In dynamic systems, some nonlinearities generate special connection problems of non-Z2 symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2 symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2 heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2 symmetric nonlinear quintic systems (orbit with one cusp); and Z2 symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.
Heteroclinic orbits (HOs) play a central role in nonlinear dynamics and many scholars have undertaken the study of HOs and heteroclinic bifurcation [
Heteroclinic connection is a path joining two different equilibrium points.
Existing research of HOs is focused mainly on classic low-order Z2 symmetric system. The study of Z2 symmetric HOs in an unperturbed system has resulted in the development of a number of analytical methods. Some examples include the generalized harmonic multiple scales method [
In the aforementioned techniques, the HOs all meet the characteristics that the saddle points are symmetric about a center point. The more general asymmetric case remains to be further explored where the non-Z2 symmetric HOs are neither symmetric about
This paper is organized as follows: in Section
Consider the following general non-Z2 symmetric nonlinear dynamical system:
HOs of system (
When a small perturbation is applied (
The non-Z2 symmetric HOs of simple autonomous system have been widely studied, as shown by the solid line in Figure
However, in some complex systems, the unperturbed system will also result in a non-Z2 symmetric HO phenomenon where the saddle points satisfy the following condition:
A trajectory that does not possess Z2 symmetry characteristics is classified as non-Z2 symmetry. This means that the saddle points are asymmetric,
Comparison of the asymmetric heteroclinic orbits found using (1) classic analytical techniques (◯) and (2) numerical computation (solid).
In order to resolve this problem, the Padé approximation method [
In this section, the solving process of this analytical method is proposed. The differential equation, including the entire nonlinear part, is solved directly. In addition, the magnitude of the perturbation parameter
A generic case, such as the solid line orbits shown in Figure
First, without loss of generality, system (
Take a series solution of system (
The rest of the parameters of (
Usually, the initial point is taken as one of the maximum points on the trajectory where it also fulfills its tangent perpendicular to the
If system (
According to the abovementioned initial conditions, we consider a new heteroclinic solution expression given by
Expression (
When
The boundary condition for (a) the upper orbit and (b) the lower orbit.
By equating the heteroclinic solution
By applying boundary conditions (
The higher the approximation order, the better the accuracy. Ordinarily, a calculation to second-order approximation (QPA2) would meet the accuracy requirements (error range is
In this section, the analytical approach proposed is applied to several classical nonlinear systems.
The HO in this system meets the characteristics that the two distances between each saddle point and the center are not equal (condition (
The Nagumo system [
The phase diagram of Nagumo system (
The maximum point of the trajectory of autonomous system (
Comparison of HO calculated using the proposed analytical method (∘) and a numerical Runge-Kutta algorithm (solid).
The HOs (with one cusp) in this system meet the characteristics that the two distances between each saddle point and the center are not equal (condition (
Consider the following autonomous system containing non-Z2 symmetric term:
Firstly, we analyze unperturbed system (
Comparison of the orbits calculated using the analytical method (∘) and a numerical Runge-Kutta technique (solid), for
Comparison of the orbits calculated using the analytical method (∘) and a numerical Runge-Kutta technique (solid), for
By applying this method, the initial values
In the perturbed system (
Comparison of HOs for the present results and the Runge-Kutta procedure.
The HOs (with two cusps) in this system meet the characteristics that the distances between each saddle point and the center are equal (condition (
Consider the following triple-well Z2 symmetric autonomous system:
HOs for the present results and the Runge-Kutta procedure comparison.
When
Heteroclinic bifurcation curve comparison.
HOs for the present results and the Runge-Kutta procedure comparison.
The results obtained from all of the examples presented demonstrate the very high precision of the analytical method developed. This is reflected in the phase diagram with the very close superposition of the analytic and numerical orbits.
The undetermined Padé approximation method is proposed to construct general non-Z2 symmetric homoclinic and heteroclinic orbits. The geometry characteristics of the orbits and the original frequency effects due to nonlinear terms are all considered. An undetermined frequency parameter and new analytical expression are given. The convergence rate of progressive approximations is accelerated and the amount of computation time required is greatly reduced. The characteristics of the special heteroclinic orbits are discussed in detail and analytical expressions of those orbits are obtained by the method developed in this paper. Consequently, irrespective of whether the orbit is single or in pairs, Z2 or non-Z2 symmetric, and in a conservative or autonomous system, the proposed method demonstrates its superiority over other existing solutions and the scope of its application has been extended. In addition, when compared with numerical results, the analytic orbits and the values of the bifurcation parameters obtained by the method presented in this paper are almost identical.
Consider the following
The authors declare that they have no competing interests.
This project is supported by National Natural Science Foundation of China (Grant nos. 11372210 and 11402172) and Tianjin Research Program of Application Foundation and Advanced Technology (16JCQNJC04700).