| Chaos is a unique form of motion in nonlinear dynamics,which is widely used in image encryption,secure communication,biology and other fields.Chaotic attractors are divided into self-excited and hidden attractors.The previous stage was mainly to research on self-excited or hidden systems.In recent years,researchers have discovered that chaotic systems with both self-excited and hidden attractors have potential value in theoretical research and practical engineering applications,thus arousing their interest in this topic.In this paper,we propose two chaotic systems with both selfexcited and hidden attractors,and analyze their dynamics.Finally,we design the corresponding simulation circuit based on Multisim software to verify it.The main research contents of the paper are as follows:(1)Generating self-excited and hidden attractors with complex dynamics in a memristor-based Jerk system.we report a special memristor-based Jerk system in which self-excited and hidden attractors can be generated by adjusting one decisive system parameter.With increasing the parameter from negative to positive,the system has a transition from unstable equilibriums to no equilibrium point,and thus leading to the occurrence of coexisting self-excited and hidden attractors in the modified Jerk system simulatenously.More interestingly,the memristor parameters play an important role in the coexistence of attractors and the formation of the Feigenbaum remerging trees,and these two dynamic behaviors can be found in both hidden and self-excited attractor regions.In addition,the study of state-switching proves that the memristor’s internal initial state is very sensitive to the system.In order to verify the complex dynamic behavior of the system,an analog circuit simulation based on Multisim is developed and two types of attractors and their coexisting attractors are successfully captured.(2)Dynamic analysis and experimental verification of 4-D chaotic system with self-excited and two types of hidden attractors.a special system with piecewise linear function in which hidden and self-excited attractors coexist is proposed.The proposed system has the coexistence of hidden attractor with no equilibrium,hidden attractor with line of equilibria and self-excited attractors.These two types of hidden attractors and self-excited attractors can be switched only adjusting one parameter.We verify the existence of hidden attractor through the time series and Poincaré map.It is worth noting that the hidden attractors with line of equilibria is proved again from the perspective of the hidden definition by drawing the basin of attraction,and it is judged as an unstable line of equilibria through theoretical analysis.we use some powerful tools like the bifurcation diagram,Lyapunov exponents,phase diagram,time series and basin of attraction to analyze many interesting dynamic phenomena,such as,multistability is observed both the hidden region and self-excited region,and the complete Feigenbaum remerging trees are described.In addition,the simulation circuit based on Multisim is implemented,and two types of hidden attractors and self-excited attractors are successfully captured.The circuit makes the practical application of the system possible. |
