Strong and Weak Chaos in Nonlinear Networks with Time-Delayed Couplings

Abstract

We study chaotic synchronization in networks with time-delayed coupling. We introduce the notion of strong and weak chaos, distinguished by the scaling properties of the maximum Lyapunov exponent within the synchronization manifold for large delay times, and relate this to the condition for stable or unstable chaotic synchronization, respectively. In simulations of laser models and experiments with electronic circuits, we identify transitions from weak to strong and back to weak chaos upon monotonically increasing the coupling strength.

Chaos in networks with time-delayed couplings

“…Although this is a linear equation for a small deviation δs between two trajectories, the delay term leads to a complex behaviour which usually can be investigated only numerically. In the limit of large delay times τ , however, some general properties can be derived [8]. Roughly speaking, if the first term of equation (2.2) explodes, then the second term cannot avoid this.…”

“…Precise scaling relations have been derived in Heiligenthal et al [8]. However, already an intuitive argument shows the properties of equation (2.2): small deviations separate from each other with δs(t) ∝ exp(λ m t).…”

“…We call it sub-LE; previously, it was called anomalous [12] or instantaneous LE [8]. Note that λ 0 still depends on the feedback, because the The sign of the sub-LE λ 0 determines the type of chaos.…”

“…Chaos is extremely sensitive to initial conditions, and the Lyapunov exponents (LEs) describe how fast two nearby trajectories separate from each other. Recently, two types of chaos have been reported for time-delayed systems: strong and weak, depending on how the LE scales with the delay time [8]. It was shown that systems with weak chaos can only synchronize completely.…”

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

“…Time delays are always present in coupled systems due to the finite signal propagation time. These time lags give rise to complex dynamics and have been shown to play a key role in the synchronization behavior of systems [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], see also the review [43]. In Ref.…”