Dynamics of Nonlinear Systems:
Integrable and Chaotic Solutions
K. S. Sreelatha
Abstract Nonlinear dynamics deals with dynamical system whose behaviour is
not linear as time evolves. Most of the natural systems are inherently nonlinear. To
reduce the complexity of nature, all systems are approximated as linear. Dynamical
systems can be modelled using differential equations. Nonlinear systems are usually
classified as chaotic and integrable systems based on their easiness to find solutions of
these dynamical equations. Integrable systems are exactly solvable and they possess
infinite number of constants of motion. Solution for nonlinear integrable equations
can be easily found. For chaotic systems, it is not possible to find exact solution.
Approximate behaviour can be studied using numerical methods. But they are highly
dependent on initial conditions. A very small change in initial condition can change
the entire behaviour of the system. We have investigated the dynamics of integrable
systems during my doctoral studies. Recently, we are investigating the use of chaos
theory in cipher texting in cryptography and in biological systems. Also trying the
possibility of applying chaos in economics. This article is a brief review of the work
done during and after my doctoral studies.
Keywords Integrability · Solitons · 2D lattice · Nonlinear waveguides · Chaos
1 Introduction
Dynamical systems are usually modelled using differential equations. Physical systems are in general classified into linear and nonlinear systems based on their overall
behaviour as time goes. Systems for which any change in its parameter causes a corresponding change in its behaviour is called a linear system. In nonlinear systems, a
small change in one parameter will lead to an unexpected variation in the behaviour
of the system.
Most of the natural systems are nonlinear. Any dynamical system can be defined
using a collection of configurational coordinates and equations of motion obeyed
K. S. Sreelatha (B)
Department of Physics, Government College Kottayam, Kerala, India
e-mail: drsreelathaks@gmail.com
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_12
165
Integrable and Chaotic Solutions
K. S. Sreelatha
Abstract Nonlinear dynamics deals with dynamical system whose behaviour is
not linear as time evolves. Most of the natural systems are inherently nonlinear. To
reduce the complexity of nature, all systems are approximated as linear. Dynamical
systems can be modelled using differential equations. Nonlinear systems are usually
classified as chaotic and integrable systems based on their easiness to find solutions of
these dynamical equations. Integrable systems are exactly solvable and they possess
infinite number of constants of motion. Solution for nonlinear integrable equations
can be easily found. For chaotic systems, it is not possible to find exact solution.
Approximate behaviour can be studied using numerical methods. But they are highly
dependent on initial conditions. A very small change in initial condition can change
the entire behaviour of the system. We have investigated the dynamics of integrable
systems during my doctoral studies. Recently, we are investigating the use of chaos
theory in cipher texting in cryptography and in biological systems. Also trying the
possibility of applying chaos in economics. This article is a brief review of the work
done during and after my doctoral studies.
Keywords Integrability · Solitons · 2D lattice · Nonlinear waveguides · Chaos
1 Introduction
Dynamical systems are usually modelled using differential equations. Physical systems are in general classified into linear and nonlinear systems based on their overall
behaviour as time goes. Systems for which any change in its parameter causes a corresponding change in its behaviour is called a linear system. In nonlinear systems, a
small change in one parameter will lead to an unexpected variation in the behaviour
of the system.
Most of the natural systems are nonlinear. Any dynamical system can be defined
using a collection of configurational coordinates and equations of motion obeyed
K. S. Sreelatha (B)
Department of Physics, Government College Kottayam, Kerala, India
e-mail: drsreelathaks@gmail.com
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_12
165