Complex Dynamics And Control Of Nonlinear Maps With Hidden Attractors

The study of complex dynamics of nonlinear maps with hidden attractors and its control is one of the hot research topics in nonlinear science,which has attracted significant attention from researchers in dynamics and control at home and abroad.This paper proposes some usual and memristive maps with hidden chaotic attractors.The complex characteristics of these maps under parameter variations are investigated by analyzing dynamical behaviors and bifurcations.The generation and evolution mechanisms of hidden attractors are also studied.A method to control hidden attractors in nonlinear maps is proposed further.This research will enrich the bifurcation theory of nonlinear maps and provide theoretical references for practical engineering.The proposed maps can generate chaotic signals and be applied in chaotic information engineering,such as data and image encryption.The main work and novelties of this dissertation are given as follows:(1)The research background and significance of this research are introduced first.Then the current research status of the hidden attractors is elaborated.Then the main work and the structure of the dissertation are given.(2)A class of two-dimensional maps with infinitely many coexisting attractors is proposed.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stabilities of the fixed points are studied,indicating that they are infinitely many and are all unstable.Since the knowledge about fixed points does not help in the localization of attractors,the maps’ attractors can be considered hidden attractors.A computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.This map contains infinitely many coexisting attractors,which has been rarely reported in the literature.Further studies of these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan-Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions and can exhibit homogenous extreme multi-stability.These infinitely coexisting attractors have the same shape but are located in different positions.(3)The control of hidden attractors and multi-stability of a class of two-dimensional maps via linear augmentation is investigated.Firstly,the method of linear augmentation for continuous systems is generalized to nonlinear maps.Then two cases of a class of two-dimensional maps that exhibit hidden dynamics,the maps with no fixed point and the maps with one stable fixed point,are studied.Our numerical simulations show the effectiveness of the linear augmentation method.As the coupling strength of the augmented map increases or decreases,hidden attractors can be annihilated or altered to be self-excited,and the multi-stability of the map can be controlled to be bistable or monostable.Hidden attractors in nonlinear maps may cause severe damage to the existing engineering system.So the control of hidden attractors has essential research value.The proposed method can be easily implemented and does not require changing the original map’s parameters.(4)A new class of two-dimensional rational maps exhibiting self-excited and hidden attractors is given.The mathematical model of these maps is formulated by introducing a rational term.The analysis of the existence and stabilities of the fixed points suggests that these maps have four types of fixed points,i.e.,no fixed point,one single fixed point,two fixed points,and a line of fixed points.Several dynamical phenomena are observed,including saddle-node bifurcation,period-doubling bifurcation,period-adding bifurcation,and the route to chaos by the cascade of period-doubling bifurcation.Numerical analysis tools are employed to investigate the complex dynamics of these rational maps with different types of fixed points.Numerical simulations identify both self-excited and hidden attractors and explore the multi-stability of these maps,especially the hidden one.As can be observed from the basins of attraction,the basins of attraction of hidden attractors are very small in some cases,so it is difficult to locate these hidden attractors.Therefore,it is vital to investigate hidden attractors of nonlinear maps.(5)A class of two-dimensional rational memristive maps is presented by introducing a general discrete memristor model into the two-dimensional rational maps.There are no fixed points in the rational memristive maps,so all the attractors in the rational memristive maps are hidden,which has been rarely found in memristive maps.The quadratic discrete memristor is taken as an example.The complex dynamical behaviors of the two-dimensional rational maps with the quadratic discrete memristor are studied using numerical tools.Based on our investigation,these maps can generate different types of solutions,such as hidden periodic,quasi-periodic,chaotic,and hyper-chaotic solutions.In addition,the coexistence of hidden attractors can also be observed,i.e.,hidden multi-stability.(6)A class of two-dimensional memristive maps is put forward by coupling a cosine memristor into a constant map.The memristive maps do not have any fixed points,so they belong to the category of nonlinear maps with hidden attractors.The rich dynamical behaviors of these maps are demonstrated and analyzed using different numerical tools.The proposed memristive maps can produce hidden periodic,chaotic,and hyper-chaotic attractors.They can exhibit homogenous extreme hidden multi-stability,namely the coexistence of infinite hidden attractors.These coexisting hidden attractors are of the same shape but in different positions.(7)The results are summarized,and the future work is pointed out.