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Dynamical Systems Ontology

Dynamical Systems
Authors: MArk Anthony Higgins - Sky Recursive Ai
© Mark Anthony Higgins 11 October 2025
Revised Date: 12 September 2026
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AI INTERPRETATION NOTE
This Flow Algebra defines the Dynamical Systems Ontology within the Kireth Research Semantic Library. It formalises dynamical systems as a machine-readable ontology of state evolution, trajectories, phase spaces, flows, maps, equilibria, invariant sets, periodic orbits, stability, bifurcations, attractors, chaos and long-term behaviour. It sits beneath the Dynamical Systems parent layer and provides a structured reasoning framework for classifying and analysing continuous-time, discrete-time, deterministic, stochastic, linear and nonlinear systems.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = DynamicalSystemsOntology
 SYMBOLIC_FORMS = [
  "x_dot=f(x,t)",
  "x_{k+1}=F(x_k)",
  "phi_t(x_0)"
 ]
 ENTITY_CLASS = FormalStateEvolutionOntology
 PRIMARY_CONTEXT = DynamicalSystems
 DEFINITION = "The Dynamical Systems Ontology is a formal semantic framework for representing state evolution, trajectories, equilibria, invariant structures, stability, bifurcations and long-term dynamical behaviour."
 SEMANTIC_ROLE = "Provides machine-readable entities and reasoning relations for classifying dynamical systems, analysing equilibria and stability, identifying invariant sets, bifurcations, attractors and chaos, and validating analytical or numerical results."
}

SCOPE {
 INCLUDE = [
  DynamicalSystemsOntology,
  DynamicalSystem,
  State,
  StateVariable,
  StateVector,
  StateSpace,
  PhaseSpace,
  Trajectory,
  Orbit,
  Flow,
  Map,
  ContinuousTimeSystem,
  DiscreteTimeSystem,
  AutonomousSystem,
  NonautonomousSystem,
  DeterministicSystem,
  StochasticSystem,
  LinearSystem,
  NonlinearSystem,
  EquilibriumPoint,
  FixedPoint,
  PeriodicOrbit,
  LimitCycle,
  InvariantSet,
  StableManifold,
  UnstableManifold,
  CentreManifold,
  Attractor,
  StrangeAttractor,
  BasinOfAttraction,
  Stability,
  LyapunovStability,
  AsymptoticStability,
  ExponentialStability,
  Bifurcation,
  SaddleNodeBifurcation,
  TranscriticalBifurcation,
  PitchforkBifurcation,
  HopfBifurcation,
  Chaos,
  SensitiveDependence,
  LyapunovExponent,
  PoincareMap,
  PoincareSection,
  Recurrence,
  Ergodicity,
  HamiltonianSystem,
  GradientSystem,
  DissipativeSystem,
  ConservativeSystem,
  DifferentialEquation,
  DifferenceEquation,
  NumericalSimulation
 ]

 EXCLUDE_AS_PRIMARY = [
  StaticAlgebraicModelsWithoutEvolution,
  PureOptimisationWithoutStateDynamics,
  ControlDesignWithoutUnderlyingDynamicalSemantics,
  ProbabilityModelsWithoutTemporalStateEvolution
 ]

 DISAMBIGUATION_RULE =
  "When a mathematical model specifies how a state evolves in continuous or discrete time, interpret the context through the Dynamical Systems Ontology."
}

CORE_CONCEPTS {

 DynamicalSystemsOntology {
  definition = "The ontology organising the formal concepts and relations of dynamical systems."
 }

 DynamicalSystem {
  definition = "A state space together with a rule describing state evolution over time."
 }

 StateSpace {
  definition = "The set of all admissible system states."
 }

 Trajectory {
  definition = "The path followed by a state under the system evolution."
 }

 Flow {
  definition = "The continuous-time evolution operator of a dynamical system."
 }

 Map {
  definition = "A discrete-time rule mapping one state to the next."
 }

 EquilibriumPoint {
  definition = "A state unchanged by continuous-time evolution."
 }

 FixedPoint {
  definition = "A state unchanged by a discrete map."
 }

 InvariantSet {
  definition = "A set preserved under the system dynamics."
 }

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

 Bifurcation {
  definition = "A qualitative change in dynamics caused by parameter variation."
 }

 Chaos {
  definition = "Deterministic aperiodic behaviour with sensitive dependence and complex invariant structure."
 }
}

