By the end of this lesson, you should be able to:
- Define a dynamical system in terms of state and evolution rules.
- Distinguish a trajectory from a phase-space portrait.
- Interpret equilibrium points as states that do not change under the dynamics.
The idea of a state
A dynamical system begins with a state: the minimum collection of variables needed to describe the system at one instant. A rule then specifies how that state changes with time or iteration.
For a continuous system, we often write \(\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})\). For a discrete map, we write \(\mathbf{x}_{n+1}=\mathbf{F}(\mathbf{x}_n)\).
Trajectories and phase space
As the state evolves, it traces a trajectory through phase space. The geometry of this trajectory can reveal equilibria, cycles, attractors, and transitions that are difficult to see from a single time series.
Fixed points
A fixed point is a state that remains unchanged under the evolution rule. In a map, \(x^*=f(x^*)\). In a flow, \(\mathbf{F}(\mathbf{x}^*)=0\).
What should stay with you
- A dynamical system is a rule for how state changes with time or iteration.
- Trajectories are paths through state space.
- Geometry in phase space often reveals behavior more clearly than the raw time series.
What does a point in phase space represent?
A phase-space point contains the state variables needed to specify the system at that instant.