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 φ6 Van der Pol oscillators, φ6 Duffing oscillators, and a φ6 Van der Pol oscillator with a φ6 Duffing oscillator in the triple-well configuration of the φ6 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.

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In this paper, recursive and active backstepping nonlinear techniques are employed to design control functions for the respective, control, and synchronization of the new hyperchaotic Liu system. The designed recursive backstepping nonlinear controllers are capable of stabilizing the hyperchaotic Liu system at any position as well as controlling it to track any trajectory that is a smooth function of time. The designed active backstepping nonlinear controllers are effective in globally synchronizing two identical hyperchaotic Liu systems evolving from different initial conditions. The results are all validated by numerical simulations.

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