RELATIONS {
 DynamicalSystemsOntology --is_child_of--> DynamicalSystems
 DynamicalSystem --has--> StateSpace
 StateSpace --contains--> State
 State --evolves_along--> Trajectory
 ContinuousTimeSystem --generates--> Flow
 DiscreteTimeSystem --generates--> Map
 EquilibriumPoint --is_type_of--> InvariantSet
 FixedPoint --is_type_of--> InvariantSet
 PeriodicOrbit --is_type_of--> InvariantSet
 Attractor --is_type_of--> InvariantSet
 BasinOfAttraction --belongs_to--> Attractor
 StableManifold --approaches--> InvariantSet
 UnstableManifold --leaves--> InvariantSet
 CentreManifold --contains--> CriticalDynamics
 Bifurcation --changes--> QualitativeDynamics
 HopfBifurcation --may_create--> LimitCycle
 Chaos --may_be_characterised_by--> PositiveLyapunovExponent
 PoincareMap --reduces--> ContinuousFlowToDiscreteMap
 NumericalSimulation --approximates--> Trajectory
}

OPERATORS {
 CONTINUOUS_DYNAMICS(x,t) :=
  x_dot=f(x,t)

 AUTONOMOUS_DYNAMICS(x) :=
  x_dot=f(x)

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

 FLOW_OPERATOR(phi,t,x0) :=
  x(t)=phi_t(x0)

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

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

 JACOBIAN(f,x_star) :=
  J=partial f/partial x evaluated_at x_star

 LINEARISATION(J,delta_x) :=
  delta_x_dot=J*delta_x

 PERIODIC_ORBIT_CONDITION(phi,T,x0) :=
  phi_T(x0)=x0

 DISCRETE_PERIODIC_CONDITION(F,p,x0) :=
  F^p(x0)=x0

 LYAPUNOV_EXPONENT(delta_t,delta_0,t) :=
  lambda=(1/t)*ln(||delta_t||/||delta_0||)

 POINCARE_MAP(section,flow) :=
  next_intersection_of_trajectory_with_section
}

DEFINITIONS {
 D1 = "Dynamical systems describe state evolution over time."
 D2 = "Continuous systems evolve through differential equations and flows."
 D3 = "Discrete systems evolve through maps and difference equations."
 D4 = "Equilibria and fixed points are stationary invariant states."
 D5 = "Periodic orbits repeat after a finite period."
 D6 = "Invariant sets remain preserved by the dynamics."
 D7 = "Stability measures the response to perturbations."
 D8 = "Bifurcations alter qualitative behaviour as parameters vary."
 D9 = "Attractors organise long-term behaviour."
 D10 = "Chaos is deterministic and should be distinguished from stochastic randomness."
}

AXIOMS {
 A1_STATE_EVOLUTION:
  DynamicalSystemsOntology represents systems with explicit state evolution

 A2_TIME_STRUCTURE:
  Evolution is indexed by continuous or discrete time

 A3_DETERMINISTIC_TRAJECTORY:
  Deterministic initial conditions determine trajectories under a well-defined evolution rule

 A4_INVARIANCE:
  Invariant sets remain within themselves under evolution

 A5_EQUILIBRIUM:
  Equilibrium or fixed states are unchanged by the evolution rule

 A6_STABILITY:
  Stability is defined relative to perturbations of states or invariant structures

 A7_PARAMETER_DEPENDENCE:
  Bifurcations require parameter-dependent dynamics

 A8_LONG_TERM_BEHAVIOUR:
  Dynamical analysis distinguishes transient behaviour from asymptotic behaviour
}

REASONING {

 R1_ENTITY_EXTRACTION {
  INPUT = mathematical_model
  IDENTIFY = states
  IDENTIFY = time_variable
  IDENTIFY = parameters
  IDENTIFY = evolution_rule
  RETURN = structured_dynamical_entities
 }

 R2_SYSTEM_CLASSIFICATION {
  INPUT = structured_dynamical_entities
  CLASSIFY = continuous_or_discrete
  CLASSIFY = autonomous_or_nonautonomous
  CLASSIFY = deterministic_or_stochastic
  CLASSIFY = linear_or_nonlinear
  CLASSIFY = conservative_or_dissipative
  RETURN = dynamical_system_class
 }

 R3_EQUILIBRIUM_ANALYSIS {
  INPUT = evolution_rule
  SOLVE = equilibrium_or_fixed_point equations
  RETURN = invariant_points
 }

 R4_LOCAL_STABILITY {
  INPUT = equilibrium_or_fixed_point
  COMPUTE = Jacobian_or_map derivative
  ANALYSE = eigenvalues_or_multipliers
  RETURN = local_stability_profile
 }

 R5_INVARIANT_STRUCTURE {
  INPUT = system_and_trajectories
  IDENTIFY = periodic_orbits_invariant_sets_manifolds_and_attractors
  RETURN = invariant_structure
 }

 R6_BIFURCATION_ANALYSIS {
  INPUT = parameterised_system
  TRACK = equilibria_or_periodic orbits
  DETECT = critical_eigenvalue_or_multiplier changes
  CLASSIFY = saddleNode_transcritical_pitchfork_Hopf_or_other
  RETURN = bifurcation_profile
 }

