Generalised projective synchronisation of novel 3-D chaotic systems with an exponential non-linearity via active and adaptive control

Abstract

Generalised projective synchronisation (GPS) of chaotic systems is a general type of synchronisation, which includes known synchronisation types such as complete synchronisation, anti-synchronisation, hybrid synchronisation and projective synchronisation as special cases. This research work also introduces a novel 3-D chaotic system with an exponential non-linearity. Phase portraits of the strange chaotic attractor for the novel chaotic system are described. The novel chaotic system is a dissipative system with fractional Lyapunov dimension. The novel chaotic system has two saddle-foci equilibrium points, which are both unstable. Since the maximal Lyapunov exponent (MLE) for the novel chaotic system has a large value, viz. L1 = 15.4249, the novel 3-D chaotic system exhibits strong chaotic behaviour. New results are derived for the GPS of identical novel chaotic systems using Lyapunov stability theory. First, active control method is used for deriving new results for the GPS of novel chaotic systems with known parameters. Then, adaptive control method is used for derived new results for the GPS of novel chaotic systems with unknown system parameters. All the main results are established using Lyapunov stability theory. Numerical simulations are shown using MATLAB to validate and demonstrate the GPS results derived in this paper for the novel chaotic systems with an exponential non-linearity.