Different types of scaling in the dynamics of period–doubling maps under external periodic driving

N Yu Ivank’Ov,S P Kuznetsov

doi:10.1155/s1026022600000546

N Yu Ivank’Ov, S P Kuznetsov

Open Access

https://doi.org/10.1155/s1026022600000546

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Abstract

Based on the renormalization group approach developed by Kuznetsov and Pikovsky (Phys. Lett., A140, 1989, 166) several types of scaling are discussed, which can be observed in a neighborhood of Feigenbaum’s critical point at small amplitudes of the driving. The type of scaling behavior depends on a structure of binary representation of the frequency parameter:F-scaling (Feigenbaum’s) for finite binary fractions,P- andQ-scaling (periodic and quasiperiodic) for periodic binary fractions, andS-scaling (statistical) for non-periodic binary fractions. All types of scaling are illustrated by parameter-plane diagrams for the rescaled Lyapunov exponent.

Highlights

It is known that many dissipative nonlinear systems of different nature demonstrate the onset of chaos via a period-doubling bifurcation cascade
According to the concept of universality suggested by Feigenbaum [1,2], details of this behavior for the whole class of systems may be described quantitatively using the simplest representative of the class, say, the logistic map
In this paper we have discussed scaling properties of the parameter space for the forced logistic map near Feigenbaum’s critical point, which follow from the Renormalization group (RG) analysis [9]

Summary

INTRODUCTION

It is known that many dissipative nonlinear systems of different nature demonstrate the onset of chaos via a period-doubling bifurcation cascade. According to the above discussion, one can use the forced logistic map to study many aspects of dynamics near the onset of chaos in these systems. There are two essentially different cases: the first, when the frequency parameter w is rational, i.e., a period of the forcing contains an integer number of time steps, and the second, when w is irrational, and one should speak about quasiperiodic rather than periodic driving. In this paper the scaling properties of the parameter space for the forced logistic map near the Feigenbaum critical point are studied. In continuous-time autonomous perioddoubling systems the dynamics under external driving is more complicated because of presence of additional variable, the phase measured along the trajectory. This class of systems has to be studied separately.

RENORMALIZATION GROUP ANALYSIS

RATIONAL FREQUENCIES

CONCLUSION