Bifurcation Theory Of Dynamical Systems Research Articles

BIFURCATIONS OF TRAVELING WAVE SOLUTIONS FOR FOUR CLASSES OF NONLINEAR WAVE EQUATIONS

Four large classes of nonlinear wave equations are studied, and the existence of solitary wave, kink and anti-kink wave, and uncountably many periodic wave solutions is proved. The analysis is based on the bifurcation theory of dynamical systems. Under some parametric conditions, various sufficient conditions for the existence of the aforementioned wave solutions are derived. Moreover, all possible exact parametric representations of solitary wave, kink and anti-kink wave, and periodic wave solutions are obtained and classified.

Bifurcations of travelling wave solutions for a coupled nonlinear wave system

By using the bifurcation theory of dynamical systems to the coupled nonlinear wave equations, the existence and stability of periodic wave solutions by Hopf bifurcations are obtained. Theory of travelling wave was applied to transform a kind of coupled nonlinear wave equations into three-dimension dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence and stability of the above solutions are given.

Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation

The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation ( B ( m , n ) equation), u tt − u xx − a ( u n ) xx + b ( u m ) xxxx =0, is studied by using bifurcation theory of dynamical system . As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave , kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given.

Travelling wave solutions in the generalized Hirota–Satsuma coupled KdV system

By using the bifurcation theory of dynamical systems to the generalized Hirota–Satsuma coupled KdV system, bifurcation parameter sets are shown. Numbers of solitary waves, kink waves and periodic waves are given. Under various parameter conditions, all explicit formulas of solitary wave solutions and kink wave solutions and periodic wave solutions are listed.

Early evolution of the planetary system around PSR B1257+12

For the plane motion we are completely analyzing the differential equations systems of gravitational capture of the exoplanet at the resonance rotation with action of gravitational and tidal torques by qualitative analysis and bifurcation theory of dynamical systems (DS). The separation of 3-dimensional parameter space of dynamical system by bifurcation surfaces is obtained. The gallery of more than thirty phase portraits of gravitational capture extends the known scenario of cosmogonical evolution of the exoplanet on the early time, when the tidal interaction is very important.

TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.

An Organizing Center for Wave Bifurcation in Multiphase Flow Models

We consider a one-parameter family of nonstrictly hyperbolic systems of conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter. In this paper we use singularity theory and bifurcation theory of dynamical systems, including Melnikov's method, to find all nearby shock waves that are admissible according to the viscous profile criterion. We use these results to construct a unique solution of the Riemann problem for each left and right state and parameter value in a neighborhood of the unperturbed shock wave solution; together with previous numerical work, this construction completes the solution of the three-phase flow model. In the bifurcation analysis, the unperturbed shock wave acts as an organizing center for the waves appearing in Riemann solutions.

Amplification properties of dynamic instabilities: Hopf bifurcation in semiconductor lasers

The small-signal amplification properties of a semiconductor laser are quantified within the framework of dynamical-systems bifurcation theory. Calculations are performed at or near a Hopf bifurcation associated with a modal switching instability in the device. The work is directed at finding practical applications for nonlinear dynamical phenomena. In addition the work underlines the importance of the laser diode as a convenient optical system for studying the properties of general dynamical systems. It is argued that the laser diode may thus play a unique role both in stimulating the need for new insights into dynamical systems and also in permitting implementation of novel nonlinear phenomena for device applications.