Autonomous Van der Pol–Duffing snap oscillator: analysis, synchronization and applications to real-time image encryption

In this paper, we investigate a four dimensional autonomous Van der Pol–Duffing (VdPD) oscillator which is built by converting an autonomous two-dimensional VdPD oscillator to a snap oscillator. The basic dynamical properties of the proposed autonomous VdPD snap oscillator are discussed in terms of its parameters by using standard tools of nonlinear analysis. The bifurcation analysis reveals rich and interesting features of the proposed autonomous VdPD snap oscillator such as period-doubling bifurcations, bistabe chaotic behavior, double scroll chaotic behavior, periodic windows, symmetry breaking and recovering bifurcations. One of the main findings of this paper is the presence of a window in the parameter space in which the proposed autonomous VdPD snap oscillator displays the striking feature of coexisting of multiple attractors. An electronic circuit model of the proposed autonomous VdPD snap oscillator is designed and its simulations are performed using an ORCAD-PSpice software. A good qualitative agreement is obtained between the numerical and ORCAD-PSpice results. Adaptive sliding mode control method is successfully used to achieve the synchronization of unidirectional coupled proposed autonomous VdPD snap oscillators in double scroll chaotic regime. Finally using LabVIEW, chaotic synchronization of unidirectional coupled proposed autonomous VdPD snap oscillators is used to develop and implement a real-time image encryption algorithm with a secondary image based on chaos masking technique.