Oscillations, Waves, and Interactions - GWDG

424 U. Parlitz
tion [69] of two uni-directionally coupled Rössler systems (20) and (21)
α ˙x1 = 2 + x1(x2 − 4)
α ˙x2 = −x1 − ω1x3 (20)
α ˙x3 = ω1x2 + 0.412x3
α ˙y1 = 2 + y1(y2 − 4)
α ˙y2 = −y1 − ω2y3 (21)
α ˙y3 = ω2y2 + 0.412y3 + c(x3 − y3).
Both Rössler systems exhibit chaotic oscillations when uncoupled (c = 0), but with
different mean frequencies given by the parameters ω1 = 1 and ω2 = 1.1. The
parameter α = 0.013 is a (time) scaling factor due to the hardware implementation.
In order to obtain a description in terms of phase variables, attractors have been
reconstructed from time series (16 bit resolution, 1 kHz sampling frequency) of the
x2 and the y2 variable using the method of delays (see Sect. 4.1) and are shown for
c = 0 in Fig. 14. From these reconstructions phases (angles) φ1(t) and φ2(t) and
mean rotation frequencies
φi(t)
Ωi = lim
t→∞
were computed using polar coordinates centered in the ‘hole’ of each reconstructed
attractor. If both Rössler systems are uncoupled their mean rotation frequencies
Ω1 and Ω2 are different due to the different parameters ω1 = 1 and ω2 = 1.1 in
Eqs. (20) and (21). This difference still exists for sufficiently small values of the
coupling parameter c as can be seen in Fig. 15 where the mean rotation frequencies
Ω1 (dashed) and Ω2 (solid) are plotted vs. c. At c ≈ 0.18 the response system
undergoes a transition to a new phase synchronised state where the mean rotation
frequencies of the drive (20) and the response system (21) coincide.
(a) (b)
Figure 14. Delay reconstruction of the attractors of the drive (a) and the response system
(b) given by Eqs.(20) and (21), respectively. Both time series were generated experimentally
using an analog computer. The mean rotation frequencies are Ω1 = 11.82 Hz (a) and
Ω2 = 13.62 Hz (b) [69].
t
(22)