CIM Thematic Term on Mathematics and the Environment
School and Workshop on Dynamical Systems
and Applications
Minicourse
Chaotic dynamics and applications
Celso Grebogi
(Univ.
Lecture
1. Introduction and basic concepts.
Chaotic behavior
is possible for a wide variety of systems. This lecture will begin by an illustration
of this fact using examples from several situations where chaos occurs.
Following that, important background material for subsequent lectures will be
introduced including the following: deterministic dynamical systems, phase
space, the Poincaré- surface of section technique,
maps and flows, and the concept of attractors. Before the advent of chaotic
dynamics it was widely believed that the possible configurations of attracting
motions in phase space were geometrically rather simple (for example, a single
point or a closed loop). It is now apparent, however, that the attractors of
commonly encountered chaotic systems can have extremely singular structure.
These sets are fractals and have the property that their dimension can be
non-integer. Hence these attractors have been called strange attractors. From
the practical point of view, chaos has an even more drastic effect. Namely, it
leads to extreme sensitivity of the motion to small changes in initial
conditions. The important consequence of this is that chaos can make the
forecasting of the system state beyond a certain time virtually impossible.
Lecture
2. Fractal basin boundaries.
It is extremely common that
a system can evolve to different final states (attractors) depending on the
initial conditions that the system starts from. The region in the phase space
corresponding to initial conditions which eventually result in a given final
state is the basin of attraction for that final state. As a consequence of
chaotic dynamics, the boundaries separating different basins can be very
complicated, and, in fact, they can be fractal. This can occur even when the
attractors themselves are not chaotic. Examples illustrating this for simple
systems will be given, and the important consequence of the fractal nature of
these boundaries as an obstruction to predictability will be illustrated.
Lecture
3. Obstruction to modelling and shadowing.
Scientists attempt to
understand physical phenomena by constructing models. A model serves as a link
between scientists and nature, and one fundamental goal is to develop models
whose solutions accurately reflect the nature of the physical process. A
dynamical model uses simplifying assumptions and approximations in the hope of
capturing the essential characteristics of how a physical system evolves with
time. The question of whether a model accurately reflects nature is one
constantly faced by scientists. Recently, we have discovered that there exists
a new level of mathematical difficulty, brought from the theory of dynamical
systems, which can limit our ability to represent nature using deterministic
models. Specifically, we have discovered that certain chaotic systems found in
nature cannot be modelled, particularly higher dimensional chaotic systems. No
model of such a system produces solutions of reasonable length that are
realized by nature. (Furthermore, for these processes, the numerical solutions
of the models do not approximate any actual model solutions.)
Lecture
4. Chaos:
control and communication.
It is common for systems to
evolve with time in a chaotic way. In practice, however, it is often desired
that chaos be avoided or modified for the system to be optimized with respect
to some performance criterion. Given a system which behaves chaotically, one
approach might be to make some large (and possibly costly) alteration in the
system which completely changes its dynamics in such a way as to achieve the
desired objectives. Here we assume that this avenue is not available. Thus we
address the following question: Given a chaotic system, how can we obtain
improved performance and achieve a desired behavior
by making only small controlling temporal perturbations in an accessible system
parameter. Controlled chaotic systems offer an advantage in flexibility in that
any one of a number of different behaviors, chaotic
or not, it can be stabilized by the small control, and the choice can be
switched from one to another depending on the current desired system
performance. I will give many relevant applications to the sciences and
engineering including applications to biological systems. In particular, I will
show that we can use the close connection between the theory of chaotic systems
and information theory in a way that is more than purely formal. I will show
that small perturbations can be utilized to cause the symbolic dynamics of a
chaotic system to track a prescribed symbol sequence thus allowing us to encode
any desired message in the signal from a chaotic oscillator. The natural
complexity of chaos thus provides a vehicle for information transmission in the
usual sense. Furthermore, I will argue that this method of communication will
often have technological advantages. Finally, I will present results of an
experiment which demonstrates that chaos can be used to transmit information.
In it the symbolic dynamics of a chaotic electrical oscillator is controlled to
carry some desired message by using small perturbing current pulses. I will
show a movie in which the communication experiment is done in real time.
Lecture
5. The plankton paradox and other issues.
Nature is permeated by
phenomena in which active processes, such as chemical reactions and biological
interactions, take place in environmental flows. They include the dynamics of
growing population of plankton in the oceans and the evolving distribution of
ozone in the polar stratosphere. I will show that if the dynamics of active
particles in environmental flows is chaotic, then necessarily the concentration
of particles have the observed fractal filamentary structures. These
structures, in turn, are the skeletons and the dynamic catalysts of the active
processes, yielding an unusual singularly enhanced productivity. I will then
suggest that this singular productivity could be the hydrodynamic explanation
for the plankton paradox, in which an extremely large number of species are
able to coexist, negating the competitive exclusion principle that asserts the
survival of only the most perfectly adapted to each limiting resource.