Abstract
We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer n in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.
Highlights
The Burgers–Huxley equation is a well-known nonlinear partial differential equation simulating nonlinear wave phenomena in physics, biology, economics and ecology
In this paper we focus on the existence of isolated periodic travelling trains which are caused by the presence of limit cycles
Due to the practical background, the value of u in model (1.2) is nonnegative, we only investigate the dynamical behavior near the equilibrium points with v ≥ 0 for system (1.5)
Summary
The Burgers–Huxley equation is a well-known nonlinear partial differential equation simulating nonlinear wave phenomena in physics, biology, economics and ecology. Latter on computing the singular point quantities of the non-degenerate center-focus type equilibrium the authors of [26] proved the existence of the isolated periodic wave solution in the non-degenerate case for equation (1.1). Bifurcations of isolated periodic wave trains for the reaction-diffusion equation have been extensively studied (see [12,13,23,27,28] and references therein) These bifurcations are caused mainly by Hopf bifurcation or Poincaré bifurcation around one non-degenerate equilibrium of the corresponding planar traveling wave system. If β(1 − γ) = 0 the singular point is degenerate since at least one of its two eigenvalues is zero In this case, when c = ∑in=0 αi, the singular point is semi-hyperbolic, and when c = ∑in=0 αi, the equilibrium is a nilpotent critical p√oint and limit cycle bifurcation may happen at (1, 0).
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