Abstract

We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field in . 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 . 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.

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 where . Through the variable transforming , system (1) is then in an ODE form: It has an orbit homoclinic to the equilibrium 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 , , 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 -homoclinic orbit to a -homoclinic orbit, exists extensively in systems with flips; see [36] 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 [1012]. 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: where is sufficiently smooth and . Suppose that the system (5) has an orbit of codimension-1 homoclinic to a saddle equilibrium at . Let be a certain time, such that and are in some small neighborhood of . Then we can take two sections vertical to :

Generally, if the small parameter , the homoclinic orbit will not exist. But the system (5) must have solutions with the properties where and are the stable and unstable manifolds of the equilibrium , and . Notice that if the gap in the transverse section , 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 to and from to to be and , respectively; there are Actually is a regular orbit and will be periodic if the gap ; namely, the orbit starting in will return to after the time ; see Figure 1(b).

From above, we see that the gap in the transverse section 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.The spectrum , and .As , the homoclinic orbit along the strong stable manifold .Vectors in the strong unstable (resp., stable) manifold (resp., ) return to the saddle in the direction along (resp., ).

We know that the discontinuity of the functions or is confined in a special position in . Since , the space is of one dimension, where can be taken as the solution of the linear variational system and , based on the assumptions of and ; refer to [13] for the details.

Beside this, by the theories of matrix, the other three solutions denoted by , , and of the system (10) can also be taken in the following ways: satisfying

Now take a transformation where and . Then system (5) becomes the following ODE in the new variable ; namely, By (14), gives where .

Notice that in (13), represents in some meaning the deviation in the normal direction of the manifolds , so and ; in other words, (16) maps a point in to a point in .

On the other hand, from assumption (), system (5) admits a local linearization where .

Suppose and . Rescale the time and write , , and . Without loss of generality, we assume for sufficiently small . Otherwise, we can set and . Clearly, .

Then by the linear approximation solutions of system (17), we have thereby Formula (18) indeed maps a point in to a point in in some subset of if we substitute by .

Now take as the initial point. System (5) must have an orbit starting at , passing through with an intersection , and finally returning to at some point ; see Figure 1. From (13), (16), and (18), we can derive , where Denote , , and . Then the gap between the points and can be represented by (refer to [1315]) Obviously, means that system (5) has closed orbits.

3. Saddle-Node Bifurcations

From this section, we analyze bifurcation construction of system (5). Firstly, set . Then equals or

Define . When , is the leading term in (21), so (21) has a small solution ; but as , . No matter which case, a periodic orbit of system (5) exists.

Theorem 1. Under and for , 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 . Consider Solving (23) for , there is 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 and for , 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 of a triple 1-periodic 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 .

Now we define a surface in the parameter space of : On the surface , (21) equals Clearly, it has a zero solution . 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 .

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 and for , system (5) has a homoclinic-saddle-node bifurcation of a 1-homoclinic orbit and a double 1-periodic orbit confined on while, for , 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 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. 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 and . That is, Then we get in the region defined by .

To find a 2-periodic orbit, the gap functions will have two positive solutions and . We suppose that after the reparametrization . Then there are Subtracting the two equations, there is Finally, we get the 2-periodic orbit bifurcation:

Remark 6. Obviously, if , the 2-periodic orbit is close to the double 1-periodic orbit perturbed from the saddle-node bifurcation. This is true by taking limit 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 -homoclinic orbit and the periodic-doubling bifurcation of -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 and , , 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 and for , system (5) has a homoclinic-doubling bifurcation of -homoclinic orbit and a periodic-doubling bifurcation of -periodic orbit defined by (37) and (40), respectively, in the parameter region .

From the above analysis, one may see that all of these bifurcation surfaces have the same order except and are tangent to . To be clear, we illustrate these bifurcation surfaces in the parameter plane in Figure 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge the support from the National Natural Science Foundation of China (11271260) and the Innovation Program of Shanghai Municipal Education Commission (13ZZ116).