Bifurcation of an Orbit Homoclinic to a Hyperbolic Saddle of a Vector Field in R 4

We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field inR. We give an expression of the gap between returning points in a transverse section by renormalizing system, through which we find the existence of homoclinicdoubling bifurcation in the case 1 + α > β > ]. 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.


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 where   () = ( − )( − 1).Through the variable transforming  =  + , system (1) is then in an ODE form: It has an orbit homoclinic to the equilibrium (, V, ) = 0 which corresponds to a solitary wave (, )(, ) = (, )() of system (1).The authors detected how homoclinic branch converted a 1-homoclinic orbit to a -homoclinic orbit.
In [2], a reversible water wave model was studied: The system admits a flip orbit for  > 2,  > 0, and shows the existence of the -homoclinic orbit in some circumstances on two sides of the flip bifurcation.
In fact, the homoclinic-doubling bifurcation, which switches a 2 −1 -homoclinic orbit to a 2  -homoclinic orbit, exists extensively in systems with flips; see [3][4][5][6] and the references therein.A simple and analytic model permitting these flips was initially given by Sandstede in a threedimensional 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][11][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.
Notice that if the gap ‖ − (, )− + (, )‖ = 0 in the transverse section  0 , it means that the homoclinic orbit is kept (see Figure 1(a)) but it may not be of codimension-1.Moreover, the system (5) still has other solutions   (, );  is a natural number.Set the time of the orbit   (, ) from  0 to  1 and from  1 to  0 to be   and   , respectively; there are Actually   (, ) is a regular orbit and will be periodic if the gap ‖  ( +   +   , ) −   (, )‖ = 0; namely, the orbit starting in  0 will return to  0 after the time   +   ; see Figure 1(b).
From above, we see that the gap in the transverse section  0 of some orbits is crucial to study bifurcations of the system.So in the next section we try to quantitate the gap size.

Main Method
To well carry out our discussion, we give some hypotheses for the system (5) here.
To look for saddle-node bifurcations of 1-periodic orbits, it is enough to differentiate (22) with respect to .Consider ) . (24) Then substituting (24) into ( 22), an asymptotic expression for a saddle-node bifurcation is given by Furthermore, if we continue to differentiate (23), there is Equation ( 26) is solvable for  > ] with ) . ( 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): where Theorem 2. Under ( 1 )-( 3 ) and for 1 +  > , system (5) has a saddle-node bifurcation  of a double 1-periodic orbit given by (25) in the parameter space; moreover, for  > ], system (5) has a saddle-node bifurcation  2 of a triple 1periodic orbit given by (28).
Remark 3.For the case  < ], (26) has no sufficiently small positive solution, so there does not exist -multiple 1-periodic orbit bifurcation for  ≥ 3.
Now we define a surface in the parameter space of : On the surface  1 , (21) equals Clearly, it has a zero solution  1 = 0.If we differentiate the part in the parentheses in (31) for , we get It has a solution for  > ]: ) . ( Then we obtain another saddle-node bifurcation similarly: where () =  4 / 3 .
Notice that (32) has no solution for  < ].But from (31), a small positive solution in the form ) exists.So we can conclude the following.Theorem 4.Under ( 1 )-( 3 ) and for 1 +  >  > ], system (5) has a homoclinic-saddle-node bifurcation  of a 1-homoclinic orbit and a double 1-periodic orbit confined on  1 ∩ () while, for  < ], system (5) has only a 1-homoclinic orbit and a 1-periodic orbit in the parameter space and the 1homoclinic orbit is of codimension-1.Remark 5.In Theorem 4, 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  4  = 0.

Homoclinic-Doubling and Periodic-Doubling Bifurcations
Now we focus on 2-homoclinic orbits and 2-periodic orbits.Correspondingly, the gap functions are To find a 2-homoclinic orbit, the above two equations must have a kind of solutions with  1 = 0 and  2 > 0. That is, ) , in the region defined by ‖ 1 ‖ ≪ ‖ 3 ‖ /(−]) .To find a 2-periodic orbit, the gap functions will have two positive solutions  1 and  2 .We suppose that  2 = (1 + ) 1 after the reparametrization  = ( 1 , ).Then there are Subtracting the two equations, there is ) . (39) Finally, we get the 2-periodic orbit bifurcation: Remark 6. Obviously, if  1 =  2 , the 2-periodic orbit is close to the double 1-periodic orbit perturbed from the saddle-node bifurcation.This is true by taking limit  → 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 of 2  -homoclinic orbit and the periodic-doubling bifurcation of 2  -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 We need only to consider solutions  1 = 0 and   > 0,  = 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.

Figure 2 :
Figure 2: Bifurcation surfaces for 1 +  > , ] >  in (a) and for 1 +  >  > ] in (b).0 means no periodic orbits and  means  periodic orbits.Chaos occurs in the region bounded by HD 2  and PD 2  .