What variables summarize the system now?
Choose the state vector.
A good state contains enough information for the evolution rule to operate.
Side 98
Dynamical
Systems
A study of how system states evolve under fixed rules. Dynamical systems focuses less on solving one equation and more on the geometry of behavior: equilibria, stability, attractors, cycles, bifurcations and chaos.
state→rule→trajectory→stability→regime
06dynamic lenses
05stability questions
05regime shifts
98Side
A dynamical system begins with a state sufficient to determine what happens next.
The state may contain positions and velocities, population levels, concentrations, beliefs or any variables needed by the update rule.
How does state update?
Continuous or discrete?
Differential equations and difference equations are two common forms.
Where does the state go?
Path through state space.
Each initial condition generates a trajectory through the set of possible states.
Which states are reachable?
Invariant sets.
Constraints and conserved quantities can restrict motion to particular regions.
What long-run behavior appears?
Rest, cycle, divergence, chaos?
Qualitative regimes often matter more than exact formulas.
Equilibria are states that reproduce themselves under the dynamics.
Finding them is only the beginning; the key question is what happens nearby.
No net change.
The update rule returns the same state.
Several resting states coexist.
Initial conditions can determine which one is approached.
One component stops changing.
Intersecting nullclines often reveal equilibria in two-dimensional systems.
Some quantities remain fixed.
Conserved structure restricts possible equilibrium movement.
Some systems move down an energy-like landscape.
Minima can correspond to stable equilibria.
Equilibrium can be structural, not optimal.
A system may settle because motion balances, not because the outcome is desirable.
Stability asks what small disturbances do.
An equilibrium can exist yet be practically irrelevant if almost any perturbation drives the system away.
Stable
Nearby trajectories remain nearby.
Small disturbances do not produce large departures.
Asymptotic
Nearby trajectories return.
The equilibrium actively attracts nearby states over time.
Unstable
Small deviations grow.
Nearby trajectories move away from the equilibrium.
Linearization
Local derivatives approximate local behavior.
Eigenvalues often determine stability near an equilibrium.
Basin
Which initial states converge here?
The basin of attraction defines the region captured by a stable attractor.
Long-run behavior can organize around more than fixed points.
Attractors describe sets toward which nearby trajectories evolve.
| Attractor | Behavior | Signature |
|---|---|---|
| Fixed point | Settles to one state | Rest |
| Limit cycle | Repeats a closed orbit | Persistent oscillation |
| Torus / quasiperiodic | Several incommensurate cycles | Nonrepeating regular motion |
| Strange attractor | Aperiodic bounded motion | Chaotic structure |
| Transient | Long-lived but not final pattern | Apparent regime before escape |
A bifurcation is a qualitative change in dynamics caused by parameter change.
The same system can pass from one equilibrium to two, from rest to oscillation, or from stable motion to chaos.
Two equilibria appear or disappear.
A stable and unstable state can collide as a parameter crosses a threshold.
Symmetry breaks.
One state can lose stability while two alternatives emerge.
Oscillation appears.
A stable equilibrium can give way to a limit cycle.
Cycles repeatedly split.
A route to chaos can emerge through successive doubling of periodic behavior.
Return path differs from departure path.
Once a regime changes, reversing the parameter may not immediately restore the prior state.
Gradual pressure creates abrupt change.
Bifurcation structure gives one rigorous meaning to a tipping point.
Deterministic rules can generate behavior that is practically unpredictable.
Chaos is not randomness; it is structured sensitivity in nonlinear dynamics.
Determinism
The same exact initial state follows the same exact rule.
Sensitivity
Tiny differences in initial conditions can grow exponentially.
Boundedness
Chaotic trajectories can remain confined to a finite region.
Aperiodicity
The trajectory does not settle into exact repetition.
Prediction horizon
Finite measurement precision creates a practical limit to long-range forecasting.
Nonlinear Dynamics and ChaosSteven Strogatz · geometry and intuition
Differential Equations and Dynamical Systemsequilibria and stability
Chaossensitivity and strange attractors
Bifurcation Theoryqualitative regime change