14)M1and bifurcates no1-periodic orbit on the opposite side.

(3)If M4=0,

(a)(1.1)has exactly a(resp.not any)1-periodic orbit for w14M1μ<0(resp.w14M1μ>0and w14M3μ>0);

(b)forRank(M1,M3)=2,w14M1μ>0and w14M3μ<0,there exists a double1-periodic orbit bifurcation surface

with a normal vector M1at μ=0,which bifurcates two1-periodic orbits when μlocates on the side of directed by−(sgnw14)M1and bifurcates no1-periodic orbit on the opposite side.

(4)If has exactly a1-homoclinic orbit and a1-periodic orbit for w42w44M4μ>0or w14M3μ<0respectively.

(5)If has only a1-periodic orbit for

(6)If(1.1)has only a1-homoclinic orbit.

ProofRecall that forM1=0,(3.1)becomes

Obviously,it has a zero solution that corresponds to a 1-homoclinic orbit.Setand

Similar to the proof of Thereom 3.2,it is easy to get a 1-periodic orbit asowing to the relative position of the lineW=P1(h,μ)and the curveW=Q1(h,μ).Thereby the claim(1)(a)holds.

Further,equationsPandexactly determine the double 1-periodic orbit bifurcation surfaceforandwith the corresponding tangent pointWithout difficulty,on the side ofwhich points tothere appear two 1-periodic orbits.Now(1)is complete.

For the cases(2)and(3),the proofs are very close to those in Theorems 3.2–3.3 or as above.We omit the details here and only give the tangent points of the corresponding double 1-periodic orbit bifurcation surface.WhenM3=0,set

In the caseM4=0,one may set another kind of curves

whereand the relevant tangent point is0.

The last claim is very clear.We finish the proof here.

From now on,we try to study the homoclinic doubling bifurcation.To begin with,we need to get the second returning successor functionF◦F(q0)−q0.Resetτ1andτ2to be the time going fromand fromtorespectively,and.Repeat the process of the establishment of(2.10),can be expressed as

Whenw33=0 andeliminatingandu3fromandj=3,4,and putting them intoand,we obtain

Notice that a 2-homoclinic orbit Γ2means that the orbit returns twice near the singular point in finite and in finite time respectively,which corresponds to the solutions1=0 ands2>0 ors1>0 ands2=0 of(3.3)and(3.4).So it is sufficient to seek the small solutions ofs1=0 ands2>0 by the symmetry ofG2.Therefore

forandwhich has a normal vectorM1atμ=0.

Going through the same procedure of findingH2,one can still find a 4-homoclinic orbit bifurcation surfaceH4(see[9–10]).Concretely,we first have,parallel to(3.3)and(3.4),

Without loss of generality,it is enough to consider the solutions1=0 andsi>0,i=2,3,4.Then(3.7)is

With all these values ofsifori=2,3,4,(3.9) finally defines the 4-homoclinic orbit bifurcation surface with the same principal part ofH2:

forM1μM3μ<0 andwhich has a vectorM1atμ=0.

Theorem 3.5Suppose thatRankand w33=0hold.Then in the neighborhood of the origin of μ space,there exists a2n-homoclinic orbit bifur-cation surfaceforand M1μM3μ<0,which has the normal vector M1at μ=0.

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