Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Ling-Hao Zhang; Wei Wang

doi:10.3934/dcds.2017024

Article Contents

2017, Volume 37, Issue 1: 591-604. Doi: 10.3934/dcds.2017024

This issue Previous Article The attractors for 2nd-order stochastic delay lattice systems Next Article Bound state solutions of Schrödinger-Poisson system with critical exponent

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Ling-Hao Zhang^1, and
Wei Wang^2, ,

1.
Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China
2.
School of Computing and Engineering, Huddersfield University, HD 4, the United Kingdom

* Corresponding author: wangweifrancistju@yahoo.com
* Corresponding author: wangweifrancistju@yahoo.com

Received: June 30, 2015

Revised: August 31, 2016

Published: September 2017

This work is supported by the National Natural Science Foundation of China (Grand No. 11102127,11372210) and Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 12JCYBJC12500)

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Abstract

In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

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Mathematics Subject Classification: 37L10, 74H60.

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Figure 1. (a) The phase portrait of homoclinic orbit; (b) The phase portrait of heteroclinic orbit

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Figure 2. (a) The phase portrait of heteroclinic orbit; (b) The curve of $u$; (c) The curve of $\dot u$; 1-without perturbation, 2-with perturbation

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Figure 3. The flow chart of the new approach

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Figure 4. Heteroclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method

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Figure 5. Heteroclinic orbit of the oscillator; 1-Second-order direct method, 2-Numerical simulation method

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Figure 6. Critical bifurcation values; 1-Second-order direct method, 2-Numerical simulation method, 3-Melnikov integration method

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Figure 7. Homoclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method

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Table 1. Comparison of the variables obtained by different methods

Group	Parameter values				$\omega_{10} $	$\theta $	$\mu $
$G_{i}$	$\omega_{0} $	$\varepsilon $	$\alpha $	$\gamma_{2,1} $	$M_{1}$	$M_{1}$	$M_{1}$	$M_{2}$	$M_{3}$
1	1.5	2	2	1	1.154	-0.192	0.220	0.228	0.225
2	2	1.5	5	3	1.596	-0.230	0.481	0.488	0.480
3	2	1	5	4	1.556	-0.205	0.630	0.648	0.640
4	2	2	5	3	1.745	-0.298	0.506	0.493	0.480
Note: $M_{1}$-Direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Melnikov integration method

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Table 2. Comparison of the variables obtained by different methods

Group	Parameter values				$\omega_{10} $	$\theta $	$\mu $
$G_{i}$	$\omega_{0} $	$\varepsilon $	$\alpha $	$\gamma_{2,1} $	$M_{1}$	$M_{1}$	$M_{1}$	$M_{2}$	$M_{3}$
3	2	1	5	4	1.559	-0.211	0.630	0.648	0.645
4	2	2	5	3	1.737	-0.291	0.506	0.493	0.494
Note: $M_{1}$-First-order direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Second-order direct method

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