Synchronization via active control of parametrically and externally excited Φ ⁶ Van der Pol and Duffing oscillators and application to secure communications

In this paper active controllers based on the Lyapunov stability theory and the Routh-Hurwitz criteria, are designed to completely synchronize two parametrically and externally excited φ 6 Van der Pol oscillators, φ 6 Duffing oscillators, and a φ 6 Van der Pol oscillator with a φ 6 Duffing oscillator in the triple-well configuration of the φ 6 potential. The coefficient matrix of the error dynamics between each pair of synchronized systems is chosen such that the number of active control functions reduces from two to one, thereby significantly reducing controller complexity in the design. The designed controllers enable the state variables of the response system to synchronize with those of the master system in both the identical and nonidentical cases. The results are validated using numerical simulations. Application to secure communications is computationally demonstrated.