Foundations
The core vocabulary of dynamical systems — states, evolution rules, and the geometry of how systems change over time.
Foundations
A dynamical system is anything whose state evolves in time according to a fixed rule. That deceptively simple idea — a state plus a law of motion — underlies planetary orbits, swinging pendulums, electrical circuits, ecosystems, and the weather. The foundations collected here are the shared grammar you need before tackling Oscillations, Chaos, or Complex Systems: they tell you how to describe where a system is, how to predict where it goes next, and how to read the qualitative shape of its long-term behavior.
The thread running through every page is geometry. Rather than chase exact solutions, we picture the set of all possible states as a State Space and watch trajectories sweep through it. From that vantage point the important questions become visual: Where does motion come to rest at a Fixed Point? Is that rest stable or precarious? Does the system settle onto an Attractor, and how violently does it react to a nudge — a question answered by the Lyapunov Exponent? And how does the whole picture reorganize when you turn a knob, the phenomenon of Bifurcation?
Start here
- Dynamical System — state plus evolution rule; continuous vs. discrete time.
- State Space — the arena of all possible states and the trajectories within it.
- Phase Portrait — reading a system's behavior from the picture of its flow.
- Fixed Point — equilibria and how to classify them.
- Stability — whether small disturbances grow or decay.
- Bifurcation — qualitative change as a parameter varies.
- Attractor — the sets that long-term motion settles onto.
- Lyapunov Exponent — the rate at which nearby trajectories separate.
- Flows and Maps — continuous flows versus iterated maps.