On real and complex dynamical models with hidden attractors and their synchronization

Abstract

Abstract In this work, we propose three chaotic (or hyperchaotic) models. These models are real or complex with one stable equilibrium point (hidden attractor). Based on a modified Sprott E model, three versions were introduced: the complex integer order, the real fractional order, and the complex fractional order. The basic properties of these models have been studied. We discover that the complex integer-order version has chaotic and hyperchaotic multi-scroll hidden attractors (MSHAs) by computing Lyapunov exponents (LEs). By making a small change to a model parameter, different MSHA values can be produced for this version. The dynamics of the real fractional version are investigated through a bifurcation diagram and LEs. It has chaotic hidden attractors for various fractional-order q values. Through varying the model parameters of the complex fractional-order (FO) version, different numbers of chaotic MSHAs can be generated. Due to the complex dynamic behaviours of the MSHAs, these models may have several applications in physics, secure communications, social relations and image encryption. A new kind of combination synchronization (CS) between one integer-order drive model and two FO response models with different dimensions is proposed. The tracking control method is used to investigate a scheme for this type of synchronization. As an example, we used our three models to test the validity of this scheme, and an agreement between the analytical and numerical results was found.