Second order adaptive time varying sliding mode control for synchronization of hidden chaotic orbits in a new uncertain 4-D conservative chaotic system

The objectives of the paper are (i) to develop a new 4-D conservative chaotic system with hidden chaotic orbits, (ii) to design a second order adaptive time varying sliding mode control for the synchronization between two identical proposed chaotic systems in the presence of matched disturbances and (iii) to compare the performances of the proposed controller with two available controllers which have been published recently. The chaotic nature of the proposed system is validated using theoretical and numerical tools like divergence property, Lyapunov exponents, Lyapunov spectrum, bifurcation diagram, phase portrait, Poincaré map and a frequency spectrum. The new conservative chaotic system exhibits the coexistence of hidden chaotic orbits with no equilibrium point. The new system is synchronized with itself using the proposed second-order adaptive time varying sliding mode control technique in the presence of matched disturbances and by considering different initial conditions. During synchronization, the parameters of both the systems, gains of the first order and second order sliding surfaces and the gains of the switching laws are considered as unknown and estimated adaptively. Only two control inputs are used to synchronize all the four states of the system. The effectiveness of the proposed controller is compared with two available controllers for the synchronization of chaotic systems and it is found that the proposed controller performs much better than the two available controllers.