Oscillations, Waves, and Interactions - GWDG

Oscillations, Waves and Interactions, pp. 405–434
edited by T. Kurz, U. Parlitz, and U. Kaatze
Universitätsverlag Göttingen (2007) ISBN 978–3–938616–96–3
urn:nbn:de:gbv:7-verlag-1-15-7
Complex dynamics of nonlinear systems
Ulrich Parlitz
Drittes Physikalisches Institut, Universität Göttingen
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
Abstract. Different nonlinear systems are presented exhibiting various dynamical phenomena,
including bifurcations, chaotic dynamics, and synchronisation. Furthermore, methods
for analysing, modelling and controlling complex dynamics are discussed. All these topics
are illustrated with examples from work on nonlinear systems conducted at the DPI 1 .
1 Introduction
Nonlinearity introduces a wealth of genuine dynamical phenomena, including multistability,
different kinds of bifurcations, chaos and synchronisation, to mention some
of the most important ones. Investigations on chaotic dynamics have shed new light
on the notions of predictability and determinism, because the deterministic dynamics
of even simple systems may be notoriously difficult to predict on short time scales and
even impossible to forecast in the long run. The reason for this behaviour is sensitive
dependence on initial conditions, where arbitrarily small perturbations of the system
grow exponentially and thus become macroscopically relevant after some finite time.
However, in spite of this extreme sensitivity chaotic systems can synchronise their
aperiodic motion and, on the other hand, there exist different ways to suppress chaos
by appropriate control methods.
Many aspects of nonlinear chaotic dynamics have been studied at the DPI 1 during
the past decades. This research was initiated by Werner Lauterborn in the 1970s
and 1980s when he published his seminal work on nonlinear bubble oscillations and
chaos [1–3]. Later, many other nonlinear oscillators have been investigated in detail,
and as a representative example we shall present some typical dynamical features of
the driven Duffing oscillator in Sect. 2.1.
Another class of physical systems with very interesting nonlinear behaviour are
lasers. Semiconductor lasers, for example, exhibit very complex chaotic dynamics
when their emitted light is partly fed back by an external reflection. This phenomenon
will be illustrated in Sect. 2.2 and in Sect. 5 synchronisation of two optically
coupled chaotic semicondutor lasers is shown. Another type of lasers showing chaotic
dynamics above some pump power threshold are frequency-doubled solid-state lasers.
Since for many technical applications irregular intensity fluctuations are unwanted
1 DPI = Drittes Physikalisches Institut = Third Physical Institute