Dynamics MOC

Fixed point

A fixed point , also called an equilibrium solution, of a dynamical system is a phase point which remains constant for all time, i.e. a system in this state will remain this way. #m/def/dynamics

• For a differential system this means ; whereas
• For an iterated map this means .

A fixed point corresponds precisely to an Invariant set of cardinality one.

Classification

Stability

A fixed point may be

• stable, meaning small perturbations decay back to the point
  • Lyapunov stable if for every neighbourhood of there exists a neighbourhood of such that for all and .
  • Asymptotically stable if it Lyapunov stable of in addition to being Lyapunov stable, there exists a neighbourhood of such that for all .
• unstable, meaning small perturbations increase away from the point
  • Lyapunov unstable if for every neighbourhood of there exists a neighbourhood of such that for all and .
  • Asymptotically unstable if it Lyapunov stable of in addition to being Lyapunov stable, there exists a neighbourhood of such that for all .
• A centre.

These can be identified qualitatively in a Phase portrait. Quantitatively it is described by

• Linear stability analysis
• Lyapunov stability criterion

Contraction

The dynamics at a fixed point is said to be contracting iff an infinitesimal ball of initial conditions around a fixed point shrink over time. This can occur for both unstable and half-stable points. Using Linear stability analysis, contracting dynamics corresponds to a negative Jacobian trace in the continuous case and a Jacobian determinant less than one in the discrete case. A system that is contracting at all points (fixed or not) is called a Dissipative system.

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