Synchronized chaotic swinging of parametrically driven pendulums

We study the problem of complete synchronization of a pendulums' array attached to a common, externally forced structure. The presented analysis is focused on chaotic synchronization of the swinging, parametric pendulums. We propose a simplified criterion of synchronization for an arbitrary number of parametrically driven oscillators, which is based on the idea of a so-called autonomous driver decomposition and conditional Lyapunov exponents. According to this proposal, complete synchronization of any number of parametric oscillators driven via a common transmitter of an external signal (the structure) can be analyzed using a three degrees of freedom system in a master-slave configuration. The reduced system consists of an equivalent synchronous oscillator coupled with its virtual replica via the linking structure. It is shown that introduction of the intermediate structure allows chaotic synchronization of parametric pendulums, even in the presence of parameter's mismatch. The idea of an intermediate transmitter and the methodology of verification of stability of the synchronous state can be applied for any set of dynamical systems with a common external drive.