# ME8230 - Nonlinear Dynamics

**Graduate course on nonlinear dynamical systems: Spring 2016 version**.

Topics include linear stability analysis and classification of equilibria, qualitative dynamics and phase portraits in 1D and 2D, various bifurcations, Lyapunov stability, Lyapunov functions, limit cycles, Floquet theory and Poincare maps, parametric excitation, discrete dynamical systems, chaos and sensitive dependence on initial conditions, periodic forcing of systems with various nonlinearities, perturbation methods and approximate analytical methods (multiple scales, Lindstedt-Poincare, harmonic balance, averaging), numerical methods for finding, analyzing, and tracking equilibria & periodic motions, Hamiltonian systems, non-smooth/hybrid systems, etc.

These classical mathematical topics will be discussed mainly in the context of mechanical systems, but are (of course) applicable to nonlinear systems arising in any context.

In addition to the HW and exams, each student completes a course project, involving analysis of a non-trivial nonlinear dynamical systems, either through techniques developed in the lectures or through other experimental or analytical techniques not considered here.

## Lecture notes

(roughly arranged in lecture order, mostly hand-written and informal) Informal introduction and some nonlinear phenomena. PDF.

A nonlinear spring is easily made from linear springs. PDF.

Global dynamics in one dimension: stability, linear stability, and bifurcations. PDF.

Equivalence of ODEs written in different forms. PDF

Linear stability analysis of equilibria in n-D. PDF.

Dynamics in 2D, classification of fixed points, phase portraits, bifurcations. PDF.

A fast-slow system. First order system as a singular (zero-inertia) limit of a second order ODE. PDF.

Lyapunov stability, asymptotic stability, and exponential stability. PDF.

Establishing stability using Lyapunov functions. PDF.

Stable equilibria as potential energy minima. PDF.

Limit cycles in autonomous systems. An aeroelastic oscillator and other examples. PDF.

Discrete dynamical systems. PDF.

Poincare maps. Limit cycle stability. PDF.

Elementary computational methods. How accurate is your finite difference derivative. PDF.

Stability of limit cycles, linear time-periodic systems, Floquet theory. PDF.

Parametric excitation (Mathieu's equation) + piecewise-linear system that can be stable or unstable. PDF.

Computing sensitivity to parameters or initial conditions. Chaos, Lyapunov exponents, Lorenz system. PDF.

Simulating non-smooth/hybrid systems. PDF.

Nonlinear systems with an external periodic forcing: frequency response, numerical explorations. PDF.

Perturbation methods: Basics, Lindstedt-Poincare method. PDF.

Approximate methods: Method of multiple scales. PDF. (from Richard Rand's lecture notes)

Approximate methods: Averaging. PDF.

Approximate methods: Harmonic balance. PDF.

Continuation techniques for tracking equilibria, periodic motions, and bifurcations. PDF.

Basin of attraction. (see HW5)

Miscellany. PDF. etc. (not exhaustive)

## Homework

HW1, HW2, HW3, HW4, HW5.## MATLAB codes from lecture

Finding fixed points and bifurcations in 1D using fsolve (simple 'tracking'): Code Simple ODE solution using ode45: Code

2D phase portraits: phase trajectories from clicking on a figure window: Code

Numerical computation of Jacobians: Code

Limit cycles: finding them using root find, Poincare maps, and stability: Code

Parametric excitation: Mathieu's equations, tongues of instability: Code

Chaos in the logistic map. Period-doubling route to chaos. Orbit diagram: Code

Double pendulum simulation: Code

Lyapunov exponents. Sensitivity to Initial Conditions. Lorenz system: Code

Non-smooth system: two stiffness oscillator: Code

Forced nonlinear oscillators. Frequency content, multiple steady states (Ueda/Duffing equation): Code

Forced nonlinear oscillators. Frequency response. Duffing equation: Code

Perturbation methods. Code

Simple continuation (homotopy) of fixed points/equilibria.

Basin of attraction.

etc. (more upcoming)