Study On Unstable Periodic Orbits Of Several Low-dimensional Dissipative Chaotic Systems

In this paper,we investigated the unstable periodic orbits of five common three-dimensional chaotic systems with the variational method and the evolution of orbits with parameter values and bifurcation behaviors.According to the topological structure of periodic orbits,in each system an appropriate symbolic dynamics are established respectively,so that each orbit is included without repetition.First of all,we systematically investigated the organization of the unstable cycles for the Chen and the Lü systems.The multivalued and non-invertible return maps show the complexity of the two dissipative flows,and a new designed method for building one-dimensional symbolic dynamics based on the orbital topological structure is proposed.We employed the variational approach for the calculations of unstable periodic orbits and used two orbital fragments for initialization.The homotopy evolution of periodic orbits provides evolution rules for the periods.For a chaotic flow,current work supplies an interesting framework for the classification of periodic orbits.Secondly,the systematical calculations of the unstable cycles for the Burke–Shaw system（BSS）are presented.In contrast to the Poincaré section method used in previous studies,we used the variational method for the cycle search and established appropriate symbolic dynamics on the basis of the topological structure of the cycles.The variational approach made it easy to continuously track the periodic orbits when the parameters were varied.Structure of the whole cycle in the dissipative system demonstrated that the methodology could be effective in most low-dimensional chaotic systems.Then what we investigated the unstable periodic orbits of a nonlinear chaotic generalized Lorenz-type system.By means of the variational method,appropriate symbolic dynamics are put forward,and the homotopy evolution approach,which can be used in the initialization of the cycle search,is introduced.Fourteen short unstable periodic orbits with different topological lengths,under specific parameter values,are calculated.We also explored the continuous deformation for part of the orbits while changing the parameter values,which provides a new approach to observe various bifurcations.The scale transformation of the generalized Lorenz-type system leads to a single parameter system known as the diffusionless Lorenz equations.By systematically calculating the periodic orbits in the diffusionless Lorenz equations,our research shows the efficiency of this topological classification method for the periodic solutions in the variants of a classical Lorenz system.Finally,we find that when parameter values change,the topology of the generalized Lorenz-type system will also change greatly.Under some parameter values,we only need to establish one-dimensional symbolic dynamics,and use two symbols to represent periodic orbits.However,under a set of typical parameters,the generalized Lorentz-type system is correlated with BSS,and the system property is more complex.It is necessary to divide the attractor inside and outside parts and establish symbolic dynamics of four symbols.The study of the periodic orbits of these systems by using variational periodic orbits directly proves the effectiveness and simplicity of the variational method in locating the periodic orbits of low-dimensional dissipative chaotic systems.