Analysis And Control Of Two Chaotic Systems With Coexistence Attractors

In recent years,due to the application of chaos in practical engineering such as motor control,power system protection,secure communication and image encryption,chaos has become a hot research direction.With the in-depth study of chaos,it has been found that chaotic systems with coexisting attractors have more abundant dynamic behavior,and have greater application value.At the same time,the fractional-order chaotic systems describe complex physical phenomena,which are more in line with the nature of physical phenomena,and suitable for more practical systems.However,most of the chaotic systems with coexistence attractors studied before have overlapping coexistence attractors,and the research on fractional-order chaotic systems with coexistence attractors needs to be improved,too.Therefore,the study of chaotic systems with independent coexistence attractors or multi-scroll coexistence attractors,including integer-order chaotic systems and fractional-order chaotic systems,can enrich the chaotic theory and enhance the practical value of chaotic research to a certain extent.Firstly,a chaotic system with two independent attractors is proposed.By using the typical methods of dissipation,Lyapunov exponent spectrum and dimension,bifurcation diagram and attractor phase diagram,it is found that the system has independent two "positive attractor" and "negative attractor".And by using the theory of topological horseshoe,the topological horseshoe and the topological entropy are obtained,the chaos of the new system is strictly verified.Furthermore,the integer-order system is extended to fractional-order form,and the maximum Lyapunov exponent of the corresponding fractional-order system is obtained.It is found that chaos exists in the order 0.8477?q?1.The results show that the corresponding fractional-order system is similar to the integer-order system,and has coexisting "positive attractor" and "negative attractor".Secondly,a memristor chaotic system with multiple-scroll attractors is also proposed.By using the typical methods of equilibrium point,Lyapunov exponent spectrum,bifurcation diagram and attractor phase diagram,it is found that the system has two-scroll,three-scroll and four-scroll attractors.And by using the theory of topological horseshoe,the topological horseshoe and the topological entropy are obtained,the chaos of the new system is strictly verified.Furthermore,the integer-order system with four-scroll attractor is extended to fractional-order form,and the maximum Lyapunov exponent of the corresponding fractional-order system is obtained.It is found that chaos exists in the order 0.92?p?1.Comparing the integer-order system,the corresponding fractional-order system also has coexisting two kinds of three-scroll chaotic attractors and coexisting three-scroll and four-scroll chaotic attractors.Finally,based on the stability theory of fractional-order systems,single state variable linear controllers are designed for the two fractional-order chaotic systems with coexisting attractors.The rationality is proved by the stability lemma of the nonlinear fractional-order system,and the effectiveness is verified by the MATLAB simulation results.