Dynamics and synchronization of a novel 4D-hyperjerk autonomous chaotic system with a Van der Pol nonlinearity

In the literature, hyperjerk systems raised up meaningful interest due to their simple and elegant structure as well as their complex dynamical features. In this work, we propose a novel 4D autonomous hyperjerk system which the particularity resides on the type of its nonlinearity namely the Van der Pol nonlinearity. The dynamics of this hyperjerk system is assessed thanks to the well-known nonlinear dynamic tools such as time series, bifurcation diagrams, Lyapunov exponent spectrum, two-parameter phase diagram, and phase portraits. As important result, it is established that the system presents a particular phenomenon of hysteretic dynamics that leads to the coexistence of attractors. Next, through the calculation of the Hamiltonian energy, we show that this latter depends on all the variables of the novel hyperjerk system. Furthermore, basing on an adaptive backstepping method whose target is a function of the states of the error system, a new controller is designed to carry out from t = 30, complete chaotic synchronization of the identical novel hyperjerk chaotic systems. Likewise, PSpice (9.2 full version) based simulations are presented in detail to confirm the feasibility of the theoretical model. One of the key points of this work is the designing in PSpice environment of this new adaptive backstepping controller to validate both theoretical and numerical synchronization results. Finally, our experimental measurements in the laboratory are in good agreement with the numerical and analog results.