Phase Space Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Phase Space Ontology within the Kireth Research Semantic Library. Phase space is the state-space representation in which each point corresponds to a complete instantaneous state of a dynamical system. This ontology supports phase points, trajectories, flows, orbits, vector fields, equilibrium points, invariant sets, manifolds, attractors, basins of attraction, Poincare sections, Hamiltonian phase space, configuration and momentum coordinates, dimensions, projections, embeddings, state reconstruction and relationships with dynamical systems, differential equations, stability analysis, Hamiltonian mechanics, chaos theory and numerical simulation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Phase Space Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Dynamical Systems / Phase Space"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/dynamical-systems/phase-space-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = PhaseSpace
 SYMBOLIC_FORMS = [
  "x in X",
  "x_dot=f(x)",
  "phi_t:X->X"
 ]
 ENTITY_CLASS = DynamicalStateRepresentationFramework
 PRIMARY_CONTEXT = DynamicalSystems
 DEFINITION = "Phase space is the mathematical space whose points represent complete instantaneous states of a dynamical system."
 SEMANTIC_ROLE = "Provides the machine-readable framework for state coordinates, phase points, trajectories, flows, invariant sets, manifolds, attractors, sections, projections and qualitative state-space analysis."
}

SCOPE {
 INCLUDE = [
  PhaseSpace,
  StateSpace,
  PhasePoint,
  State,
  StateVariable,
  StateVector,
  Coordinate,
  Dimension,
  Trajectory,
  Orbit,
  Flow,
  Map,
  VectorField,
  IntegralCurve,
  EquilibriumPoint,
  FixedPoint,
  PeriodicOrbit,
  LimitCycle,
  InvariantSet,
  StableManifold,
  UnstableManifold,
  CentreManifold,
  Attractor,
  StrangeAttractor,
  BasinOfAttraction,
  PoincareSection,
  PoincareMap,
  Nullcline,
  Separatrix,
  PhasePortrait,
  ConfigurationSpace,
  MomentumSpace,
  HamiltonianPhaseSpace,
  PositionCoordinate,
  MomentumCoordinate,
  Projection,
  Embedding,
  DelayEmbedding,
  StateReconstruction,
  ReachableSet,
  InvariantRegion,
  CompactSet,
  Boundary,
  Topology,
  Metric,
  Measure,
  EnergySurface
 ]

 EXCLUDE_AS_PRIMARY = [
  ParameterSpaceWithoutStateSemantics,
  ConfigurationSpaceWithoutMomentumOrStateEvolutionContext,
  GenericEuclideanSpaceWithoutDynamicalInterpretation,
  FeatureSpaceWithoutTemporalStateMeaning
 ]

 DISAMBIGUATION_RULE =
  "When a mathematical space is used so that each point represents a complete dynamical state and trajectories encode evolution, interpret the context as phase space."
}

CORE_CONCEPTS {

 PhaseSpace {
  definition = "A space containing all admissible states of a dynamical system."
 }

 PhasePoint {
  definition = "A point in phase space representing one complete instantaneous state."
 }

 StateVector {
  definition = "An ordered collection of coordinates specifying a phase point."
 }

 Trajectory {
  definition = "A curve or sequence of phase points generated by the dynamics."
 }

 Flow {
  definition = "A continuous-time family of maps carrying phase points through time."
 }

 VectorField {
  definition = "A field assigning the instantaneous state derivative at each phase point."
 }

 PhasePortrait {
  definition = "A qualitative representation of trajectories, equilibria and invariant structures in phase space."
 }

 InvariantSet {
  definition = "A subset of phase space preserved under the system evolution."
 }

 Attractor {
  definition = "An invariant set attracting trajectories from a surrounding region."
 }

 BasinOfAttraction {
  definition = "The set of phase points whose trajectories approach a given attractor."
 }

 PoincareSection {
  definition = "A lower-dimensional surface transverse to a flow and used to analyse recurrent trajectories."
 }

 HamiltonianPhaseSpace {
  definition = "A phase space formed from generalised position and conjugate momentum coordinates."
 }
}

