Syllabus

Introduction (Module-I, 6 Hours): Phase space, deterministic versus stochastic modeling, finite vs infinite-dimensional models, linear vs non-linear, autonomous vs non-autonomous systems. geometric approach to dynamical systems, fixed points, linearization, and stability

Dynamical systems (Module-II, 6 Hours): Continuous vs discrete-time systems, conservative vs dissipative systems, Existence, uniqueness, and smooth dependence of solutions of ODE's on initial conditions and parameters.

Bifurcations in one and two-dimensional systems (Module-III, 6 Hours): Local vs global bifurcations, Implicit function theorem, classification of bifurcations, Some generalities: center manifold and normal form, symmetry and symmetry breaking, relation to catastrophes and sudden transitions.

Nonlinear systems Analysis (Module-IV, 10 Hours): Stable and unstable manifolds, conservative systems, reversible systems, Solution of (fully nonlinear) damped pendulum equation, Limit cycles, relaxation oscillations, weakly nonlinear oscillators, averaging method, and two time-scales, Hopf bifurcation and oscillating power electrics systems, quasiperiodicity, coupled oscillators systems, nonlinear resonance and frequency locking.

Chaos and Fractal Analysis (Module-V, 6 Hours): Introduction, fixed points and cobweb diagram, Numerics and analysis of the logistic map, periodic window, Lyapunov exponent, strange attractors, and example. cantor sets, probabilistic constructions of fractals, fractals from deterministic systems, fractal basin boundaries, fractal dimension, Correlation Dimension

To introduce and describe nonlinear phenomena in physical and engineering systems

To comprehend the basic traits of chaotic systems and their modeling techniques

To develop the ability to analyze nonlinear systems using phase-plane diagrams, stable and unstable manifolds, and bifurcation theory

To explore the practical applications of chaotic and limit cycle oscillations in real-life electrical systems for control or synchronization

1. Gain basic knowledge of nonlinear differential equations and iterative maps.
2. Understand the properties of various strange attractors in both continuous and discrete-time systems.
3. Develop analysis skills for different nonlinear systems design.
4. Explore various applications of chaotic attractors in real-life systems.
5. Examine practical uses and limitations of nonlinear models to analyze power electronics systems in industrial applications.

S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry And Engineering”, Perscus Book Publishing Group

Kathleen T. Alligood,? Tim D. Sauer and,? James A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer

H. B. Stewart, J. M. T. Thompson, Nonlinear Dynamics and Chaos, Wiley and Sons, NY, USA

Robert C. Hilborn, Chaos and Nonlinear Dynamics, Oxford University Press