Stability Analysis of Nonlinear Systems

Course Aim

Extend your understanding of the world from linear to nonlinear and unstable systems

Course Description

Though well instructed in the central themes of physics – quantum mechanics, thermodynamics, solid state etc. – I have come to realize that students only incidentally meet examples of instabilities, and of these the most atypical, phase changes, are usually the most emphasized. A serious discussion of non-linear response is almost entirely neglected. Yet the world is a very nonlinear place, and research in nearly all branches of science and engineering now contends with instabilities as well as non-linear responses.

This course is primarily concerned with the response of systems in equilibrium to perturbing forces, and the general theory underlying their behavior. When a system is in equilibrium it can remain motionless indefinitely, until it is disturbed. Then it may sink back to its original state, or vibrate about the position of rest, or fall over. Also, if the conditions governing the system are slowly changed, the system will adjust itself to the alteration in a smooth fashion, except at critical points, where a tiny change of conditions may lead to a major alteration, as when a drip of water suddenly detaches itself from a tap. Important modern concepts to which serious attention is given include linear response functions, bifurcation and chaos in the response of driven nonlinear systems, elementary catastrophe theory, and phase changes, especially at critical points and lambda-transitions.

Target students are Graduate Students in Science and Engineering

Course Contents

Week 1 Introduction; Harmonic Oscillator
Week 2 Generalized Linear Systems and stability criteria
Week 3 Response of linear systems
Week 4 Periodically driven non-linear systems
Week 5 Elementary types of Catastrophe
Week 6 Phase Transitions
Week 7 Broken Symmetry
Week 8 Hydrodynamic Stability: Introduction + Examples
Week 9 Squire Theorem & Orr-Sommerfeld Squire Equation
Week 10 Critical layers & Boundary layer stability
Week 11 Non-modal Stability Analysis
Week 12 Nonlinear Analysis

The final 25% of the course is on hydrodynamic stability, as requested by a few students. This final 25% part of the course will be co-taught together with Dr. Prabal Negi from Touber Unit.


Weekly written and reading assignments

Prerequisites or Prior Knowledge

Graduate classical/analytical mechanics


Alternate years course, AY2024

Research Specialties