TY  - JOUR
A1  - Zillmer, Rüdiger
A1  - Ahlers, Volker
A1  - Pikovskij, Arkadij
T1  - Scaling of Lyapunov exponents of coupled chaotic systems
N2  - 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 continuous-time 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.
Y1  - 2000
ER  - 
TY  - JOUR
A1  - Ahlers, Volker
A1  - Zillmer, Rüdiger
A1  - Pikovskij, Arkadij
T1  - Statistical theory for the coupling sensitivity of chaos
Y1  - 2000
SN  - 1-563-96915-7
ER  - 
TY  - JOUR
A1  - Zillmer, Rüdiger
A1  - Ahlers, Volker
A1  - Pikovskij, Arkadij
T1  - Stochastic approach to Lapunov exponents in coupled chaotic systems
Y1  - 2000
SN  - 3-540-41074-0
ER  - 
TY  - JOUR
A1  - Ahlers, Volker
A1  - Zillmer, Rüdiger
A1  - Pikovskij, Arkadij
T1  - Lyapunov exponents in disordered chaotic systems : avoided crossing and level statistics
N2  - The behavior of the Lyapunov exponents (LEs) of a disordered system consisting of mutually coupled chaotic maps with different parameters is studied. The LEs are demonstrated to exhibit avoided crossing and level repulsion, qualitatively similar to the behavior of energy levels in quantum chaos. Recent results for the coupling dependence of the LEs of two coupled chaotic systems are used to explain the phenomenon and to derive an approximate expression for the distribution functions of LE spacings. The depletion of the level spacing distribution is shown to be exponentially strong at small values. The results are interpreted in terms of the random matrix theory.
Y1  - 2001
ER  - 
TY  - JOUR
A1  - Zillmer, Rüdiger
A1  - Pikovskij, Arkadij
T1  - Continuous approach for the random-field Ising chain
N2  - We study the random-field Ising chain in the limit of strong exchange coupling. In order to calculate the free energy we apply a continuous Langevin-type approach. This continuous model can be solved exactly, whereupon we are able to locate the crossover between an exponential and a power-law decay of the free energy with increasing coupling strength. In terms of magnetization, this crossover restricts the validity of the linear scaling. The known analytical results for the free energy are recovered in the corresponding limits. The outcomes of numerical computations for the free energy are presented, which confirm the results of the continuous approach. We also discuss the validity of the replica method which we then utilize to investigate the sample-to-sample fluctuations of the finite size free energy
Y1  - 2005
ER  -