Abstract
This work extends the high-order Melnikov method established by FJ Chen and QD Wang to heteroclinic orbits, and it is used to prove, under a certain class of perturbations, the heteroclinic orbit in a planar vector field that remains unbroken. Perturbations which have this property together form the heteroclinic persisting space. The Van der Pol system is analysed as an application.
1. Introduction
Since the pioneering work of Poincaré on celestial mechanics [1, 2], homoclinic tangles created by transversal intersection of the stable and unstable manifolds have been playing a prominent role in the progress of chaos theory. In 1967, Smale observed that horseshoe chaos occurs in all homoclinic tangles [3]. For the past few years, the studies in [4–7] presented that periodic sinks and H nonlike attractors are also residing in homoclinic and heteroclinic tangles. The H nonlike attractors provide observable chaos in the sense of Sinai–Ruelle–Bowen (SRB) measure [8, 9].
In general, the standard Melnikov function is enough to verify whether the homoclinic tangle or heteroclinic tangle occurs [10, 11]. However, there exist perturbations which make the standard Melnikov function identically equal to zero. In such a case, no simple zero exists. As a result, we need to calculate high-order Melnikov functions for further investigation. There have been many previous works to refer high-order Melnikov method (see [12–19]). We focus on the method presented in [13] and extends the high-order Melnikov functions for homoclinic orbits to heteroclinic orbits.
Naturally, another question arises: does there exist a perturbation such that all the high-order Melnikov functions identically equal to zero? This means the homoclinic or heteroclinic solution remains unbroken. Through the high-order Melnikov method, we prove that the answer of the above question is positive. The union of such perturbations will be defined as heteroclinic persisting space.
This paper is built up as follows: In Section 2, we outline the main results. Section 3 is devoted to focus on the derivation of high-order Melnikov functions for heteroclinic orbits. Then, in Section 4, we prove the heteroclinic orbit remains unbroken under a certain class of autonomous perturbations. Finally, Van der Pol system is considered in Section 5.
2. Statement of Results
We start with a planar system:
Assume that (1) has two hyperbolic saddle fixed points and with a heteroclinic solutionsuch that
The functions and are real analytic near .
Adding time-periodic perturbations to (1) yieldswhere is a small parameter. The periodic functions are real analytic with respect to , and they are terms of second order and higher at and . Let be the unstable manifold of and be the stable manifold of . In general, for . We let be the splitting distance, which can be expressed in the form ofwhere is equivalent to the standard Melnikov function in [10, 11] and , are the high-order Melnikov functions. Moreover, we write , , , , , and for easy of notations. We also introducewhere
In what follows, we give the formulas of and .
Theorem 1. In (5), we have the following:(i)Integral formula for :(ii)Integral formula for :
On the other hand, we consider the question whether there exist any perturbations such that the heteroclinic solution remains unbroken, which implies the splitting distance vanishes identically. By using the high-order Melnikov functions above, such class of perturbations are obtained. Before giving our main theorem, we first introduce some definitions.
Definition 1. Let be a smooth manifold and and be vector fields on it. We assume there exists a heteroclinic orbit of and a heteroclinic orbit of , where is a small parameter. We call a heteroclinic persisting perturbation of ifin the sense of Hausdorff limit.
Definition 2. All heteroclinic persisting perturbations of are referred to as the heteroclinic persisting space of , which we denote by .
It is not difficult to find a heteroclinic persisting perturbation. Let in (4) be satisfyingThen, we have
It follows that is also a heteroclinic solution of (4). By Definition 1, we claim is a heteroclinic persisting perturbation.
However, this heteroclinic persisting perturbation is trivial, since the heteroclinic orbit is just the previous one. We would like to quest for more general ones. For this purpose, we focus on an autonomous perturbed system:
Then, splitting distance (5) can be rewritten aswhere is the -th order Melnikov integral, The following fact is trivial.
The fact is that is a heteroclinic persisting perturbation if and only if for all , where is sufficiently small.
Note that the geometric shape of heteroclinic orbits in planar vector fields are often symmetric which contributes to the computations of high-order Melnikov integrals. We obtain the following theorem.
Theorem 2. For the given vector field defined by (1), we assume is odd and is even in and is even and is odd in . Then, we have
The example considered in Section 5 will show that the parity assumption made in Theorem 2 is without loss of generality.
3. High-Order Melnikov Method
This section is intended to derive the high-order Melnikov functions for equation (4). The method here is based on the homoclinic case introduced in [13], but there still exist some nonnegligible differences between them.
3.1. Canonical Equation around the Heteroclinic Orbit
We introduce new phase variables by letting
Differentiating both sides of (16) with respect to gives
By Cramer’s Rule and (4), we havewhere
We suppose to be a stable solution or an unstable solution of system (18) such that and . Let and ; then, is well-defined for all or , and it is the solution of the systemsatisfying . To differ from the other solutions, we call them the primary stable and unstable solutions.
3.2. Integral of Primary Solutions
We letas one more change of variables for system (20). Then, we have
Expanding , , , and at , we obtain
It then follows that
We solve (24) to obtain the primary stable solutionand the primary unstable solutionin which and are the initial values.
Before proceeding further, we need to consider the asymptotic behaviors of and as which are crucial to get the integral equations for primary solutions. Observe thatand due to the local linearization theory, there exists a small number such thatwhere and are terms of second order and higher at and and are terms of second order and higher at . We assume that are the eigenvalues of and are the eigenvalues of . There exist two invertible matrices and such that
In what follows, we introduce two coordinate transformations:
Unperturbed system (1) is transformed to
In the case , our problem is about the unstable solution in a small neighborhood of . Let be sufficiently large. For , we can find a near-identity coordinate transformation:where and are homogeneous polynomials of degree in , such that the unperturbed system near is linearized to
Assume that the power series in (33) are convergent on . It is not difficult to verify thatwhere be such that and . We solve (35) to obtain
On the other hand, let . The problem is then about the stable solution in a small neighborhood of . We similarly havewhere and are homogeneous polynomials of degree .
