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K. S. Sreelatha
5 Chaos Theory—An Introduction
In the first session of this article, I have mentioned that nonlinear dynamical systems
can in general be classified to integrable systems and chaotic systems. My research
work in the area of integrable systems has been explained in the previous sections.
In this section, I will briefly explain about chaos theory and its potential applications
in economics, cryptography, biological sciences and music; these are the areas in
which I am working now [8].
The discovery of apparently random behaviour of certain dynamical systems
turned out to be quite revolutionary leading to many issues interconnecting stability theory, new geometrical features and new signatures characterizing dynamical
performances. Systems which are basically nonlinear and exhibiting an apparently
random behaviour for certain range of values of system parameters are referred to as
chaotic. However, the solutions or trajectories of the system remain bounded within
the phase space. Mixing: It is a characteristic of a system in which a small interval
of initial conditions gets spread over the full phase space in its asymptotic evolution.
In a chaotic system, an arbitrary interval of initial conditions spread over the part
(attractor) of the phase space to which the trajectory asymptotically confines. Thus
any region gets into every other region of the spatial attractor of phase space. The
chaos theory, also called the complexity theory is a scientific discipline which is
based on the study of nonlinear systems. Chaos theory can be considered as a mathematical method that allows us to extract beautifully ordered structures from a group
of chaotic systems—complex natural systems such as the beating of the human heart
and the trajectories of asteroids.
5.1 Lorentz System
A meteorologist named Edward Lorenz in 1961 made a profound discovery while
trying to find methods to predict weather using computational techniques. It was a
continuous time nonlinear system exhibiting chaotic trajectories for specific values
of system parameters. The system consists of a set of three ordinary differential
equations to model a thermally induced fluid convection in the atmosphere given by
dx
dt
= σ (y − x);
dy
dt
= Rx − y − xz;
dz
dt
= x y − βz
(46)
These equations are well known as Lorenz equations. This model could explain the
uncertainties observed in weather predictions. The phase space of this system can be
plotted and the event is being referred to as Butterfly effect. According to Lorenz,
unpredictability in complex systems is called “sensitivity to initial conditions”. This
means that, in a complex, nonlinear system, a tiny difference in starting position
can lead to greatly varied results. In his own words “if a butterfly is flapping its
wings in Argentina and we cannot take that action into account in our weather
K. S. Sreelatha
5 Chaos Theory—An Introduction
In the first session of this article, I have mentioned that nonlinear dynamical systems
can in general be classified to integrable systems and chaotic systems. My research
work in the area of integrable systems has been explained in the previous sections.
In this section, I will briefly explain about chaos theory and its potential applications
in economics, cryptography, biological sciences and music; these are the areas in
which I am working now [8].
The discovery of apparently random behaviour of certain dynamical systems
turned out to be quite revolutionary leading to many issues interconnecting stability theory, new geometrical features and new signatures characterizing dynamical
performances. Systems which are basically nonlinear and exhibiting an apparently
random behaviour for certain range of values of system parameters are referred to as
chaotic. However, the solutions or trajectories of the system remain bounded within
the phase space. Mixing: It is a characteristic of a system in which a small interval
of initial conditions gets spread over the full phase space in its asymptotic evolution.
In a chaotic system, an arbitrary interval of initial conditions spread over the part
(attractor) of the phase space to which the trajectory asymptotically confines. Thus
any region gets into every other region of the spatial attractor of phase space. The
chaos theory, also called the complexity theory is a scientific discipline which is
based on the study of nonlinear systems. Chaos theory can be considered as a mathematical method that allows us to extract beautifully ordered structures from a group
of chaotic systems—complex natural systems such as the beating of the human heart
and the trajectories of asteroids.
5.1 Lorentz System
A meteorologist named Edward Lorenz in 1961 made a profound discovery while
trying to find methods to predict weather using computational techniques. It was a
continuous time nonlinear system exhibiting chaotic trajectories for specific values
of system parameters. The system consists of a set of three ordinary differential
equations to model a thermally induced fluid convection in the atmosphere given by
dx
dt
= σ (y − x);
dy
dt
= Rx − y − xz;
dz
dt
= x y − βz
(46)
These equations are well known as Lorenz equations. This model could explain the
uncertainties observed in weather predictions. The phase space of this system can be
plotted and the event is being referred to as Butterfly effect. According to Lorenz,
unpredictability in complex systems is called “sensitivity to initial conditions”. This
means that, in a complex, nonlinear system, a tiny difference in starting position
can lead to greatly varied results. In his own words “if a butterfly is flapping its
wings in Argentina and we cannot take that action into account in our weather