Dynamical Behavior And Symbolic Dynamics Analysis Of Ordinary Differential Equations

Since 1950s Nonlinear Science has developed so widely that now it already penetrates into a lot of new subjects such as Neural Networks(NN),Cellular Automata(CA),Complex Networks(CN)and so on.So it is very significant to study some nonlinear systems,especially the Lorenz system which is a touchstone for many new ideas in chaotic dynamics.At present the main methods to study Nonlinear Science are analytical means,numerical work and experimentations.Because of the complication of the chaotic phenomenon,the traditional methods are not enough to find new properties and new laws.Symbolic dynamics is turned to account and it is useful to yield global results on chaotic and periodic regions that are by far not a simple job to accomplish neither by purely analytical means nor by numerical work alone.Thus,some dynamic behaviors and symbolic dynamic analysis of two nonlinear system are studied in the paper which contain three aspects as follows:Firstly,dynamic behaviors of the Lorenz system in the internal bifurcation parameters space are discussed.The properties of fixed points in the ordinary differential equations influence heavily to its global behaviors.Take several fixed points as the internal parameters and combine it with some primary parameters,a new parameter space-internal bifurcation parameter space is constructed in which we can study dynamic behaviors of the system on the Poincarésurface of section by changing the location of fixed points.According to this idea a new internal bifurcation parameter space is constructed firstly for the Lorenz system and then the numerical analysis is took in the wide parameter range.Many new interesting phenomena are found,so to say,there are abundant co-existences of a)the fixed point and a periodic or chaotic attractor:b)a periodic orbit and a sequence of period-doubling bifurcation to chaos.The work supplements the research for the Lorenz system and a new path is broken for our later work.Secondly,the symbolic dynamic analysis of the Lorenz system in the internal bifurcation parameters space are studied.Symbolic dynamic is a powerful method to study complex dynamics with finite precision which associate the topology with the numerical calculation together.One- and especially two-dimensional symbolic dynamics of the Lorenz system are investigated in the internal bifurcation parameter space.And all the admissible periodic sequences up to period 6 and their relevant locations are given too.The symbolic dynamics analysis of the Lorenz equations in the primary parameter space had been done already by Prof.Liu.But in the new parameter space we can find a perfect controlling parameter so that two-dimensional structures and the fundamental forbidden zone in two-dimensional symbolic dynamics are more obvious than that in the primary parameter space.Especially, although the structure of Poincarémapping in the internal bifurcation parameter space at x~*=3.0,z~*=17.0 is the same with that in the primary parameter space at r= 28,the former has forward contracting foliations and back contracting foliations and two line of their tangent points partition the phase space into four parts.So its symbolic description wants not two words as usually but four words.This outcomes break the old viewpoint that only two words are wanted in description the global chaotic attractors like ones in the primary parameter space at r= 28.Moreover,the local chaotic attractor in the internal bifurcation parameter space at x~*= 2.0,z~*= 25.3 whose Poincarémapping locates only in the first quarter of phase plan is also partitioned into two parts so that two words of symbolic dynamics must be constructed.Such are another new outcomes and enrich the symbolic dynamics.Finally,dynamic behaviors and ordering rule of symbolic sequences in a new chaotic system are investigated.Since Lorenz found the first canonical chaotic attractor in a simple three-dimensional autonomous system in 1963,the study about chaotic dynamics of nonlinear systems is very thoroughly,especially the controlling and application of chaos attract a lot of researchers recently.In 1999 Chen attained Chen system by the chaotic feedback controlling method which is not topological with Lorenz equations though they have the same nonlinear terms.With the same method Lüand Chen obtained Lüsystem in 2002 which is a bridge between Lorenz system and Chen system.Then another new chaotic system—Liu system is discussed that the forming mechanism of butterfly attractors' compound structure can be obtained by merging together two simple attractors after performing one mirror operation. Add a 2-term cross product to the third equation of Liu system and a new chaotic system are constructed.The structure of its chaotic attractor in Poincarémap which includes two fixed points but zero is richer and 2-dimensional structures such as hooks,layers are more obvious then Lorenz system.But the ordering rule coincides with each other.