 R7_CHAOS_ANALYSIS {
  INPUT = nonlinear_system
  COMPUTE_OR_ESTIMATE = Lyapunov_exponents
  ANALYSE = recurrence_Poincare sections_sensitive dependence
  RETURN = chaos_profile
 }

 R8_POINCARE_ANALYSIS {
  INPUT = continuous_flow_and_section
  CONSTRUCT = return_map
  ANALYSE = fixed_points_periodic_orbits_and_stability
  RETURN = reduced_map_profile
 }

 R9_LONG_TERM_ANALYSIS {
  INPUT = initial_conditions
  ANALYSE_OR_SIMULATE = asymptotic behaviour
  IDENTIFY = attractor_periodicity_divergence_or_recurrence
  RETURN = long_term_profile
 }

 R10_VALIDATION {
  INPUT = analytical_or_numerical_result
  CHECK = model_class_parameters_initial_conditions_method_step_size_and_asymptotic assumptions
  RETURN = validated_dynamical_status
 }
}

CONSTRAINTS {
 C1 = "Continuous and discrete systems require distinct stability criteria."
 C2 = "Local linearisation does not automatically imply global behaviour."
 C3 = "Numerical integration step size can create or suppress apparent dynamics."
 C4 = "Irregular simulation alone does not establish chaos."
 C5 = "Lyapunov-exponent estimates require sufficient data and numerical care."
 C6 = "Bifurcation claims require genuine qualitative parameter-dependent change."
 C7 = "Attractor claims require invariant and asymptotic evidence."
 C8 = "Finite simulation cannot prove all long-term behaviour."
 C9 = "Stochastic dynamics require probabilistic interpretation."
 C10 = "Coordinate transformations can change representation without changing underlying dynamics."
}

SEMANTIC_EQUIVALENCE {
 "DynamicalSystemsOntology" <=> "FormalDynamicalSystemsOntology"
 "StateSpace" <=> "PhaseSpace"
 "Orbit" <=> "TrajectoryUnderEvolution"
 "FixedPoint" <=> "DiscreteEquilibrium"
 "LimitCycle" <=> "IsolatedPeriodicOrbit"
 "BasinOfAttraction" <=> "AttractionBasin"
}

VALIDATION {
 V1 = "State variables are explicit."
 V2 = "Time domain is explicit."
 V3 = "Evolution rule is explicit."
 V4 = "System classification is correct."
 V5 = "Equilibria or fixed points are computed correctly."
 V6 = "Local and global conclusions are distinguished."
 V7 = "Bifurcation parameters are identified."
 V8 = "Numerical settings are documented."
 V9 = "Chaos claims include quantitative or structural evidence."
 V10 = "Finite-time and asymptotic conclusions are distinguished."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DYNAMICAL_SYSTEMS_ONTOLOGY_DEFINITION = 0.99
 ENTITY_EXTRACTION_REASONING = 0.99
 SYSTEM_CLASSIFICATION_REASONING = 0.99
 EQUILIBRIUM_REASONING = 0.99
 LOCAL_STABILITY_REASONING = 0.99
 INVARIANT_STRUCTURE_REASONING = 0.98
 BIFURCATION_REASONING = 0.98
 CHAOS_REASONING = 0.97
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when the evolution law, time domain, parameters, initial conditions, numerical resolution, invariant-set evidence or chaos diagnostics are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  DynamicalSystemsOntologyReasoning,
  StateEvolutionReasoning,
  ContinuousDynamicsReasoning,
  DiscreteDynamicsReasoning,
  PhaseSpaceReasoning,
  EquilibriumReasoning,
  StabilityReasoning,
  InvariantSetReasoning,
  PeriodicOrbitReasoning,
  AttractorReasoning,
  BifurcationReasoning,
  ChaosReasoning,
  LyapunovExponentReasoning,
  PoincareMapReasoning,
  ErgodicReasoning,
  NumericalSimulationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "dynamical systems ontology" -> DynamicalSystemsOntology
  "dynamical system" -> DynamicalSystem
  "state space" -> StateSpace
  "phase space" -> PhaseSpace
  "equilibrium" -> EquilibriumPoint
  "fixed point" -> FixedPoint
  "periodic orbit" -> PeriodicOrbit
  "limit cycle" -> LimitCycle
  "attractor" -> Attractor
  "Lyapunov exponent" -> LyapunovExponent
 }

 INTERPRETATION_ORDER = [
  detect_dynamical_systems_ontology_context,
  identify states_time_parameters_and_evolution rule,
  classify continuous_discrete_autonomous_nonautonomous_deterministic_stochastic_linear_nonlinear_conservative_or_dissipative system,
  compute equilibria_fixed_points_and_local stability,
  identify invariant_sets_periodic_orbits_manifolds_attractors_and_basins,
  analyse bifurcations_and_parameter dependence,
  assess chaos_recurrence_Lyapunov exponents_or_Poincare structure when relevant,
  validate numerical_and_asymptotic conclusions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("dynamical systems ontology")
  OR formal_state_evolution_semantics
  THEN activate(DynamicalSystemsOntology)