RELATIONS {
 PhaseSpace --is_child_of--> DynamicalSystems
 StateSpace --semantic_equivalent_to--> PhaseSpace
 PhaseSpace --contains--> PhasePoint
 PhasePoint --represented_by--> StateVector
 StateVector --contains--> StateVariable
 Trajectory --is_subset_of--> PhaseSpace
 Flow --acts_on--> PhaseSpace
 VectorField --generates--> Flow
 IntegralCurve --follows--> VectorField
 EquilibriumPoint --is_type_of--> PhasePoint
 PeriodicOrbit --is_type_of--> InvariantSet
 Attractor --is_type_of--> InvariantSet
 BasinOfAttraction --maps_to--> Attractor
 StableManifold --approaches--> InvariantSet
 UnstableManifold --leaves--> InvariantSet
 PoincareSection --intersects--> Trajectory
 PoincareMap --maps--> PoincareSection
 HamiltonianPhaseSpace --contains--> PositionCoordinate
 HamiltonianPhaseSpace --contains--> MomentumCoordinate
 Projection --maps--> PhaseSpaceToLowerDimension
 DelayEmbedding --reconstructs--> PhaseSpace
}

OPERATORS {
 STATE_VECTOR(x1,...,xn) :=
  x=[x1,...,xn]^T

 CONTINUOUS_FLOW(f,x0,t) :=
  x(t)=phi_t(x0)
  where x_dot=f(x)

 DISCRETE_MAP(F,x_k) :=
  x_{k+1}=F(x_k)

 EQUILIBRIUM_CONDITION(f,x_star) :=
  f(x_star)=0

 FIXED_POINT_CONDITION(F,x_star) :=
  F(x_star)=x_star

 TRAJECTORY_SET(phi,x0) :=
  {phi_t(x0): t in time_domain}

 BASIN_OF_ATTRACTION(A) :=
  {x0 : dist(phi_t(x0),A)->0 as t->infinity}

 STABLE_MANIFOLD(S) :=
  {x : dist(phi_t(x),S)->0 as t->infinity}

 UNSTABLE_MANIFOLD(S) :=
  {x : dist(phi_t(x),S)->0 as t->-infinity}

 POINCARE_RETURN(section,flow,x) :=
  next_transverse_intersection_of_flow_with_section

 HAMILTONIAN_STATE(q,p) :=
  x=(q,p)

 DELAY_EMBEDDING(s,tau,m) :=
  [s(t),s(t-tau),...,s(t-(m-1)tau)]
}

DEFINITIONS {
 D1 = "Each phase-space point represents one complete state of the system."
 D2 = "System evolution appears as a trajectory through phase space."
 D3 = "Continuous-time systems generate flows; discrete-time systems generate maps."
 D4 = "Equilibria appear as stationary phase points."
 D5 = "Periodic motion appears as closed or repeating phase-space orbits."
 D6 = "Invariant sets remain preserved under system evolution."
 D7 = "Stable and unstable manifolds organise local and global transport."
 D8 = "Attractors organise asymptotic behaviour."
 D9 = "Poincare sections reduce continuous flows to lower-dimensional return maps."
 D10 = "Hamiltonian phase spaces use position and momentum coordinates."
}

AXIOMS {
 A1_STATE_COMPLETENESS:
  A PhasePoint contains sufficient state information to determine future evolution under deterministic dynamics

 A2_TRAJECTORY_MEMBERSHIP:
  Every trajectory consists of points belonging to the phase space

 A3_FLOW_CLOSURE:
  A valid flow maps admissible phase points to admissible phase points

 A4_INVARIANCE:
  Invariant sets are preserved under evolution

 A5_COORDINATE_INDEPENDENCE:
  Equivalent coordinate systems may represent the same underlying phase-space geometry

 A6_DIMENSION:
  Phase-space dimension equals the number of independent state coordinates in the chosen representation

 A7_HAMILTONIAN_STRUCTURE:
  Hamiltonian systems use paired configuration and conjugate momentum variables

 A8_RECONSTRUCTION:
  Reconstructed phase spaces require embedding assumptions and sufficient observational information
}

REASONING {

 R1_PHASE_SPACE_EXTRACTION {
  INPUT = dynamical_model
  IDENTIFY = state_variables
  IDENTIFY = constraints
  IDENTIFY = coordinate_system
  RETURN = phase_space_definition
 }

