##  [Phase Portrait](/phase-portrait-0) 

 Definition

A geometric diagram representing the set of trajectories of a dynamical system in its state space, showing equilibrium points, limit sets, and typical orbit behavior without explicit time parametrization.

 

 

 

 

 

 





## Principle

Principle

Trajectories are integral curves of the system's vector field; their configuration and invariant sets reveal qualitative dynamics and stability properties.

 

 

 

 

 





## Demonstration

Demonstration

Plotting solution curves of a two-dimensional autonomous system x' = Ax with A having eigenvalues of opposite sign produces a saddle phase portrait with stable and unstable manifolds crossing at the equilibrium.

 

 

 

 

## Misapplication

Misapplication

Inferring global system properties from a single numerically computed trajectory or from a phase portrait of a reduced projection can mislead if invariant manifolds or higher-dimensional structure are omitted.

 

 

 

 

 





## Consequence

Consequence

Enables classification of long-term behaviour (attractors, repellers, basins) and guides qualitative analysis without solving trajectories analytically.

 

 

 

 

## Reversal

Reversal

A time-series plot emphasizes variable values versus time with explicit parametrization; reversing to that view recovers temporal information lost in the purely geometric portrait.

 

 

 

 

 





## Boundary

Boundary

Applies to deterministic finite-dimensional continuous- or discrete-time autonomous systems where trajectories are well-defined; excludes stochastic sample paths and systems whose relevant dynamics are infinite-dimensional or nonautonomous unless an autonomous embedding is used.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Closely related to but distinct from return maps or bifurcation diagrams, which emphasize discrete crossings or parameter dependence rather than the continuous geometry of orbits.

 

 

 

 

 





## Synthesis

Synthesis

A visualization capturing the geometry of a dynamical system's orbits and invariant structures in state space, used to infer qualitative behaviour and stability independent of explicit time parametrization.