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:
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1
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A First Course in Chaotic Dynamical Systems, Robert L Devaney
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Series on : Studies in Nonlinearity edited by Robert L Devaney, The Advanced Book Program, Perseus Books Publishing, LLC, USA, 1992
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2
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Non-linear dynamics and chaos, Steven H. Strogatz
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Westview Press, USA; marketed in India by Levant Books, Kolkata, 2007
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3
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An Exploration of Chaos, J. Argyris, G Faust, M. Hasse
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North Holland, Amsterdam, 1994
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4
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Introduction to Applied Non-linear Dynamical Systems and Chaos, Stephen Wiggins
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Springer, NY, 2003
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5
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An Introduction to Difference Equations, Saber Elyadi
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Springer, NY,2005
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6
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Economic Dynamics: Ronald Shone
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Cambridge University Press, 2002, Cambridge, New York
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7
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Introduction to Chaos: H Nagashima and Y Baba
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Kinki University Higashi-Osaka Japan, Overseas Press, Delhi, 2005
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8
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Understanding Nonlinear Dynamics: Daniel Kaplan and Leon Glass
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Springer-Verlag, NY 1995
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9
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Nonlinear Dynamical Economics and Chaotic Motion: Hans-Walter Loerenz
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Volkwirtschafliches Seminar Georg-August –Universitat, Gottingen Germany, 1992
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10
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Lecture Notes on Dynamical Systems, Chaos, and Fractal Geometry: Geoffrrey R. Goodson
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Townson University Mathematics Department, Spring 2013
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11
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Chaos: an Introduction to Dynamical Systems: Kathleen T. Alligwood Tim D. Sauer, James A. Yorke
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Springer-Verlag, New York, 1996
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12
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Introduction to Dynamical Systems and Chaos: G.C.Layek
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Springer New Delhi, 2015
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13
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Chao in Dynamical Systems: Edward Ott
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Cambridge Uni versity Press, 1993
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