 
Summary: Scaling of Lyapunov exponents of coupled chaotic systems
Ru¨diger Zillmer, Volker Ahlers, and Arkady Pikovsky
Department of Physics, University of Potsdam, Postfach 601553, D14415 Potsdam, Germany
Received 20 May 1999
We develop a statistical theory of the coupling sensitivity of chaos. The effect was first described by Daido
Prog. Theor. Phys. 72, 853 1984 ; it appears as a logarithmic singularity in the Lyapunov exponent in
coupled chaotic systems at very small couplings. Using a continuoustime stochastic model for the coupled
systems we derive a scaling relation for the largest Lyapunov exponent. The singularity is shown to depend on
the coupling and the systems' mismatch. Generalizations to the cases of asymmetrical coupling and three
interacting oscillators are considered, too. The analytical results are confirmed by numerical simulations.
PACS number s : 05.45.Xt
I. INTRODUCTION
The dynamics of coupled chaotic systems attracted large
interest recently. Many interesting phenomena, in particular
different kinds of synchronization, can already be observed
in the simplest cases of two interacting chaotic attractors
13 . While the synchronization occurs for couplings large
enough to suppress a chaosinduced tendency to desynchro
nization, an interesting anomality in the dynamics happens
for very small couplings already. This is the effect of cou
