In this section we briefly explained many of our theoretical research interests and, if possible, we provide some references in the general science literature. For works of a more technical nature please refer to the papers section or click the appropriate paper icon below for the topic of your choice. For a brief overview of chaos we refer you to
The following list should come in handy for navigational purposes:
Chaotic Scattering
In the classical dynamics potential scattering problem one considers a
Hamiltonian H = p![]()
Control and Targeting
Two fundamental aspects of chaos are the following:
Crises Another type of transition to a chaotic attractor is the crises. Basically, what
happens in this case is that the unstable chaotic set responsible for a chaotic transient becomes stable as the parameter
p is increased through a critical crisis value p![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Dimensions
Perhaps the most basic aspect of an attractor is its dimension. While it is clear
that the dimension of a fixed point in phase space is zero and that of the
limit cycle is one, it is also the case that invariant sets arising in dynamical
systems (such as chaotic attractors) often have structure on arbitrarily fine
scale, and the determination of the dimension of such sets is nontrivial. In such
cases the assignment of a dimension value gives a much needed quantitative
characterization of the geometrical structure of a complicated object.
Fast Magnetic Dynamos Magnetic fields are pervasive in the Universe. A natural
approach to explain their prevalence is the kinematic dynamo problem: Given a flow of a conducting fluid, will a small
seed magnetic field amplify exponentially with time? If the answer is yes then the zero magnetic field state is
unnatural, and the flow will self-generate its own magnetic field. Recently it has been shown that chaos in the Lagrangian
dynamics of the underlying flow is the key consideration for answering the question posed by the kinematic dynamo problem.
Fractal Basin Boundaries
Basin boundaries arise in dissipative dynamical systems when two, or
more, attractors are present. In such situations each attractor has a
basin of initial conditions which lead asymptotically to that
attractor. The basin boundaries are the sets which separate different
basins. It is very common for basin boundaries to contain unstable
chaotic sets. In such cases the basin boundaries can have very
complicated fractal structure. Because of this complicated very
fine-scaled structure, fractal basin boundaries can pose an impediment
to predicting long-term behavior. In particular, if an initial
condition is specified with only finite precision, it may be very
difficult a priori to determine in which basin it lies if the
boundaries are fractal.
Networks
The interconnection of large numbers of dynamical units is common across a variety
of fields including such applications as the internet, the world wide web, gene networks,
and the web of social interactions between people, corporations, and animals.
Recent research in this area has focused on characterizing network properties, on the
evolution and growth of networks, and on the relation of network topology to the
dynamics and functioning of networked systems.
Pattern Formation
Pattern formation in non-equilibrium systems has drawn intensive
attention and
has been the subject of much rigorous investigation in recent years.
Belousov-Zhabontinskii(BZ) chemical reaction, colonies of social
amoebaem, electical excitation propagation in the heart are situations where considerations of spatiotemporal pattern
dynamics are important.
As the simplest description of such systems, we have investigated various aspects of the Complex Ginzburg
Landau Equation, especially the formation of spiral waves.
Quantum Chaos
According to the correspondence principle, there is a limit where
classical behavior as described by Hamilton's equations becomes
similar, in some suitable sense, to quantum behavior as described by
the appropriate wave equation. Formally, one can take this limit to be
h approaching zero, where h is Planck's constant; alternatively, one
can look at successively higher energy levels, etc. Such limits are
referred to as "semiclassical". It has been found that the
semiclassical limit can be highly nontrivial when the classical
problem is chaotic. The study of how quantum systems, whose classical
counterparts are chaotic, behave in the semiclassical limit has been
called quantum chaos. More generally, these considerations also apply
to elliptic partial differential equations that are physically
unrelated to quantum considerations. For example, the same questions
arise in relating classical acoustic waves to their corresponding ray
equations. Among recent results in quantum chaos is a prediction
relating the chaos in the classical problem to the statistics of
energy-level spacings in the semiclassical quantum regime.
Riddled Basins
The notion of determinism in classical dynamics has eroded since
Poincaré's work led to recognition that dynamical systems can exhibit
chaos: small perturbations grow exponentially fast. Hence,
physically ubiquitous measurement errors, noise, and computer
roundoff strongly limit the time over which, given an initial
condition, one can predict the detailed state of a chaotic
system. Practically speaking, such systems are nondeterministic.
Notwithstanding the quantitative uncertainty caused by perturbations, the
system state is confined in phase space (on an "attractor") so at least
its qualitative behavior is predictable. Another challenge to
determinism arises when systems have competing attractors. With a
boundary (possibly geometrically convoluted ) between sets of
initial conditions tending to distinct attractors ("basins of
attraction"), perturbations make it difficult to determine the fate
of initial conditions near the boundary. Recently, mathematical
mappings were found that are still worse: an attractor's entire basin
is riddled with holes on arbitrarily fine scales. Here,
perturbations globally render even qualitative outcomes
uncertain; experiments lose reproducibility.
Shadowing
Chaotic processes have the property that relatively small numerical
errors tend to grow exponentially fast. In an iterated process, if
errors double each iterate and numerical calculations have 48-bit (or
15-digit) accuracy, a true orbit through a point can be expected to
have no correlation with a numerical orbit after 50 iterates. One may
therefore question the validity of a computer study over times longer than this relatively short time. A relevant result in
this regard is that of Anosov and Bowen who showed that systems which
are uniformly hyperbolic will have the shadowing property: a numerical
(or noisy) orbit will stay close to (shadow) a true orbit for all
time. Unfortunately, chaotic processes typically studied do not have
the requisite uniform hyperbolicity, and the Anosov-Bowen result does
not apply. Recent results of our group apply to nonhyperbolic situations.
Unstable Chaotic SetsAttractors refer to sets which "attract" orbits and hence determine typical long-term behavior. It is also possible to have sets in phase space on which the dynamics can be exceedingly complicated, but which are not attracting. In such cases orbits placed exactly on the set stay there forever, but typical neighboring orbits eventually leave the neighborhood of the set, never to return. One indication of the possibility of complex behavior on such nonattracting (unstable) sets is the presence within these sets of periodic orbits whose number increases exponentially with their period, as well as the presence of an uncountable number of nonperiodic orbits. Nonattracting unstable chaotic sets can have important observable macroscopic consequences. Three such consequences are the phenomena of chaotic transients, fractal basin boundaries, and chaotic scattering.
|