Recurrence Analysis Of Bifurcation Of Henon Map

Bifurcation is used to describe any sudden change that occurs while parameters are being smoothly varied in nonlinear systems. Bifurcation theory which studies the characteristics and mechanisms of bifurcation phenomenon is of great importance to many practical systems.Recurrence refers that trajectory of the system returns to the position where nearby the initial location after a period of time, it is more realistic than the periodicity. Bifurcation behavior of dynamic systems can be analyzed by the method of recurrence. Recurrence plots is a tool that can visualize the recurrence of trajectory, it is the graphical representation of a binary matrix which encodes the times when two states are in close proximity by pairwise comparison. Recurrence measure is a series of measures which based on recurrence plots,different values of these measures may reflect different dynamics. Recurrence is a basic properties of dynamic system. Research on recurrence helps to improve and deepen the understanding of the dynamic system and the nature.This paper is to study the famous two-dimensional nonlinear system–Henon map, and analysis the bifurcation of the Henon map by the method of recurrence. The recurrence analysis of Henon map mainly study the recurrence measures change with the parameter ’a’in the case of fixing the parameter ’b’ from three perspectives(b = 0, |b| < 1and|b| = 1),comparing the values of recurrence measures with the bifurcation diagram, and analyzing the relationship between the recurrence measures and the bifurcation point. The results showed that recurrence measures were significantly fluctuating at the bifurcation points,then bifurcation points can be detected by these fluctuations, and those fluctuations are more intuitive than the bifurcation diagram. Furthermore, this paper presented a new measure of recurrence–another determinism, analyzed its meaning, and compared it with the bifurcation diagram and other recurrence measures. Our research indicated that the new measure were significantly fluctuating at the bifurcation points, and it was more intuitive to show the period-doubling bifurcation and other dynamic changes. At last, we summarized the main contents of this paper, and prospected the research tendency of the bifurcation phenomenon analyzed by recurrence.