# Berlin 2012 – wissenschaftliches Programm

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# DY: Fachverband Dynamik und Statistische Physik

## DY 22: Posters I

### DY 22.49: Poster

### Mittwoch, 28. März 2012, 17:00–19:00, Poster A

**Attractor dimension at the transition to chaos synchronization for networks with time-delayed couplings** — •Steffen Zeeb and Wolfgang Kinzel — Theoretische Physik III, Universität Würzburg

A network of nonlinear units interacting by time-delayed couplings can synchronize to a common chaotic trajectory. Although the transmission time may be very long the units are completely synchronized without time shift.

We investigate the attractor dimension at this transition to complete chaos synchronization. In particular, for networks of iterated maps we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents which is calculated analytically for Bernoulli maps and numerically for Tent maps. However, we argue that the Kaplan-Yorke conjecture cannot be true at the transition. For the synchronized state the Lyapunov exponents perpendicular to the synchronization manifold cannot contribute to the attractor dimension. Consequently, the Kaplan-Yorke dimension has to be discontinuous at the transition. We calculate the magnitude of this jump for different networks.

The Kaplan-Yorke dimension is an upper bound for the correlation dimension. Using the method of Grassberger & Procaccia we calculate the correlation dimension for networks of iterated Bernoulli and Tent maps. For Bernoulli networks the correlation dimension jumps at the transition to synchronization whereas for Tent maps the correlations dimension is continuous. We conclude that for some systems the Kaplan-Yorke conjecture yields qualitatively incorrect results.