Definition
Dynamic systems (often used interchangeably with dynamical systems) are mathematical models describing how a system’s state evolves over time according to a fixed rule. These models use either differential equations (for continuous time) or difference equations (for discrete time) to map the current state to a future state.
How to read: The time derivative of state vector x equals a function f of state x, time t, and input vector u. Meaning / when to use: This state-space equation represents a continuous-time dynamic system, used to compute how the system’s internal state changes given its current state and external inputs.
Why It Matters
Everything in the universe, from orbital mechanics and chemical reaction rates to economic cycles and neural networks, is a dynamic system. Understanding them allows us to predict the future behavior of complex phenomena. Without dynamic systems theory, we could only describe static snapshots of reality; with it, we can design control systems (like autopilot software) that actively steer systems toward desired outcomes.
Core Concepts
- State Variables: The minimum set of variables needed to fully describe the system at any given moment.
- Time Evolution: The core characteristic is that the system’s output depends not just on current inputs, but on the system’s history (its current state).
- Linear vs. Non-linear: Linear systems obey the principle of superposition and are analytically solvable. Non-linear systems can exhibit chaos and typically require numerical simulation.
- Equilibrium Points: States where the system experiences no change ().