Types of dynamical systems; Linear and nonlinear difference and differential equations : basic properties and methods of solution for simple systems; Converting differential equations to difference equations, deriving difference and differential equations from experimental data; Characteristics of chaos, examples of chaos in nature; One dimensional systems; Fixed and periodic points; Properties of tent-map; Properties of logistic map; Estimating Lyapunov Exponent for quadratic family; Different types of bifurcations: saddle node, pitch fork, trans-critical and their variations; Routes and transition to chaos; Feigenbaum's constant, Sarkovskii's theorem; Period 3 window; The role of critical orbits: the Schwarzian derivative, basins of attraction; Examples two dimensional systems: nonlinear dynamics with two variables with an example of Hopf bifurcation; More examples of nonlinear dynamics with two variables; Supercritical, subcritical, and degenerate types of bifurcations; Three dimensional systems, strange attractors, limit cycles; Chaos analysis in time series through Introduction to fractals; Chaos analysis in time series through recurrent quantification analysis (RQA);The examples of chaos in engineering, physical sciences, medical engineering, economics, demand supply models will be included at appropriate points during the course; Methods of controlling chaos: continuous time feed forward control, piecewise constant dither control; OGY method of control.
Suggested basic books:
1
|
A First Course in Chaotic Dynamical Systems, Robert L Devaney
|
Series on : Studies in Nonlinearity edited by Robert L Devaney, The Advanced Book Program, Perseus Books Publishing, LLC, USA, 1992
|
2
|
Non-linear dynamics and chaos, Steven H. Strogatz
|
Westview Press, USA; marketed in India by Levant Books, Kolkata, 2007
|
3
|
An Exploration of Chaos, J. Argyris, G Faust, M. Hasse
|
North Holland, Amsterdam, 1994
|
4
|
Introduction to Applied Non-linear Dynamical Systems and Chaos, Stephen Wiggins
|
Springer, NY, 2003
|
5
|
An Introduction to Difference Equations, Saber Elyadi
|
Springer, NY,2005
|
6
|
Economic Dynamics: Ronald Shone
|
Cambridge University Press, 2002, Cambridge, New York
|
7
|
Introduction to Chaos: H Nagashima and Y Baba
|
Kinki University Higashi-Osaka Japan, Overseas Press, Delhi, 2005
|
8
|
Understanding Nonlinear Dynamics: Daniel Kaplan and Leon Glass
|
Springer-Verlag, NY 1995
|
9
|
Nonlinear Dynamical Economics and Chaotic Motion: Hans-Walter Loerenz
|
Volkwirtschafliches Seminar Georg-August –Universitat, Gottingen Germany, 1992
|
10
|
Lecture Notes on Dynamical Systems, Chaos, and Fractal Geometry: Geoffrrey R. Goodson
|
Townson University Mathematics Department, Spring 2013
|
11
|
Chaos: an Introduction to Dynamical Systems: Kathleen T. Alligwood Tim D. Sauer, James A. Yorke
|
Springer-Verlag, New York, 1996
|
12
|
Introduction to Dynamical Systems and Chaos: G.C.Layek
|
Springer New Delhi, 2015
|
13
|
Chao in Dynamical Systems: Edward Ott
|
Cambridge Uni versity Press, 1993
|
|