The Symbolic Analysis Of Transition Between U-Sequence And Farey Sequence

U-sequence and Farey sequence can be founded in one-dimensional(1D) unimodal mapand 1D circle map.For Ordinary Di?erential Equation,e.g.,the periodically forced Brus-selator,the poincare′map of the attractor is close to 1D map because of shrinkage of phasevolume.In a direction of a parameter space,U-sequence is found,in accordance with 1Dunimodal map.When the coupling strength between linear oscillator and limit cycle is veryweak,the phenomenon of phase-locking happens.The adjacent Winding numbers array byFarey construction.When a parameter changes,the dynamic behavior of the poincare′mapalso changes.The transition undergoes from 1D interval map to 2D interval map,and then2D annular map,and then 1D circle map.The periodically forced Brusselator is a goodexample to discuss transition between U-sequence and Farey sequence.Symbolic dynamicsis the feasible and e?ective method to analyze the transition.The main task of the thesis is as follows:At first,in a direction of a parameter space,we carry out symbolic dynamics analysisof the system corresponding to 2D interval map and 2D annular map for the periodicallyforced Brusselator.If we trace a period orbit during the process,then the transition ofU-sequence and Farey sequence can be founded.If the transition exists,the relations of twodi?erent period sequences are established.The partition line C? divides the attractor into twoparts,denoted by letters L and R ,meaning the right and left region of C?,respectively.Forannular map,the partition lines ?S,?G,and the preimage ?D of ?S divide the attractorinto three parts,denoted by the letters L,R and N.For the periodically forced Brusselatorsystem,Whenω=0.775,A=0.48-0.2ω,B=1.2,letαvary from 0.035 to 0.025,the transition from U-sequence to Farey sequence is discussed.For U-sequence whichinclude common leading string of forward sequence,i.e., = RLLRRLLRRL,if C=L,itdoesn't have corresponding Farey sequence,because when the map changes from intervalto annular,some letter in the U-sequence would become N,while Farey sequence includeletters L and R only.U-sequence with C=R keeps existing even in the annular map,in whichsome sequences consist of letters L,R and N,some consist of letters L and R.Comparedwith the table of Farey sequence,some of the latter are numbers of Farey sequence.When tracing the period orbits and analyze admissibility of them,we see they are forbidden in theannular map ,not transit to circle map.Second,in several sets of parameters,transition from Farey sequence to U-sequenceis discussed.On the base of symbolic analysis of periodically forced Brusselator,there iscertain relations at all regional symbolic letters on the attractor ,so we can get symbolicsequences of interval map,and then analyze the admissibility of them.If the sequencesare allowed,Farey sequence can transit to U-sequence in a proper direction of parameterspace.Otherwise,the Farey sequence doesn't have related U-sequence in the direction.Fordi?erent sets of parameters,there are di?erent Winding number intervals.We discuss Fareysequence with di?erent Winding numbers,look for the related U-sequence.Finally,making a conclusion of all the task in the thesis,we suggest some problems tobe further studied.