Rare and hidden attractors in a periodically forced Duffing system with absolute nonlinearity

Abstract

• The composite cell coordinate system (CCCS) method is modified to non-smooth systems. • Rare and hidden attractors of a non-smooth Duffing system are studied. • The accurate global properties such as attractor and basin can be obtained. • Hidden and rare attractors can be controlled by adjusting the amplitude. • The results show the effectiveness of our proposed improvement strategy. Hidden attractors and rare attractors are two kinds of special attractors in multistable systems, studying the appearance and properties of hidden attractors and rare attractors can increase the possibility of the system remaining on the ideal attractor and reduce the risk of sudden jump to unexpected behavior. This paper presents an investigation of the rare attractors and hidden attractors of a nonautonomous Duffing-like system with absolute function. Based on modified composite cell coordinate system (CCCS) method, the attractors and the corresponding basins are shown and the coexistence of multiple attractors is observed. By calculating the probability that the system is on each attractor, we discover different rare attractors in this system whose basins of attraction are extremely small. In addition, the equilibria of the system are calculated for a given set of parameters, according to the property that the hidden attractor does not contain any equilibrium, several hidden attractors are found for three parameter ranges. With the change of the amplitude of external excitation, the rare and hidden attractors appear in a very small parameter range. The results show that the modified CCCS method is a powerful tool to research the hidden and rare attractors in the non-smooth system.