In this paper, a Van der Pol–Duffing (VdPD) jerk oscillator is designed. The proposed VdPD jerk oscillator is built by converting the autonomous two-dimensional VdPD oscillator to a jerk oscillator. Dynamical behaviours of the proposed VdPD jerk oscillator are investigated analytically, numerically and analogically. The numerical results indicate that the proposed VdPD jerk oscillator displays chaotic oscillations, symmetrical bifurcations and coexisting attractors. The physical existence of the chaotic behaviour found in the proposed VdPD jerk oscillator is verified by using Orcad-PSpice software. A good qualitative agreement is shown between thenumerical simulations and the PSpice results. Moreover, the fractional-order form of the proposed VdPD jerk oscillator is studied using stability theorem of fractional-order systems and numerical simulations. It is found that chaos, periodic oscillations and coexistence of attractors exist in the fractional-order form of the proposed jerk oscillator with order less than three. The effect of fractional-order derivative on controlling chaos is illustrated. It is shown that chaos control is achieved in fractional-order form of the proposed VdPD jerk oscillator only for the values of linear controller used. Finally, the problem of drive–response synchronisation of the fractional-order form of the chaotic proposed VdPD jerk oscillators is considered using active control technique.

This work deals with the dynamical analysis and FPGA implementation of a pendulum controlled by a delayed proportional feedback force. The system has infinity equilibrium points and so the nature of these points is established according to the value of the gain of the controller. Analysis of the effect of delay on the linear system through a stability chart revealed stability switching. The Hopf bifurcation curves are presented gradually as the delay varies and thus confirm the analytical predictions. Furthermore, these bifurcation diagrams show the existence of complex behaviours and the coexistence of infinity strange attractors for different initial values. This system can be considered as a self-producer of attractors depending on the initial conditions. For certain values of the delay, the system exhibits the multiscroll behaviour which can be observed through the phase portraits. The FPGA implementation of the system is based on the VHDL hardware description which takes into account the RK4 resolution method with calculations under the 32-bit 16Q16 fixed-point number standard.