Statistical characteristics of the Poincaré return times for a one - dimensional nonhyperbolic map
نویسنده
چکیده
Characteristics of the Poincaré return times are considered in a one-dimensional cubic map with a chaotic nonhyperbolic attractor. Two approaches, local one (Kac’s theorem) and global one related with the AP-dimension estimation of return times, are used. The return times characteristics are studied in the presence of external noise. The characteristics of Poincaré recurrences are compared with the form of probability measure and the complete correspondence of the obtained results with the mathematical theory is shown. The influence of the attractor crisis on the return time characteristics is also analyzed. The obtained results have a methodical and educational significance and can be used for solving a number of applied tasks.
منابع مشابه
Poincaré-Bendixson Theorem for Hybrid Systems
The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stabili...
متن کاملConvergence of Rare Events Point Processes to the Poisson for Billiards
We show that for planar dispersing billiards the return times distribution is in the limit Poisson for metric balls almost everywhere w.r.t. the SRB measure. Since the Poincaré return map is piecewise smooth but becomes singular at the boundaries of the partition elements, recent results on the limiting distribution of return times cannot be applied as they require the maps to have bounded seco...
متن کاملA Remark on heteroclinic bifurcations Near Steady State/Pitchfork bifurcations
We consider a bifurcation that occurs in some two-dimensional vector fields, namely a codimension-one bifurcation in which there is simultaneously the creation of a pair of equilibria via a steady state bifurcation and the destruction of a large amplitude periodic orbit. We show that this phenomenon may occur in an unfolding of the saddle-node/pitchfork normal form equations, although not near ...
متن کاملA family of pseudo-Anosov maps
We study a family of area-preserving maps of the 2-torus and show that they are pseudo-Anosov. We present a method to construct finite Markov partitions for this family which utilizes their common symmetries. Through these partitions we show explicitly that each map is a tower over a first return map, intimately linked to a toral automorphism. This enables us to calculate directly some dimensio...
متن کاملReturn Map Quantization from an Integrate-and-Fire Model with Two Periodic Inputs
In this paper, we consider the Integrate-andFire Model (ab. IFM) with two periodic inputs. The IFM outputs a pulse-train which is governed by a one dimensional return map. Using the return map, the relationship between the inputs and the output is clarified: the first input determines the global shape of the return map and the IFM outputs various periodic and chaotic pulse-trains; the second in...
متن کامل