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 tentmap; Properties of logistic map; Estimating Lyapunov Exponent for quadratic family; Different types of bifurcations: saddle node, pitch fork, transcritical 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

Nonlinear 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 Nonlinear 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 HigashiOsaka Japan, Overseas Press, Delhi, 2005

8

Understanding Nonlinear Dynamics: Daniel Kaplan and Leon Glass

SpringerVerlag, NY 1995

9

Nonlinear Dynamical Economics and Chaotic Motion: HansWalter Loerenz

Volkwirtschafliches Seminar GeorgAugust –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

SpringerVerlag, 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

