Obsidian/300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md

title	allDay	startTime	endTime	date	completed	type
Frameworks and Review	false	12:30	14:30	2024-09-09	null	single

Introduction

Where do nonlinearities come from? Well, a couple of places...

1. Geometric nonlinearities (pendulum)
2. External fields
3. Material properties So we're stuck with them. But how do we deal with noninearities?

A nonlinear equation

\dot{x} = \frac{dx}{dt} = 1-2\cos x

How do you solve this? You can't use Laplace, you can't separate... insert very long expression that Bajaj wrote. Getting an analytical solution can be a PITA to obtain. For this reason: The general case is that nonlinear equations are unsolvable. This doesn't mean we can't learn things. We can describe these systems qualitatively.

Really our options come down to:

• Solve exactly (Not likely to happen)
• Solve numerically
• Analyze qualitatively (~geometrically)
• Solve an approximation to the problem We mix and match these approaches.