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Bifurcations of Heteroclinic Loop with Twisted Conditions

Yinlai Jin

http://orcid.org/0000-0002-5728-6391

School of Mathematics and Statistics, Linyi University, Linyi 276005, Shandong, P. R. China

E-mail Address: jinyinlai@lyu.edu.cn

Corresponding author.

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Suoling Yang

School of Mathematics and Statistics, Linyi University, Linyi 276005, Shandong, P. R. China

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, Shandong, P. R. China

E-mail Address: suoer-2008@163.com

Corresponding author.

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Yuanyuan Liu

School of Mathematics and Statistics, Linyi University, Linyi 276005, Shandong, P. R. China

E-mail Address: liuyuanyuan1117@163.com

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Dandan Xie

School of Mathematics and Statistics, Linyi University, Linyi 276005, Shandong, P. R. China

E-mail Address: xiedandan01@sina.com

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, and

Nana Zhang

School of Mathematics and Statistics, Linyi University, Linyi 276005, Shandong, P. R. China

E-mail Address: zhangnana01@sina.com

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https://doi.org/10.1142/S0218127417501206Cited by:4 (Source: Crossref)

Abstract

The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case $ρ_{1}^{1} > λ_{1}^{1}$ $ρ_{1}^{1} > λ_{1}^{1}$ , $ρ_{2}^{1} < λ_{2}^{1}$ $ρ_{2}^{1} < λ_{2}^{1}$ , $ρ_{1}^{1} ρ_{2}^{1} < λ_{1}^{1} λ_{2}^{1}$ $ρ_{1}^{1} ρ_{2}^{1} < λ_{1}^{1} λ_{2}^{1}$ , where $- ρ_{i}^{1} < 0$ $- ρ_{i}^{1} < 0$ and $λ_{i}^{1} > 0$ $λ_{i}^{1} > 0$ are the pair of principal eigenvalues of unperturbed system at the critical point $p_{i}$ $p_{i}$ , $i = 1, 2$ $i = 1, 2$ . Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

This work is supported by the National Natural Science Foundation of China (No. 11601212), Shandong Province Natural Science Foundation (ZR2015AL005), Shandong Province Higher Educational Science and Technology Program (J16LI03) and the Applied Mathematics Enhancement Program of Linyi University.

Keywords:

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