What is it?
A state and a rule that moves it forward: continuous () or discrete (). The questions change from "find the formula" to "what happens in the long run?" — and every iterative algorithm, from gradient descent to a recurrent network, is a discrete dynamical system.
Formulas
Where it shows up in computing
Control is the art of shaping the dynamics of a system by feedback.
The atmosphere is a huge dynamical system; forecasting is integrating it forward.
Where it shows up in AI
Recurrent networks are discrete dynamical systems ; exploding and vanishing gradients are questions of stability.
Where is it used?
Computing topics reachable from here, through the chain of ideas that leads to them:
⚛ Physics and simulation
- Weather and climate modelling★★★★★
- Equilibria and stability→Stiffness and implicit methods→Physics engines★★★★★
- Equilibria and stability→Stiffness and implicit methods→Physics engines→N-body gravitational simulation★★★★★
- Equilibria and stability→Stiffness and implicit methods→Physics engines→Fluid dynamics and CFD★★★★★
- Equilibria and stability→Bifurcations→Population and epidemic models★★★★★
ℒ AI and machine learning
- Equilibria and stability→Gradient descent★★★★★
- Equilibria and stability→Gradient descent→Learning rate★★★★★
- Phase space→Attractors→Neural networks★★★★★
- Equilibria and stability→Gradient descent→Backpropagation★★★★★
- Equilibria and stability→Gradient descent→Stochastic gradient descent (SGD)★★★★★
- Equilibria and stability→Gradient descent→Loss landscape★★★★★
- +7
What depends on it
This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.