It is interesting that, by applying the diode bridge memristor into the original jerk circuit, authors proposed a novel hybrid diode based jerk circuit. performed an analysis of a jerk circuit including a pair of semiconductor diodes connected in antiparallel and investigated its multiple attractors. discovered the coexistence of multiple attractors and crisis route to chaos in a chaotic jerk circuit and a simple autonomous jerk oscillator with multiple attractors was introduced in. Jerk oscillators can exhibit multistability despite their simplicity. Multistability is associated with the presence of coexisting attractors with different initial conditions. Researchers have shown an increased interest in multistability. Especially, a three-dimensional novel jerk chaotic oscillator with two hyperbolic sinusoidal nonlinearities was reported in. A simple chaotic jerk circuit was used in a sound encryption scheme. An elegant chaotic oscillators based on jerk equation was implemented in a circuit where the nonlinearity was provided by a single diode. Multiscroll attractors in jerk circuits were presented by Ma et al. By using Josephson junctions, Yalcin constructed a general jerk circuit with multiscroll and hypercube attractors. Different jerk oscillators with chaos were summarized by Sprott. It is worth noting that if the scalar is denoted as a physical variable at time t, the third derivative presents the jerk. There are increasing works on chaos in jerk oscillators. There has been a significant increase in studies of chaotic oscillators over the past decades because of their complex behaviors and their promising applications.

Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method. A good qualitative agreement is illustrated between the numerical and FPGA results. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper.

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