Nonlinearity
A nonlinear system is one that cannot be described by a system of linear equations.The solutions of linear equations can in general be described as a superposition of other solutions of the same equation. This makes linear equations particularly easy to solve and reason about.
Nonlinear systems are more complex, and much harder to understand because of their lack of simple superposed solutions. In nonlinear systems the solutions to the equations do not form a vector space and cannot be superposed (added together) to produce new solutions. This makes solving the equations much harder than in linear systems.
The necessary mathematical techniques only started to be developed in the 20th century.
Examples of nonlinear systems:
- general relativity
- the Navier-Stokes equations of fluid dynamics
- systems with solitons as solutions
- nonlinear optics
- the Earth's weather system
- balancing a robot unicycle
To do:
- 20th century mathematics
- chaos theory, fractals
- Lyapunov stability and non-linear control systems
- non-linear video editing
- etc.