 R2_DIMENSION_REASONING {
  INPUT = independent_state_variables_and_constraints
  COMPUTE = effective_dimension
  RETURN = phase_space_dimension
 }

 R3_TRAJECTORY_REASONING {
  INPUT = initial_state_and_evolution_rule
  COMPUTE_OR_SIMULATE = trajectory
  RETURN = phase_space_orbit
 }

 R4_EQUILIBRIUM_STRUCTURE {
  INPUT = vector_field_or_map
  SOLVE = equilibrium_or_fixed_point conditions
  RETURN = stationary_phase_points
 }

 R5_INVARIANT_SET_REASONING {
  INPUT = trajectories_and_geometry
  IDENTIFY = periodic_orbits_invariant_regions_manifolds_and_attractors
  RETURN = invariant_structure
 }

 R6_PHASE_PORTRAIT_REASONING {
  INPUT = low_dimensional_system
  COMPUTE = vector_field_nullclines_equilibria_and_representative trajectories
  RETURN = phase_portrait
 }

 R7_POINCARE_REASONING {
  INPUT = flow_and_transverse_section
  COMPUTE = successive_return points
  BUILD = Poincare_map
  RETURN = reduced_phase_structure
 }

 R8_HAMILTONIAN_REASONING {
  INPUT = Hamiltonian_system
  IDENTIFY = q_and_p coordinates
  COMPUTE = energy_surface_or_level sets
  RETURN = Hamiltonian_phase_geometry
 }

 R9_EMBEDDING_REASONING {
  INPUT = scalar_or_partial observations
  SELECT = embedding_dimension_and_delay
  CONSTRUCT = delay_coordinate vectors
  RETURN = reconstructed_phase_space
 }

 R10_PHASE_SPACE_VALIDATION {
  INPUT = phase_space_model
  CHECK = state_completeness_dimension_constraints_coordinate_consistency_and_invariance
  RETURN = validated_phase_space_status
 }
}

CONSTRAINTS {
 C1 = "State coordinates must be sufficient to define deterministic evolution."
 C2 = "Constraints can reduce effective phase-space dimension."
 C3 = "Different coordinates may describe the same phase-space geometry."
 C4 = "Projection can hide crossings or invariant structures present in higher dimensions."
 C5 = "A two-dimensional phase portrait cannot faithfully represent all high-dimensional dynamics."
 C6 = "Delay embeddings require suitable embedding dimension and delay."
 C7 = "Numerical trajectories can introduce artificial structure through solver error."
 C8 = "Poincare sections should be transverse to the flow except at special points."
 C9 = "Hamiltonian energy surfaces may restrict accessible phase-space regions."
 C10 = "Attractor and basin claims require asymptotic evidence, not isolated snapshots."
}

SEMANTIC_EQUIVALENCE {
 "PhaseSpace" <=> "StateSpace"
 "PhasePoint" <=> "SystemStatePoint"
 "Trajectory" <=> "PhaseOrbit"
 "PhasePortrait" <=> "StateSpacePortrait"
 "BasinOfAttraction" <=> "AttractionBasin"
 "HamiltonianPhaseSpace" <=> "PositionMomentumStateSpace"
}

