Dynamics Analysis And Circuit Implementation Of Chaotic System With Self-excited Attractors And Hidden Attractors

Chaotic system is a kind of special and noteworthy nonlinear dynamical system,which has long-term unpredictability.Chaotic attractors are densely covered by numerous unstable periodic orbits.Except for equilibrium points,periodic orbits are the simplest invariant set of the system.Therefore,extracting periodic orbits plays an important role in calculating dynamic average.In this paper,the unstable periodic orbits of three chaotic systems are studied by means of variational method and coded with several symbols.This encoding method is not only applicable to the chaotic systems in this paper,but also to other similar low-dimensional dissipative systems.Particularly,it could be extended to solve a large number of concrete physical problems in different regimes,such as recolliding periodic orbits in attosecond physics,Newtonian three-body planar periodic orbits,and chaos in a hybrid optical bistable system.In addition,a variety of tools are used to explore the dynamic behaviors,bifurcations and forming mechanism of chaos,and the circuit design and simulation of the new chaotic system are realized.First,a so-called symbolic encoding method is introduced to describe and analyze the unstable periodic orbits of the Rucklidge system,which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits.In this work,the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method.The dynamics in the Rucklidge system are explored by using phase portrait analysis,Lyapunov exponents,and Poincaré first return maps.Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values.Meanwhile,the bifurcations of the periodic orbits are explored,significantly improving the understanding of the dynamics of the Rucklidge system.The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.Secondly,the short periodic orbits of the Qi system within a certain topological length are extracted by the variational method.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by using typical nonlinear analysis tools.Based on periodic orbits found with different periods and shapes,they are encoded systematically with two letters or four letters for two different sets of parameters.The periodic orbits outside the attractor with complex topology are discovered by accident.In addition,the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by analytical and numerical means.In this process,the rule of orbital period changing with parameters is also investigated.Finally,a novel system with multiple types of coexisting attractors(including chaotic,periodic,quasi-periodic,and unbound cycles)is proposed,and its dynamics is studied by Lyapunov exponents spectrum,bifurcation diagrams,basin of attraction and other methods.The mechanism of generating chaotic attractors is explored through numerical simulation.Under some conditions,the unstable periodic orbits of the new system without equilibrium points are studied using the variational method,and symbolic coding with four letters is successfully established.The flexibility of the new system and the validity of the numerical results are verified by the circuit simulation with Multisim.