Discrete Dynamics in Nature and Society
Volume 2006 (2006), Issue 3, Article ID 74723, 19 pages
doi:10.1155/DDNS/2006/74723
Universality and scaling in networks of period-doubling maps with a pacemaker
Anna S. Ivanova1
, Sergey P. Kuznetsov2
and Andrew H. Osbaldestin3
1Department of Nonlinear Processes, Saratov State University, Astrakhanskaya 83, Saratov 410026, Russia
2Laboratory of Theoretical Nonlinear Dynamics, Saratov Branch of Institute of Radio-Engineering and Electronics, Russian Academy of Sciences, Zelenaya 38, Saratov 410019, Russia
3Department of Mathematics, University of Portsmouth, Portsmouth PO1 3HE, UK
Abstract
The networks of globally coupled maps with a pacemaker have been introduced. We consider a generalization of the Kaneko model with a pacemaker represented by a single period-doubling element coupled unidirectionally with a set of other mutually coupled cells. We also investigate the dynamics of a system of two unidirectionally coupled elements, which manifests a special type of critical behaviour, known as bicriticality, at the point of simultaneous transition to chaos in both subsystems. With the help of the renormalization group (RG), we show for a case of two mutually coupled bicritical maps with a pacemaker that there are two types of coupling: dissipative and inertial. We investigate the dynamics of a network with a pacemaker with two types of global coupling and the properties of universality and scaling in this system.