Exploring Chaos and Ergodic behavior of an Inductorless Circuit driven by Stochastic Parameters

There exist extensive studies on periodic and random perturbations of various continuous maps investigating their dynamics. This paper presents a random piecewise smooth map derived from a simple inductor-less switching circuit, which is useful in chaotic communication. The
bifurcation parameter is bounded and randomly selected from a stationary distribution. Due to the stochasticity inherent in either the parameter values or the state variable, the time evolution of the state variable cannot be predicted at a specific time instant. We observe that the state variable exhibits completely ergodic behavior when the minimum value of the parameter is 2.0. However, the ensemble average of the state variable converges to a fixed value. For parameter values ranging from 2.0 to 3.5, the system demonstrates nonchaotic behavior, and the absolute value of the Lyapunov exponent increases monotonically with the asymmetry of the distribution from which the bifurcation parameter values are sampled. We determine the probability density function of the random map and verify its invariance under any initial condition. The most noteworthy result is the disappearance of chaotic behavior when the lower range of the distribution is varied while maintaining a fixed upper threshold for a particular distribution, even though the nonrandom map exhibits an array of periodic and chaotic behaviors within that range. Since, for any practical system, we cannot precisely measure the exact parameter or the state variable values, our work is helpful for the analysis of the randomness of the circuit useful in chaotic communication.