Bifurcations and Exact Bounded Solutions of Some Traveling Wave Systems Determined by Integrable Nonlinear Oscillators withq-Degree Damping

Author:

Zhang Lijun1,Chen Guanrong2,Li Jibin1ORCID

Affiliation:

1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong Province, P. R. China

2. Department of Electrical Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR, P. R. China

Abstract

For a class of nonlinear diffusion–convection–reaction equations, the corresponding traveling wave systems are well-known nonlinear oscillation type of systems. Under some parameter conditions, the first integrals of these nonlinear oscillators can be obtained. In this paper, the bifurcations, exact solutions and dynamical behavior of these nonlinear oscillators are studied by using methods of dynamical systems. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink and anti-kink wave solutions, as well as limit cycles, are obtained. Most important and interestingly, a new global bifurcation phenomenon of limit bifurcation is found: as a key parameter is varied, so that singular points (except the origin) disappear, a planar dynamical system can create a stable limit cycle.

Funder

National Natural Science Foundation of China

Natural Science Foundation of Jilin Province

Publisher

World Scientific Pub Co Pte Ltd

Subject

Applied Mathematics,Modeling and Simulation,Engineering (miscellaneous)

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