Discrete Dynamics in Nature and Society
Volume 5 (2000), Issue 3, Pages 223-232

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

N. Yu. Ivank’ov1 and S. P. Kuznetsov2

1College of Applied Sciences, Saratov State University, Astrakhanskaja 83, Saratov 410026, Russia
2lnstitute of Radio-Engineering and Electronics RAS, Saratov Division, Zelenaya 38, Saratov 410019, Russia

Copyright © 2000 N. Yu. Ivank’ov and S. P. Kuznetsov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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- and Q-scaling (periodic and quasiperiodic) for periodic binary fractions, and S-scaling (statistical) for non-periodic binary fractions. All types of scaling are illustrated by parameter-plane diagrams for the rescaled Lyapunov exponent.