Chaotic Behaviour of a Nonlinear Vibrating System with a Retarded Argument

This paper investigates the behaviour of a nonlinear vibrating system, i. e., van der Pol and Duffing's system, with a retarded argument under a harmonic stimulating force. An approximate analytical method, i.e., the averaging scheme, is used to analyse subharmonic oscillations of the order 1/2. The algorithm used to study periodic solutions and their stabilities with higher-order approximations, is presented. A computer simulation is used to obtain Poincare mapping and invariant manifolds. Through the application of the approximate analytical procedure presented here and numerical simulation, both symmetrical and unsymmetrical subharmonic solutions are observed in addition to period-doubling bifurcations and chaotic behaviour. In addition, one of the Lyapunov exponents, which is positive, has verified that the vibrating system possesses chaotic phenomena.