Oscillations, Waves, and Interactions - GWDG

Figure 15. Mean rotation frequencies
Ω1 (dashed) and Ω2 (solid)
vs. the coupling parameter c. For
c > 0.18 phase synchronisation occurs
and both rotation frequencies
coincide [69].
Complex dynamics of nonlinear systems 425
Figure 16 shows the phase difference ∆φ(t) = φ1(t)−φ2(t) as a function of time for
different values of the coupling constant c. For small coupling (c = 0.1) ∆φ increases
unbounded almost linearly in time, similar to the periodic case described by Adler’s
equation (19). If the coupling is increased above the critical value of c ≈ 0.18 chaotic
phase synchronisation occurs and ∆φ undergoes a bounded chaotic oscillation.
This kind of phase synchronisation of chaotic oscillators [70] occurs also for large
networks of coupled oscillators and may be viewed as a partial synchronisation (or
coherence) because the amplitudes of the individual oscillators remain essentially
uncorrelated. To synchronise their temporal evolution, too, stronger coupling is
required and an (almost) perfect coincidence of all state variables of the coupled
systems can, of course, be expected only if the systems are (almost) identical.
This kind of synchrony is called identical synchronisation and can be achieved by
unidirectional coupling if some appropriate coupling scheme is used [66,67]. Furthermore,
the driving chaotic system can by modulated by an external signal (a ‘message’)
and synchronisation of the response system provides all information required to
extract this signal from the (transmitted) coupling signal [67,71]. Whether (synchronising)
chaotic systems are useful potential building blocks for secure communication
systems is controversially discussed, because there are also powerful techniques from
nonlinear time series analysis to attack such an encryption. If, for example, some
part of the message input signal (plaintext) and the corresponding coupling signal
(ciphertext) are known one may ‘learn’ the underlying relation induced by the (deterministic!)
chaotic dynamics. An example for such a ‘known plaintext attack’ using
cluster weighted modelling may be found in Ref. [54].
Figure 16. Phase difference ∆φ =
φ1 − φ2 vs. time t for two representative
cases: c = 0.1, ∆φ grows
linearly, no phase synchronisation;
c = 0.2, ∆φ is bounded, phase synchronisation
[69].