Synchronization of mechanical systems driven by chaotic or random excitation

The phenomenon of synchronization in dynamical and, in particular, mechanical systems has been known for a long time. Recently, the idea of synchronization has been also adopted for chaotic systems. It has been demonstrated that two or more chaotic systems can synchronize by linking them with mutual coupling or with a common signal or signals [1–5,15]. In the case of linking a set of identical chaotic systems (the same set of ODEs and values of the system parameters) ideal synchronization can be obtained. The ideal synchronization takes place when all trajectories converge to the same value and remain in step with each other during further evolution (i.e., limt-N jxðtÞ yðtÞj 1⁄4 0 for two arbitrarily chosen trajectories xðtÞ and yðtÞ). In such a situation all subsystems of the augmented system evolve on the same attractor on which one of these subsystems evolves (the phase space is reduced to the synchronization manifold). Linking homochaotic systems (i.e., systems given by the same set of ODEs but with different values of the system parameters) can lead to practical synchronization (i.e., limt-N jxðtÞ yðtÞjpe; where e is a vector of small parameters) [6,7]. In such linked systems it can also be observed that there is a significant change of the chaotic behaviour of one or more systems. This so-called ‘‘controlling chaos by chaos’’ procedure has some potential importance for mechanical and electrical systems. An attractor of such two systems coupled by a negative feedback mechanism can be even reduced to the fixed point [8]. This paper concerns the ideal synchronization of a set of identical uncoupled mechanical oscillators (with one or more degrees of freedom) linked by common external excitation only. This problem has been described widely for oscillators with periodic excitation because such kind of driving is often met in real oscillators [1]. However, from a viewpoint of practical considerations, a non-periodic external excitation can also occur in mechanical systems. For that reason, this paper concentrates on the analysis of the ideal synchronization for non-linear oscillators forced by chaotic and stochastic external driving. The analysis presented is based on the connections