Lemma 1. For the above heteroclinic solution , we have
Proof. We recall thatand letThen,This is to implyAs , we infer from (36) thatAccordingly, we haveRecall (29), and we haveSubstituting (45) into (44) givesAs , we infer from (37) thatAfter similar derivation, we haveThis completes the proof.
By using Lemma 2, one can easily verify that
Observe that and should be bounded as , and the initial values in (25) and (26) are claimed to be
Substituting (50) and (51) into (25) and (26), we have the integral equations of the primary stable solutionand the primary unstable solution
3.3. Derivation of High-Order Melnikov Functions
The properties of here are coincident with the ones in [13]. We have the following power series:
In what follows, we substitute (54) into (52) and (53) to determine and inductively. They arefor . Here, are nonnegative integrals, and , , and . and are coefficient functions of , which have the form of
Here, is a constant depending on the specific term. In the next section, these notations contribute to the proof of Theorem 2.
Recall the definition of , and we have
Then,and for the -th order Melnikov function iswhere and are the values by letting in (56). Particularly, we have
Up to now, the formula of can be obtained.
4. Heteroclinic Persisting Perturbations
This section serves as the proof of Theorem 2. For autonomous perturbations, all the functions above having the variable are now dropped off. Before starting our proof, we need some lemmas.
Lemma 2. Under the conditions of Theorem 2, we have the following:(1)If is odd, then and are even in , while and are odd in (2)If is even, then and are odd in , while and are even in .
Proof. We will show the parity of , and the other three can be verified similarly. Recall thatSince is an odd function while is even, then and are even and and are odd. It follows thatDenote . We haveMoreover, the parity in will not change by taking a derivative with respect to . Inductively, we obtainSince differs from by a constant factor, the proof is now completed by letting .
At the end of this section, let us prove Theorem 2 with the help of the preceding preparations.
Proof of Theorem 2. To start with, we prove, for all ,By using mathematical induction, we first consider the case . Recall thatwhere is an odd function in and and are even functions in . Moreover, we haveBy means of variable substitution, we haveNow, we let be a positive integer, and suppose that (66) holds for all nonnegative integer . According to (56), we haveNotice that is the counterpart of , changing “” and “” to “” and “,” respectively. Since , we obtainThat is,From Lemma 2, we haveFinally, we obtainThe proof of (66) is completed.
Recall (60), and for autonomous perturbation, we haveDue to (14), we thus arrive the conclusion that for all small . This means the stable manifold and unstable manifold coincide. As a result, there exists a heteroclinic orbit which satisfiesTherefore, is the heteroclinic persisting perturbation of .
5. An Concrete Example: Van der Pol System
Let us consider the Van der Pol system on a torus:where . Without loss of generality, we fix a periodic region of the vector field in which we have four hyperbolic saddle fixed points:
We write and , and the heteroclinic solutions are
The vector field of (77) is displayed in Figure 1.
In order to obtain the heteroclinic persisting perturbations of these heteroclinic orbits, we need some rotation transformations of coordinates to satisfy the preconditions in Theorem 2.
5.1. Persistence of and
Introducing the change of variableswe obtain
In such a case, the saddle points are transformed to
Denote
The new vector field and heteroclinic orbits are plotted in Figure 2.
We note that the image of is
One can easily verify that is odd in while is even and is even in while is odd. For the convenience of derivation, we denote
Applying Theorem 2, we have . Now, we consider the following equations:where . Let us go back to the -plane and replace by for easy computation. It follows that
We introduce to be the set obtained by through the transformation of coordinates (80), which means given , there exists a such that
Thus, we have
Obviously, . We also have . In particular, we setThis implies
We consider the following system:
It follows that there exist two heteroclinic solutions which satisfy equality (10) in Definition 1. The perturbed vector field and the two heteroclinic orbits and are plotted in Figure 3. Figure 4 shows the other two are broken.
5.2. Persistence of and
For the heteroclinic orbits and , we introduce another transformation of coordinates:
It follows that
Denote
Similarly, the image of is transformed to
The new saddle points of (94) are
One can easily verify that is odd in while is even and is even in while is odd. Denote
It then follows from Theorem 2 that . We take , and consider the following perturbed system:
Back to plane, we have
Similarly, we obtain the set through transformation (93). Then, for any , we obtain
Consequently, we have
Since , we also have .
We takeIt follows that
Hence, we get the following perturbed system:
Figures 5 and 6 are the plots of the perturbed vector field of (105) and the two heteroclinic orbits and . The other two are broken.
5.3. Persistence of the Heteroclinic 4-Cycle
We note that in unperturbed system (77), forms a heteroclinic 4-cycle. It is natural to ask when the heteroclinic 4-cycle remains unbroken. We rewrite and as
From (80) and (93), we have the relation between phase variables and as
Thus, we have
Our goal is to figure out the elements in and so we letIt then follows that
Depending on the parities of , , , and with respect to the first variable, we can use (110) to obtain their parities with respect to the second variable. Let
We finally getin which and .
In order to visualize this result, we consider the following perturbations:
Adding them to system (77) gives
Since , all the four heteroclinic orbits of (114) remain unbroken (see Figure 7).
Data Availability
All data included in this study are available upon request to the author.
Conflicts of Interest
The author declares no conflicts of interest.