Nonlinear System Analysis

Week 1 : Why nonlinear systems? - Non-linear Models of Physical SystemsWeek 2 : Mathematical Preliminaries: Finite dimensional normed spaces, Euclidean space and its topologyWeek 3 : Infinite dimensional Banach spaces - Contraction mapping theoremWeek 4 : Existence and Uniqueness results for solutions to non linear ODEsWeek 5 : ODEs as vector fields - One dimensional systems - Phase portrait of second order linear systems - Equilibrium points, linearization and their classificationWeek 6 : Examples: Simple pendulum, Bead on a hoop, Lotka-Volterra models for predation and competition, biological transcriptional system, van der Pol oscillator and conservative systems, non linear circuits - Limit cyclesWeek 7 : Bifurcations of two dimensional flows: Saddle-node, pitchfork, transcritical and Hopf - their normal formsWeek 8 : Notions of stability - Lyapunov and LaSalle’s theoremsWeek 9 : Finding Lyapunov functions: Linear systems, variable gradient method - Center Manifold TheoremWeek 10 : Physical Non-linearities - Interconnections and feedback - Aizermann’s conjecture - PassivityWeek 11 : PR systems - Dissipation equality - Passive filtersWeek 12 : KYP Lemma - Popov and circle criterion