Oscillations, Waves, and Interactions - GWDG

(a)
U(∆ Φ)
Complex dynamics of nonlinear systems 423
∆ Φ
Figure 13. (a) Potential U(∆φ) = −∆ω∆φ − ε cos(∆φ) of the Adler Eq. (19) for weak
(|ε| < |∆ω|, blue curve) and strong coupling (|ε| > |∆ω|, red curve), where ∆φ(t) converges
to local minima. (b) Stability region (Arnol’d tongue, shaded) where ∆φ(t) converges to
some fixed value and both oscillators synchronise.
5.1 Synchronisation of periodic oscillations
Modern research on synchronisation began in the 1920s and again, it were technical
systems (vacuum tube oscillators) where synchronisation phenomena were observed
and investigated in detail by E. V. Appleton [61] and B. van der Pol [62] (based on
previous work and a patent of W. H. Eccles and J. H.Vincent) [58]. R. Adler [63]
showed in 1945 for a general pair of weakly coupled periodic oscillators that their
phase difference ∆φ = φ1 − φ2 is governed by a differential equation
d∆φ
dt
(b)
ε
∆ω
= ∆ω − ε sin ∆ϕ , (19)
where ∆ω = ω1 − ω2 denotes the (small) frequency mismatch between the freerunning
oscillators (with individual frequencies ω1 and ω2) and ε is the (small) coupling
strength. Stability analysis shows that ∆φ grows unbounded if the coupling is
weak (|ε| < |∆ω|) but converges to a fixed value if the coupling exceeds some threshold
(|ε| > |∆ω|). This dynamical behaviour can also be visualised as motion of a
particle in a potential U(∆φ) = −∆ω∆φ − ε cos(∆φ) and results in a wedge-shaped
stability region (Arnol’d tongue) in the (∆ω, ε)–parameter space where synchronisation
(i. e., entrainment) occurs (see Fig. 13). The same synchronisation analysis
holds for periodically driven systems.
5.2 Phase synchronisation of chaotic oscillations
In Adler’s equation both oscillators are described by their phases, an approximation
that is valid for weakly coupled periodic systems [58]. However, synchronisation
phenomena are not restricted to this class of dynamical systems but occur also for
coupled or driven chaotic oscillators [64–68].
As an example we investigated with Lutz Junge an analog circuit implementa-