No abstract

In this paper, we consider a simple equation which involves a parameter [Formula: see text], and its traveling wave system has a singular line. Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties: (i) In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit. (ii) When [Formula: see text], in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave. (iii) When [Formula: see text], in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation. Secondly, we perform numerical simulations to visualize the above properties. Finally, when [Formula: see text] and the constant wave speed equals [Formula: see text], we give exact expressions to the above phenomena.

This article introduces a new hyperchaotic system of four-dimensional autonomous ordinary differential equations, with only cubic cross-product nonlinearities, which can respectively display two hyperchaotic attractors with only nonhyperbolic equilibria line. Several issues such as basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new hyperchaotic and chaotic system are investigated, either theoretically or numerically. Of particular interest is the fact that the two coexisting attractors of the new hyperchaotic system are symmetrical, and this hyperchaotic system can generate plenty of complex dynamics including two coexisting chaotic or periodic attractors. Moreover, some chaotic features of the attractor are justified numerically. Finally, 0-1 test is used to analyze and describe the complex chaotic dynamic behavior of the new system.

In this paper, we study the existence and bifurcation of peakon and anti-peakon to the [Formula: see text]-degree [Formula: see text]-equation with [Formula: see text] and positive integer [Formula: see text]. Using qualitative theory and bifurcation method of dynamical systems, for positive wave speed we confirm the following properties: (1) When [Formula: see text], the equation has peakon, but no anti-peakon. (2) When [Formula: see text], the equation has not only peakon but also anti-peakon. There is a bifurcation wave speed for peakon and anti-peakon. (3) When [Formula: see text] is even, there exists a maximum wave speed for peakon. (4) When [Formula: see text] is odd, there is no maximum wave speed for peakon.