A Simple Jerk System with Piecewise Exponential Nonlinearity

Third-order explicit autonomous differential equations in one scalar variable, sometimes called jerky dynamics, constitute an interesting subclass of dynamical systems that can exhibit chaotic behavior. In this paper, we investigated a simple jerk system with a piecewise exponential nonlinearity by numerical examination as well as dynamic simulation. Using the largest Lyapunov exponent as the signature of chaos, the region of parameter space exhibiting chaos is identified. The results show that this system has a period-doubling route to chaos and a narrow chaotic region in parameter space. The rescaled system is approximately described by a one-dimensional quadratic map. The parameters are fitted to a simple function to predict the values for which chaos occurs in the case of high nonlinearity where the region in parameter space that admits chaos is relatively small.