×

Periodic attractors of complex damped nonlinear systems. (English) Zbl 1068.70525

Summary: The aim of the present paper is to investigate the dynamics of a class of complex damped nonlinear systems described by the equation \(\ddot z+\omega^2z+ \epsilon \dot z f(z,\bar z,\dot z,\dot{\bar z})P(\Omega t)=0\), where \(z(t)=x(t)+iy(t)\), \(i=\sqrt{-1}\), the bar denotes the complex conjugate and \(\epsilon\) is a small positive parameter. The periodic attractors are important in the study of these systems, since they represent stationary or repeatable behavior. This equation appears in several fields of physics, mechanics and engineering, for example, in high-energy accelerators, rotor dynamics, robots and shells. In the numerical investigation of this work we used the indicatrix method which has been introduced and extended in our previous studies to study the existence of the periodic attractors of our systems. To illustrate these periodic attractors, we constructed Poincaré plots at some of the parameter values which were obtained by the indicatrix method for the case \(\omega^2\cong 1/4\), \(f=| z|^2\) and \(P(\Omega t)=\sin 2t\) as an example. Our recent method which is based on the generalized averaging method is used to obtain approximate analytical solutions and investigate the stability properties of the solutions. We compared the analytical results of our example with the numerical results, and excellent agreement was found.

MSC:

70K40 Forced motions for nonlinear problems in mechanics
37N05 Dynamical systems in classical and celestial mechanics
PDFBibTeX XMLCite
Full Text: DOI