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Indian Institute of Science Education and Research Bhopal
PHY 411: Nonlinear Dynamics and Chaos (4)
Prerequisites: PHY 305: Classical Mechanics,
PHY 301: Mathematical Methods I
Learning Objectives:
This course introduces fundamental concepts of dynamical systems, dynamical flows, non-linearity and chaos.
Course Contents:
Introduction to Dynamical Systems: Overview, Examples and Discussion
One-dimensional flows: Flows on the line, Fixed points and stability, Population growth, Linear stability analysis, Saddle-node, Transcritical and Pitchfork bifurcations, Flow on the circle
Two-dimensional flows: Linear system: Definitions and examples, Phase portraits, Fixed points and linearization, Limit cycles, Poincare-Bendixson theorem, Lienard systems, Bifurcations revisited: Saddle-node, Transcritical and Pitchfork bifurcations, Hopf bifurcations, Oscillating chemical reactions, Poincare maps, Global bifurcation of cycles, Coupled Oscillators and Quasiperiodicity
Chaos: Lorenz equations: Properties of Lorenz equation, Lorenz Map; One-dimensioanl map: Fixed points, Logistic map, Liapunov exponent, Fractals: Countable and Uncountable Sets, Cantor Set, Dimension of Self-Similar Fractals, Box dimension, Pointwise and Correlation Dimensions; Strange Attractors: Baker’s map, Henon map Chaos in Hamiltonian systems
Suggested Books:
- Steven H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering
- Edward Ott, Chaos in dynamical systems (Cambridge University Press)
- R. C. Hilborn, Chaos and Nonlinear Dynamics (Cambridge Univ. Press. 1994)
- M. Lakshmanan and S. Rajasekar, Nonlinear dynamics: Integrability Chaos and Patterns (Springer)
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