Oscillations, Waves, and Interactions - GWDG

Complex dynamics of nonlinear systems 429
Stabilisation of the Nd:YAG laser succeeded also with Notch Filter Feedback control
[93] providing some easily implementable approximation of MDFC.
6.2 Controlling spatio-temporal chaos
Delayed feedback can also be used to locally stabilise and manipulate complex spatiotemporal
dynamics [94]. To illustrate this approach we use the two-dimensional complex
Ginzburg-Landau equation (GLE)
∂tf = (1 + ia)∇ 2 f + f − (1 + ib)f|f| 2 + u (29)
with an external control signal u(x, t). ∂t and ∇ denote the temporal and the spatial
derivative, respectively. The GLE (29) is a prototypical equation for spatio-temporal
systems close to a supercritical Hopf-bifurcation. It is solved here numerically for
periodic boundary conditions with a spectral code based on a Runge-Kutta scheme
of 4th order combined with a spectral method in space with a spatial grid of 90 × 90
elements (∆x = ∆y = 1).
The control signal is applied at a few control cells Ci, only, simulating experimental
sensors and actuators. In general, the control signal ui which is applied at cell Ci
ui(t) =
M�
m=1
kima sij (t − τim ) − kimb sij (t) (30)
is given by delayed and non-delayed input signals sij measured at other cells C where a measured signal
�
ij
sk(t) = f(z, t) dz (31)
Ck
is the averaged value of f at control cell Ck.
Again, the performance of the control scheme depends crucially on the gains
kima , kimb and delay times τim that may vary from cell to cell (as indicated by
the multiple index).
Figure 19 shows two examples where different coupling schemes (Figs. 19(b) and
19(d)) are used to stabilise plane waves (Fig. 19(a)) and to trap a spiral wave
(Fig. 19(c)).
7 Conclusion
This article is an attempt to give a tutorial overview of research in nonlinear dynamics
at the DPI. Of course, it is incomplete but we hope it motivates the reader to learn
more about this exciting interdisciplinary field that is heading now towards even more
complex systems like large networks of coupled oscillators or swarms of interacting
agents. So, stay tuned .... at DPI.
Acknowledgements. The author thanks Werner Lauterborn, Thomas Kurz, Robert Mettin
and all other coworkers, students and staff at the DPI for excellent collaboration and
support.