This is a course webpage for Nonlinear Systems offered in Sem II, 2022-2023.
Abhilash Patel, apatel[at]iitk.ac.in
Kushal Pratap Singh (kushalp20[at]iitk.ac.in), Arijit Bhowmik (arijitb21[at]iitk.ac.in)
MWF, 9.00AM-10AM, at ACES 213
| Lecture | Topics | Suggested Readings (/ implies OR) |
|---|---|---|
| 1 | Course Mechanics | FCH |
| 2 | Nonlinearity and induced dynamical behaviours | ShS 1.1/ HK 1.1-1.2.1 |
| 3 | Tools from Matrix Theory, Existence | ShS 3.2,3.4/StS 2.5 |
| 4 | Uniqueness and Existence, Parameteric and Initial Condition Dependence | HK 3.1, 3.3/ ShS 3.4,1 |
| 5 | Overview of Control-theoretic tools, Describing Function | HK 3.1, 3.3/ ShS 3.4,1 |
| 6 | Describing Function, Harmonic Balance | HK 7.2/ JW 5.1-5.3 |
| 7 | Common nonlinearity, Prediction and stability of limit cycles, Error bound in approximation | HK 7.2/ JW 5.3-5.4 | 8 | Talyor-series, Linearization, Solution of Linearized System, Role of Eigenvectors and Eigenvalues | HK 2.1/ ShS 2.2 / StS 6.3 | 9 | Hartman-Grobman Theorem, Phase Plane Analysis, Nodes, Manifold, Invariant Manifold, Stable-Unstable Manifold | HK 2.3/ ShS 2.2 / StS 5.2 | 10 | Phase Plane: Saddle, Center, Focus, Extending to Nonlinear Systems, Heteroclinic and Homoclinic Orbit | HK 2.3/ ShS 2.2 / StS 5.2 | 11 | Isoclines, Nullclines | StS 6.1.1/See Additional References | 12 | Center Manifold Theory | ShS 7.6/HK 8.1 | 13 | Bifurcations: Saddle Node, Transcritical | ShS 2.5/HK 2.7 | 14 | Bifurcations: Pitchfork, Hopf | ShS 2.5/HK 2.7 | 15 | Mapping Limit cycles to fixed points, Global Bifurcations: Fold, Sniper, Homoclinic | Sts 8.4 | 16 | Existence of limit cycles, Bendixson theorem, Poincare-Bendixson theorem | JW2.6/ HK 2.6 | 17-19 | Mid Semester Project Progress | 20 | Notions of stability & convergence | JW3.1-3.2/ HK 4.1 | 21 | Stability of Equilibrium, Lyapunov stability, Indirect & Direct approach for autonomous systems | JW3.1-3.4/ HK 4.1 | 22 | LaSalle's Invariance Theorem | JW 3.4.3/ HK 4.2 | 23 | Finding Lyapunov functions: Quadratic approach, Krasovskii theorem | JW 3.5 | 24 | Finding Lyapunov functions: Energy approach, Variable Gradient, Lyapunov's direct approach for non-autonomous systems | JW 3.5, 4.1-4.2/ HK 4.5 | 25 | Lyapunov's direct approach for non-autonomous systems, Barbalat's Lemma | JW 4.2/ HK 4.5 | 26 | Stability of Limit Cycles, Orbital Stability, Analysis: Transformation approach, Poincare Map | HK 8.4 | 27 | Floquet Theory | See Additional References | 28 | Lyapunov Approach for Limit Cycles | See Additional References | 29 | Incremental Stability, Convergent Dynamics | See Additional References | 30 | Contraction Theory, Partial Contraction, Observer Design | See Additional References | 31 | Input to State Stability, Input to Output Stability | See Additional References | 32-36 | End Semester Project Presentation |