#### Milestone Year

### 1990

#### Chaotic dynamics and its applications

Dynamical system theory describes how the states of a system change in
time according to some deterministic evolution law. It turns out that even
though the evolution laws may appear to be simple, the changes in the
states may be very complicated and possibly chaotic.
Example of systems that were considered by Weizmann scientists and
collaborators include:
particles that move due to the motion of the carrying fluid
(such as pollution in the ocean);
motion of cold atoms in a dark optical trap with a given shape
that may change in time;
motion of high-altitude weather balloons in the atmosphere;
changes in optical waves due to nonlinear effects;
and chemical reactions.

These studies lead to new scientific insights in each of the corresponding
field of science as well as to the development of new mathematical tools
for analyzing dynamical systems. For example, a noteworthy insight
to the field of chaotic advection was the realization (in 1990)
that an abstract dynamical system object called "the unstable manifold"
is in fact the observed feature in many flow visualization experiments,
and is, in many cases, the controlling structure in fluid transport and mixing.
This finding was significant in a large variety of applications
including atmospheric, oceanographic, lab-scale/mechanical devices,
nano-scale devices and marine-life motion. Noteworthy theoretical
developments include the development of a mathematical theory describing
lobe dynamics (in 1986-1994), a mathematical theory describing motion
in steep potentials (in 1996-2008),
and the discovery of a new chaotic mechanism
called the parabolic resonance (in 1997) and the development of a theory
which explains it (in 1997-2010).
These mathematical theories are now available for studying
models that emerge from a large variety of fields of science.

Evolution of a nearly flat profile under the forced nonlinear Schrodinger equation with three different forcing frequencies.