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The dynamics of a simple autonomous jerk circuit previously introduced by Sprott in 2011 are investigated. In this paper, the model is described by a three-time continuous dimensional autonomous system with an exponential nonlinearity. Using standard nonlinear techniques such as time series, bifurcation diagrams, Lyapunov exponent plots, and Poincaré sections, the dynamics of the system are characterized with respect to its parameters. Period-doubling bifurcations, periodic windows, and coexisting bifurcations are reported. As a major result of this work, it is found that the system experiences the unusual phenomenon of asymmetric bistability marked by the presence of two different attractors (e.g., screw-like Shilnikov attractor with a spiralling-like Feigenbaum attractor) for the same parameters setting, depending solely on the choice of initial states. Among few cases of lower dimensional systems capable of such type of behavior reported to date (e.g., Colpitts oscillator, Newton-Leipnik system, and hyperchaotic oscillator with gyrators), the jerk circuit/system considered in this work represents the simplest prototype. Results of theoretical analysis are perfectly reproduced by laboratory experimental measurements.

The aim of this paper is to address the problem of the electronic implementation of chaos control using a single controller and synchronization of chaotic permanent magnet synchronous motor (PMSM). Firstly, different dynamical behaviors of the PMSM including steady state, periodic and chaotic behaviors are found using numerical methods such as twodimensional largest Lyapunov exponents graph associated with two parameters of PMSM. Secondly, two simple and single controllers are designed and added to the chaotic PMSM in order to suppress chaotic behavior. The performance of the two proposed simple and single controllers is illustrated by numerical simulations. Thirdly, controllers are designed to achieve synchronization of unidirectional coupled identical chaotic PMSMs. Numerical simulations are also used to verify the effectiveness of the synchronization. Finally, the existence of chaos in PMSM and the physical feasibility of the proposed two simple and single controllers as well as the chaos synchronization are validated through the circuit implementation on OrCAD-PSpice software. The circuit implementation results comply fairly with those of the numerical simulation results and establish that the existence of chaotic behavior in the PMSM and the achievement of chaos synchronization in unidirectional coupled identical PMSMs designed and the two single controllers designed are effective and successful in suppressing chaotic behavior in PMSM.