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
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
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 modeled, 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
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.
Dynamical systems, numerical experiments and super-computing
Dynamical systems study the
evolution models of natural phenomena and the
simplified models which help to understand them.
They can be given in deterministic form, either
by means of ordinary differential equations, partial
differential equations or discrete maps. They
are useful in all domains of science and technology.
In their study tools from all
areas of mathematics are used. But for systems
with some degree of complexity it is impossible
to produce a fairly complete description of the
evolution in the space of states, and its dependence
with respect to parameters, without using numerical
techniques. They are essential for concrete applications
and very useful even for theoretical studies.
They can be seen as an experimental part of mathematics.
In the last years it has become
possible to achieve a generalisation in the systematic
use of numerical experiments, due to the availability
of large arrays of processors working in parallel
with a reduced cost. But the impact of new algorithms
has been even larger. This makes feasible to face
problems of larger and larger complexity.
In the lectures some aspects
of the general methodology and several examples
will be presented.