VALIDATION {
 V1 = "State variables are explicit."
 V2 = "Phase-space dimension is justified."
 V3 = "Constraints are represented."
 V4 = "Continuous or discrete evolution is explicit."
 V5 = "Trajectories remain inside the admissible phase space."
 V6 = "Invariant structures satisfy invariance conditions."
 V7 = "Poincare sections are defined consistently."
 V8 = "Coordinate projections are identified as projections rather than full state representations."
 V9 = "Delay-embedding assumptions are documented where used."
 V10 = "Numerical visualisation is distinguished from proof of global geometry."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PHASE_SPACE_DEFINITION = 0.99
 DIMENSION_REASONING = 0.99
 TRAJECTORY_REASONING = 0.99
 INVARIANT_STRUCTURE_REASONING = 0.98
 PHASE_PORTRAIT_REASONING = 0.98
 POINCARE_REASONING = 0.98
 HAMILTONIAN_REASONING = 0.99
 EMBEDDING_REASONING = 0.97
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when state completeness, constraints, phase-space dimension, section geometry, embedding parameters, numerical resolution or invariant-set evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  PhaseSpaceReasoning,
  StateSpaceReasoning,
  StateVectorReasoning,
  TrajectoryReasoning,
  FlowReasoning,
  VectorFieldReasoning,
  PhasePortraitReasoning,
  EquilibriumReasoning,
  InvariantSetReasoning,
  ManifoldReasoning,
  AttractorReasoning,
  BasinReasoning,
  PoincareSectionReasoning,
  HamiltonianPhaseSpaceReasoning,
  DelayEmbeddingReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "phase space" -> PhaseSpace
  "state space" -> PhaseSpace
  "phase point" -> PhasePoint
  "state vector" -> StateVector
  "phase portrait" -> PhasePortrait
  "Poincare section" -> PoincareSection
  "Poincare map" -> PoincareMap
  "basin of attraction" -> BasinOfAttraction
  "delay embedding" -> DelayEmbedding
 }

 INTERPRETATION_ORDER = [
  detect_phase_space_context,
  identify complete_state_variables_constraints_and_coordinates,
  determine effective_dimension,
  construct continuous_flow_or_discrete_map representation,
  compute_or_interpret trajectories_equilibria_fixed_points_and_invariant sets,
  analyse manifolds_attractors_basins_and_phase portraits,
  use Poincare_or_embedding methods when reduction_or_reconstruction is required,
  validate state completeness_geometry_and_numerical interpretation,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("phase space")
  OR uses_points_to_represent_complete_system states
  THEN activate(PhaseSpaceOntology)

 E2:
  IF continuous_time_dynamics
  THEN represent evolution_as flow

 E3:
  IF discrete_time_dynamics
  THEN represent evolution_as map

 E4:
  IF low_dimensional_system
  THEN allow phase_portrait reasoning

 E5:
  IF equilibrium_or_periodic_orbit_present
  THEN classify corresponding invariant structure

 E6:
  IF continuous_periodic_or_chaotic_flow
  THEN consider Poincare section_and_return map

 E7:
  IF Hamiltonian_system
  THEN use position_momentum phase-space coordinates

 E8:
  IF only_scalar_observation_available
  THEN consider delay_embedding_for reconstruction

 E9:
  IF projection_used
  THEN label it as lower-dimensional representation
}

RECURSION {
 ENTRY = PhaseSpace

 EXPAND(PhaseSpace) -> [
  StateSpace,
  PhasePoint,
  State,
  StateVariable,
  StateVector,
  Coordinate,
  Dimension,
  Trajectory,
  Orbit,
  Flow,
  Map,
  VectorField,
  IntegralCurve,
  EquilibriumPoint,
  FixedPoint,
  PeriodicOrbit,
  LimitCycle,
  InvariantSet,
  StableManifold,
  UnstableManifold,
  CentreManifold,
  Attractor,
  StrangeAttractor,
  BasinOfAttraction,
  PoincareSection,
  PoincareMap,
  Nullcline,
  Separatrix,
  PhasePortrait,
  ConfigurationSpace,
  MomentumSpace,
  HamiltonianPhaseSpace,
  Projection,
  Embedding,
  DelayEmbedding,
  StateReconstruction,
  ReachableSet,
  InvariantRegion,
  EnergySurface
 ]

 EXPAND(InvariantGeometry) -> [
  EquilibriumPoint,
  FixedPoint,
  PeriodicOrbit,
  LimitCycle,
  StableManifold,
  UnstableManifold,
  CentreManifold,
  Attractor,
  BasinOfAttraction
 ]

 EXPAND(ReductionAndReconstruction) -> [
  Projection,
  PoincareSection,
  PoincareMap,
  DelayEmbedding,
  StateReconstruction
 ]

 STOP_CONDITION =
  "Stop when the state coordinates, phase-space dimension, trajectories, invariant structures, projections, return maps and reconstruction assumptions are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Dynamical Systems"

