On the Number of Limit Cycles Bifurcating from a Compound Polycycle

This paper focuses on the number of limit cycles bifurcating from a symmetrical compound polycycle with three saddles. We use two methods, the Melnikov function method and the method of stability-changing of a homoclinic loop or a double homoclinic loop to study this problem. We find 15 limit cycles and 16 limit cycles respectively with four alien limit cycles under certain conditions.

The Bifurcation and Chaotic Characters of an Intelligent Magnetoelectroelastic Thin Plate

In this paper, an intelligent magnetoelectroelastic thin plate is coupled with a transverse magnetic field and uniformly distributed load. Considering the von Karman plate theory of large deflection and the geometric nonlinearity, the damping Duffing equation is obtained. Using the Melnikov function method, the Chaos condition of the system under the Smale horseshoe transformation is obtained. The bifurcation diagram, the wave diagram of displacement, and the phase diagram are shown here by the numerical analysis. The simulation results show the complex nonlinear vibration characters of the intelligent magneto-electro-elastic thin plate.

LIMIT CYCLE BIFURCATIONS OF TWO KINDS OF POLYNOMIAL DIFFERENTIAL SYSTEMS

In this paper, by using qualitative analysis and the first-order Melnikov function method, we consider two kinds of polynomial systems, and study the Hopf bifurcation problem, obtaining the maximum number of limit cycles.

SUPPRESSING OR INDUCING CHAOS BY WEAK RESONANT EXCITATIONS IN AN EXTERNALLY-FORCED FROUDE PENDULUM

Based on analytic and numerical investigations of chaotic vibrations and quasiperiodic rotations of the Froude pendulum, we present a sufficient condition for controlling chaos by means of a weak resonant excitation as the initial phase difference Ψ varies. It is shown via the Melnikov function method that the initial phase difference Ψ plays a vital role in suppressing or inducing chaotic motions or quasiperiodic rotations.

Evolution of long waves and chaos in magnetohydrodynamics

The evolution equation for a wavepacket travelling on the surface of a conducting fluid of finite depth in 2+1 dimensions in the long-wave approximation is studied in the context of magnetohydrodynamics. The amplitude equation thus obtained is the well-known Kadomtsev–Petviashvili equation with modified coefficients in the presence of a tangential magnetic field. By taking the double limit of this equation, Schrödinger–Poisson type equations are obtained. A nonlinear evolution equation is sought, in order to study Rayleigh–Taylor instability. It is shown that the magnetic field and surface tension have a stabilizing influence on the formation of bubbles arising owing to this instability. By incorporating forcing and damping terms in the Kadomtsev–Petviashvili equation, a condition for the existence of transverse homoclinic orbits giving rise to chaotic motions is obtained by using the Melnikov function method. It is shown that the chaotic motions can be suppressed by applying a suitably strong magnetic field. A study of subharmonic bifurcation leading to surface waves is also undertaken. The tangential magnetic field has a stabilizing influence on such motions.