Periodic attractors of complex damped non-linear systems

The aim of the present paper is to investigate the dynamics of a class of c omplex damped non-linear systems described by the equation (z) double over dot + omega(2)z + epsilon(z) over dot f(z,(z) over bar,(z) over dot,z (radical anion))P(Omega t) = 0, (*) where z(t) = x(t) + iy(t), i = root-1, the bar denotes the complex conjugat e and epsilon is a small positive parameter. The periodic attractors of Eq. (*) are important in the study of these systems, since they represent stat ionary or repeatable behavior. This equation appears in several fields of p hysics, 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 o f our systems. To illustrate these periodic attractors we constructed Poinc are plots at some of the parameter values which are obtained by the indicat rix method for the case omega(2) congruent to 1/4, f = /z/(2) and P(Omega t ) = sin 2t as an example. Our recent method which is based on the generaliz ed averaging method is used to obtain approximate analytical solutions of E q. (*) and investigate the stability properties of the solutions. We compar ed the analytical results of our example with the numerical results and exc ellent agreement is found. (C) 1999 Elsevier Science Ltd. All rights reserv ed.