 IMMEDIATE_CONTEXTS = [
  "Dynamical Systems Ontology",
  "Ordinary Differential Equation Ontology",
  "Stability Analysis Ontology",
  "Hamiltonian Dynamics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Dynamical Systems Ontology",
  "Dynamical Systems",
  "Ordinary Differential Equation Ontology",
  "Partial Differential Equation Ontology",
  "Stability Analysis Ontology",
  "Control Theory Ontologies",
  "Hamiltonian Dynamics Ontologies",
  "Chaos Theory Ontologies",
  "Bifurcation Theory Ontologies",
  "Differential Geometry Ontologies",
  "Topology Ontologies",
  "Ergodic Theory Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Phase Space -> Dynamical Systems Ontology for core state-evolution semantics",
  "Phase Space -> ODE Ontology for continuous trajectory generation",
  "Phase Space -> Stability Analysis for local phase geometry",
  "Phase Space -> Control Theory for reachable state-space structure",
  "Phase Space -> Hamiltonian Dynamics for position-momentum geometry",
  "Phase Space -> Chaos Theory for strange attractors and reconstructed phase spaces",
  "Phase Space -> Bifurcation Theory for changing invariant geometry",
  "Phase Space -> Differential Geometry and Topology for manifolds and global structure",
  "Phase Space -> Ergodic Theory for invariant measures",
  "Phase Space -> Numerical Analysis for trajectory and section computation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible phase-space entities, geometries or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve complete-state semantics.",
  "Preserve state-space and phase-space equivalence where context allows.",
  "Preserve trajectory and flow semantics.",
  "Preserve invariant-set semantics.",
  "Preserve manifold and attractor semantics.",
  "Preserve projection-versus-full-state distinction.",
  "Preserve reconstruction assumptions.",
  "Preserve machine-readable phase-space semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Phase space is the mathematical space in which every point represents a complete instantaneous state of a dynamical system."

 STATE_SUMMARY =
  "A phase point is specified by a state vector containing the coordinates required to determine system evolution."

 TRAJECTORY_SUMMARY =
  "System evolution appears as a trajectory through phase space."

 INVARIANT_SUMMARY =
  "Equilibria, periodic orbits, manifolds and attractors are invariant phase-space structures."

 PORTRAIT_SUMMARY =
  "A phase portrait visualises representative trajectories and invariant structures in low-dimensional phase spaces."

 POINCARE_SUMMARY =
  "Poincare sections and return maps reduce continuous flows to lower-dimensional discrete dynamics."

 HAMILTONIAN_SUMMARY =
  "Hamiltonian phase space is constructed from generalised positions and conjugate momenta."

 RECONSTRUCTION_SUMMARY =
  "Delay-coordinate embeddings can reconstruct phase-space geometry from suitable observations under appropriate assumptions."

 VALIDATION_SUMMARY =
  "Phase-space models should be checked for state completeness, dimension, constraints, invariance and projection or embedding limitations."
}

FLOW {
 INPUT
 -> detect_phase_space_context
 -> identify independent_state_variables_constraints_coordinates_and_time structure
 -> construct phase_point_and_state_vector representation
 -> determine phase_space_dimension_and_admissible region
 -> construct continuous_flow_or_discrete_map
 -> compute_or_interpret trajectories_equilibria_fixed points_and_periodic orbits
 -> identify invariant_sets_manifolds_attractors_and_basins
 -> if_low_dimensional construct phase_portrait
 -> if_reduction_required construct Poincare section_or_projection
 -> if_partial_observation reconstruct via delay_embedding when justified
 -> route to dynamical_systems ODE PDE stability control Hamiltonian chaos bifurcation geometry topology ergodic numerical_analysis_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Phase Space Ontology defines the state-space geometry of dynamical systems. It supports phase points, state vectors, coordinates, dimensions, trajectories, flows, maps, vector fields, equilibria, fixed points, periodic orbits, limit cycles, invariant sets, stable, unstable and centre manifolds, attractors and basins, Poincare sections and maps, nullclines, separatrices, phase portraits, configuration and momentum spaces, Hamiltonian phase space, projections, embeddings, delay-coordinate reconstruction, reachable sets, invariant regions and energy surfaces."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-PHASE-SPACE-ONTOLOGY-V1"
}

}

END_FLOW