A unique chaotic snap system with a smoothly adjustable symmetry and nonlinearity: Chaos, offset-boosting, antimonotonicity, and coexisting multiple attractors

Abstract

This work proposes and systematically investigates the dynamics of a novel snap system with a single parameterized nonlinearity in the form φk(z)=0.5(exp(kz)−exp(−z)). The form of nonlinearity is physically interesting in the sense that the corresponding circuit realization involves only off-the shelf electronic components such as resistors, semiconductor diodes and operational amplifiers. Parameter k (i.e. a control resistor) serves to smoothly adjust the nonlinearity, and hence the symmetry of the system. In particular, for k=1, the nonlinearity is a hyperbolic sine, and thus the system is point symmetry about the origin. For k ≠ 1, the system is non-symmetric. The fundamental dynamics of the system are investigated in terms of equilibria and stability, phase space trajectory plots, bifurcations diagrams, and graphs of Lyapunov exponents. When monitoring the system parameters, some striking phenomena are found including period doubling bifurcation, reverse bifurcations, merging crises, coexisting bifurcations, hysteresis and offset boosting. Several windows in the parameters space are depicted in which the novel snap system displays a plethora of coexisting attractors (i.e. two, three, four, five or six different attractors) depending solely on the choice of the initial conditions. The magnetization of the state space due to the presence of multiple competing solutions is illustrated by means of basins of attraction. Laboratory experimental results confirm the theoretical predictions.