Dynamics of Nonlinear Systems: Integrable and Chaotic Solutions
181
prediction, then we will fail to predict a thunderstorm over our home town two weeks
from now because of this dynamic” [9]. Precise dynamical properties of this system
are exploited for studying various physical, biological, chemical and environmental
systems and also in cryptographic schemes. The significance of sensitivity to initial
conditions is that if we start with a limited amount of information about the system
then after a certain time, the system would no longer be predictable. This does not
mean that one cannot assert anything about events far in the future.
To study how a system pass from being nonchaotic to being chaotic as some
parameter of the system is varied continuously, there are certain parameters could be
checked for the given system known as route to chaos. As an example, for any constant
vector field on the three-dimensional torus, we can choose a small perturbation which
results in a chaotic attractor.
Chaos can be seen in many systems such as electrical circuits, planetary bodies
orbiting each other, fluid dynamics and in chemical reactions. But many real systems
such as the weather need too many parameters to analyze precisely with computers,
which makes these systems chaotic. Important routes to chaos are the period-doubling
cascade, intermittency, crisis and quasiperiodic routes. Most commonly used model
to study the dynamics of a nonlinear system is the logistic map. It uses a nonlinear
difference equation which maps at discrete time steps. This model was first used to
map the population value at any time step to its value at the next time step:
x n+1 = λx n (1 − x n )
(47)
Using this model, it is easy to predict chaotic behaviour of a system by drawing a
graph between λ and x n values. Using these equations, it is possible to study the
dynamics of GDP variation in different countries so as to check whether the most
suitable prediction is possible in the case of GDP growth for a country [8]. The use
of nonlinear oscillator equation for creating cyphertext in cryptography for more
simple and secure data transfer is also studied [9]. Similarly, the phase space study
of Lorenz equation can be used to model the dynamics of brain and hence we can
give a prediction regarding the behaviour of an epileptic person [10–12]. We have
recently started these studies and expect that we could find a good result which can
be utilized for public reference.
6 Conclusion
This article mainly includes a review of the research work I have carried out in
the area of nonlinear dynamics during my Ph.D. under the mentorship of Dr. K.
Babu Joseph. We studied the effect of perturbation in the integrability properties of
some nonlinear dynamical systems, mainly nonlinear Schrödinger equations having
optical soliton solutions and Kadomtsev Petviashvili equations that model the wave
propogation through 2D lattice. The results were published in reputed journal and are
given in reference list [3, 4]. The research carried out by my Ph.D. student, Rosmin
181
prediction, then we will fail to predict a thunderstorm over our home town two weeks
from now because of this dynamic” [9]. Precise dynamical properties of this system
are exploited for studying various physical, biological, chemical and environmental
systems and also in cryptographic schemes. The significance of sensitivity to initial
conditions is that if we start with a limited amount of information about the system
then after a certain time, the system would no longer be predictable. This does not
mean that one cannot assert anything about events far in the future.
To study how a system pass from being nonchaotic to being chaotic as some
parameter of the system is varied continuously, there are certain parameters could be
checked for the given system known as route to chaos. As an example, for any constant
vector field on the three-dimensional torus, we can choose a small perturbation which
results in a chaotic attractor.
Chaos can be seen in many systems such as electrical circuits, planetary bodies
orbiting each other, fluid dynamics and in chemical reactions. But many real systems
such as the weather need too many parameters to analyze precisely with computers,
which makes these systems chaotic. Important routes to chaos are the period-doubling
cascade, intermittency, crisis and quasiperiodic routes. Most commonly used model
to study the dynamics of a nonlinear system is the logistic map. It uses a nonlinear
difference equation which maps at discrete time steps. This model was first used to
map the population value at any time step to its value at the next time step:
x n+1 = λx n (1 − x n )
(47)
Using this model, it is easy to predict chaotic behaviour of a system by drawing a
graph between λ and x n values. Using these equations, it is possible to study the
dynamics of GDP variation in different countries so as to check whether the most
suitable prediction is possible in the case of GDP growth for a country [8]. The use
of nonlinear oscillator equation for creating cyphertext in cryptography for more
simple and secure data transfer is also studied [9]. Similarly, the phase space study
of Lorenz equation can be used to model the dynamics of brain and hence we can
give a prediction regarding the behaviour of an epileptic person [10–12]. We have
recently started these studies and expect that we could find a good result which can
be utilized for public reference.
6 Conclusion
This article mainly includes a review of the research work I have carried out in
the area of nonlinear dynamics during my Ph.D. under the mentorship of Dr. K.
Babu Joseph. We studied the effect of perturbation in the integrability properties of
some nonlinear dynamical systems, mainly nonlinear Schrödinger equations having
optical soliton solutions and Kadomtsev Petviashvili equations that model the wave
propogation through 2D lattice. The results were published in reputed journal and are
given in reference list [3, 4]. The research carried out by my Ph.D. student, Rosmin