 |
|
Most natural phenomena are nonlinear; yet even today,
theoretical analysis of physical systems is usually based
on linear mathematical models or ones having small deviations
from linearity. Linear models are still routinely used because
they are much easier to solve than the correct nonlinear ones.
Within the last two decades, however, both theoretical and
experimental investigations of nonlinear phenomena have shown that
often behavior that appears to be random or chaotic is actually
deterministic in its orgin. Nonlinear deterministic systems under
these conditions are predictable only for short times. This
paradoxical situation exists because the deterministic solutions depend
very sensitively on initial conditions. It has also been found that
several classes of systems show universal behavior at the onset of
chaos. Thus, system as diverse as a dripping faucet and a heart in
ventricular fibrillation show many common features in their dynamics.
Generally speaking most of the issues touched here are from the point of view
of computer simulation of dynamical systems. However, there are some sections
where simple analytical methods/results are also described/mentioned.
The section on "general chaos" starts by
introducing the concepts via the Logistic map. The choice of the
system, besides historical, is because of the relative ease with which some
analytical analysis can be done. In most of the cases the material is illustrated
using the output of the programs provided.
Important as it is to understand the mathematics behind the pretty pictures, I have
tried to provide working programs that can be downloaded and played around to get an
intuitive feel of the dynamics of chaotic models. Representative models
of nonlinear systems and their analysis is illustrated
using maps (discrete time), ordinary
differential equations (ODEs), delay differential
equations(DDE). The latter is an infinite dimensional system and special care
has to be
taken to simulate it. The above mentioned approaches to model dynamical systems are
examples that has time as an independent variable (both discrete and continuous time).
Following are some of the few ways of modeling spatially extended
systems: Cellular automata (CA),
Coupled map lattice (CML) systems and
coupled ordinary differential equations, and
partial differential equations(pde).
Following is a conceptual table of some of the modeling approaches that will
be touched upon.
where D denotes discrete and C denotes continuous time models. The column about
domensionality refers to the number of independent phase variables needed
to specify the state of the system uniquely in the phase space. Thus, N
for the case of Maps and ODEs denote the independent variables needed to
specify the state of system. Thus, as an example N = 1 and N = 2 for the
case of Logistic Map and
Henon map, respectively and similary N = 3 for the
case of Rossler and
Lorenz system. The phase space dimensions
of such systems are necessarily finite integer valued. In contrast, to
ODEs, delay differential equations require
initial functions to "jump
start" the integration and are actually infinite dimensional systems, even
if there is only one dependent variable (you need to specify infinite
number of initial conditions between [-delay_time, current_time]). By
similar argument PDEs are infinite dimensional
systems (which require
an initial function that specifies the state of the system at the
boundaries). The dimensionality for the case of CA,
CML, and coupled ODEs
diverges as the
size ( number of lattice sites) of the system increases.
|
Leon O. Chua
