Unit 1

Introduction and Motivation – Examples of Nonlinear and Chaotic Systems, definition of dynamical system, state space, vector field and flow

One Dimensional Flows – Flows on the line, fixed points and their stability, linear stability analysis, impossibility ofoscillations, bifurcations in one dimensional case, saddle-node,transcritical and pitchfork, flows on the circle, examples.

Unit 2

Two Dimensional Flows – Planar linear systems, solving linear systems, eigenvalues and eigen vectors, dynamical classification based on eigenvalues, planar nonlinear systems, phase portraits, linearisation, hyperbolic fixed points and Hartman – Grobman theorem, stable, unstable and centre manifolds, limit cycles, van der pol equation, Poincare – Bendixson theorem, saddle-node, transcritical, pitchfork and Andronov-Hopf bifurcations in planar case.

Unit 3

Chaotic Dynamics – One dimensional maps, fixed points and cobwebs, logistic map, bifurcations in iterated maps and chaos, Feigenbaum universality.Three dimensional systems, Poincaresections, quasiperiodicity, routes to chaos. Quantifying chaos -Lyapunov exponents, Kolmogorov Sinai entropy, fractal dimensions. Analytical methods for nonlinear systems -Perturbation method, Secular terms, Lindsted – Poincare method, averaging method, method of multiple scales.

Course Objectives

This course is expected to enable the student

• Familiarize with nonlinear dynamics concepts for better understanding of physical systems
• Demonstrate analytical and numerical tools to analyse systems with nonlinear effects

Course Outcomes

• CO1: Apply the qualitative approach to the study of dynamical systems to analyse nonlinear systems.
• CO2: Develop theoretical and computational tools for the analysis of one-dimensional, two-dimensional and multi- dimensional nonlinear systems
• CO3: Analyse different bifurcations of practical nonlinear systems and to use them in design
• CO4: Differentiate chaotic and non-chaotic systems and to analyse mechanical engineering systems exhibiting chaotic behaviour
• CO5: Solve interdisciplinary problems in engineering, ecological, electronic, biological and financial systems using nonlinear dynamics tools

CO – PO Mapping

Textbook(s)

• Steven H. Strogatz, “Nonlinear Dynamics and Chaos”,Reading, Addison-Wesley, 1994.
• Robert C. Hilborn, “Chaos and Nonlinear Dynamics”, Second Edition, Oxford University Press, 2000.

Reference(s)

• Ali Hasan Nayfeh, “Introduction to Perturbation Techniques”, John Wiley, 1993.
• Morris W. Hirsch, Stephen Smale, and Robert L. Devaney, “Differential Equations, Dynamical Systems and an Introduction to Chaos”, Academic Press, Elsevier, 2004.
• Lakshmanan M. and Rajashekhar S., “Nonlinear Dynamics”, Springer Verlag, 2003.
• Robert L. Devaney, “An Introduction to Chaotic Systems”, Second Edition, West View Press, 2003.
• Edward Ott, “Chaos in Dynamical Systems”, Cambridge University Press, 1993.

Evaluation Pattern