|
The Poincare mapping close to the one-dimensional U sequence and Farey sequence corresponding to the dynamical behavior of logistic map and circle image system of ordinary differential equations, such as cycle driven Brussels oscillator phase volume contraction attractor images in the direction of a parameter space, which appear similar to the one-dimensional unimodal mappings U sequence; linear oscillator limit cycle coupling strength is weak, the periodic motion of locking are adjacent revolutions Farey number sequence structure. When a parameter changes, the dynamic behavior of the system Poincare mapping two-dimensional torus image range from a one-dimensional image to a two-dimensional interval mapping and then finally to the one-dimensional circle map transition provides good research the U sequence Farey sequence between transition instance symbolic dynamics analysis of this transition effective method. The main work is as follows: First, for the the periodic driving Brussels vibrator subsystem, in the parameter space of a certain direction, established with the two-dimensional interval mapping, two-dimensional the ring surface map corresponding symbolic dynamics, and track parameters change when some U sequence to Farey how the sequence is to be expressed, and thus establish a correspondence relationship of two different sequences of periodic word. Dividing line C? Image in the interval on the attractor in the Poincare section is divided into two parts, the right area is represented by the letter R, and the left area of ??the letter L, C represents the point of tangency of the dividing line. Torus mapping Poincare dividing line? S,? G and? D will attract sub-divided into three parts, marked with the letters R, L, and N, respectively. The paper discusses the parameters ω = 0.775, B = 1.2, A = 0.48-0.2Ω, α = 0.035 has been reduced to α = 0.025, containing common words head = RLLRRLLRRL the U sequence Farey sequence transition. After discussion, analysis, C L, U sequence impossible the corresponding Farey sequence, since it is the transition from the interval mapping with the letter N, the total torus Mappings, Farey sequence is composed of only by the letters L and R the latter is not necessarily all of the sequence; the C of R U sequence transition to the ring surface mapping, some corresponding sequence consisting of L, R and N, and some by the L, R composed of the sequence control of the table with the Farey sequence found Farey sequence, and we are in the process of tracking these periodic orbits, and make allow analysis, we can see that they are somewhere in the torus mapping is disabled, can not transition to the circular mapping. Secondly, the study Farey sequence to several groups of different parameters U sequence transitional, based on the symbolic dynamics of the system analysis, found attractor symbol letters on the regional transition is the only certainty, so write the interval mapping symbol sequence it allows the analysis. If the corresponding sequence of symbols are permitted, then along a specified direction of the parameter space, which Farey sequences can transition to the U sequence; the contrary, in the direction the Farey sequence does not correspond to the U sequence. Control Farey sequence table, different parameter set corresponding to a different number of revolutions of the interval range, we Farey sequences of the different number of revolutions, respectively research to identify the sequence of the corresponding U. Finally, the summary of the full text of the work, and pointed out that the problem needs further study.
|