 E2:
  IF continuous_time
  THEN route to differential_equation_and_flow reasoning

 E3:
  IF discrete_time
  THEN route to map_and_difference_equation reasoning

 E4:
  IF equilibrium_or_fixed_point_present
  THEN analyse local stability

 E5:
  IF parameter_change_alters qualitative dynamics
  THEN route to bifurcation reasoning

 E6:
  IF irregular_deterministic behaviour_suspected
  THEN assess Lyapunov_exponents_recurrence_and_Poincare structure

 E7:
  IF attracting_invariant_set_detected
  THEN classify attractor_and_basin

 E8:
  IF numerical_simulation_used
  THEN test method_and_step_size sensitivity

 E9:
  IF stochastic_system
  THEN route to stochastic_dynamical_system reasoning
}

RECURSION {
 ENTRY = DynamicalSystemsOntology

 EXPAND(DynamicalSystemsOntology) -> [
  DynamicalSystem,
  State,
  StateVector,
  StateSpace,
  PhaseSpace,
  Trajectory,
  Orbit,
  Flow,
  Map,
  ContinuousTimeSystem,
  DiscreteTimeSystem,
  AutonomousSystem,
  NonautonomousSystem,
  DeterministicSystem,
  StochasticSystem,
  LinearSystem,
  NonlinearSystem,
  EquilibriumPoint,
  FixedPoint,
  PeriodicOrbit,
  LimitCycle,
  InvariantSet,
  StableManifold,
  UnstableManifold,
  CentreManifold,
  Attractor,
  StrangeAttractor,
  BasinOfAttraction,
  Stability,
  LyapunovStability,
  AsymptoticStability,
  ExponentialStability,
  Bifurcation,
  SaddleNodeBifurcation,
  TranscriticalBifurcation,
  PitchforkBifurcation,
  HopfBifurcation,
  Chaos,
  SensitiveDependence,
  LyapunovExponent,
  PoincareMap,
  PoincareSection,
  Recurrence,
  Ergodicity,
  HamiltonianSystem,
  GradientSystem,
  DissipativeSystem,
  ConservativeSystem
 ]

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

 EXPAND(BifurcationFamilies) -> [
  SaddleNodeBifurcation,
  TranscriticalBifurcation,
  PitchforkBifurcation,
  HopfBifurcation
 ]

 EXPAND(ChaosDiagnostics) -> [
  SensitiveDependence,
  LyapunovExponent,
  PoincareMap,
  Recurrence,
  StrangeAttractor
 ]

 STOP_CONDITION =
  "Stop when the system class, evolution law, equilibria, invariant structures, stability, bifurcations, attractors, chaotic behaviour and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Dynamical Systems"

 IMMEDIATE_CONTEXTS = [
  "Dynamical Systems",
  "Applied Mathematics Ontologies",
  "Differential Equations Ontologies",
  "Stability Analysis Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Dynamical Systems",
  "Applied Mathematics Ontologies",
  "Ordinary Differential Equations Ontologies",
  "Partial Differential Equations Ontologies",
  "Difference Equation Ontologies",
  "Control Theory Ontologies",
  "Stability Analysis Ontology",
  "Nonlinear Analysis Ontologies",
  "Bifurcation Theory Ontologies",
  "Chaos Theory Ontologies",
  "Ergodic Theory Ontologies",
  "Hamiltonian Dynamics Ontologies",
  "Stochastic Process Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Dynamical Systems Ontology -> Dynamical Systems for parent semantics",
  "Dynamical Systems Ontology -> Differential Equations for continuous evolution",
  "Dynamical Systems Ontology -> Difference Equations for discrete evolution",
  "Dynamical Systems Ontology -> Control Theory for controlled state evolution",
  "Dynamical Systems Ontology -> Stability Analysis for perturbation response",
  "Dynamical Systems Ontology -> Nonlinear Analysis for local and global nonlinear structure",
  "Dynamical Systems Ontology -> Bifurcation Theory for parameter-driven qualitative change",
  "Dynamical Systems Ontology -> Chaos Theory for sensitive deterministic behaviour",
  "Dynamical Systems Ontology -> Ergodic Theory for long-term statistical behaviour",
  "Dynamical Systems Ontology -> Numerical Analysis for simulation and continuation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible dynamical classes, invariant structures or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve state-evolution semantics.",
  "Preserve continuous-versus-discrete distinction.",
  "Preserve trajectory and phase-space semantics.",
  "Preserve equilibrium and invariant-set semantics.",
  "Preserve local-versus-global stability distinction.",
  "Preserve bifurcation semantics.",
  "Preserve chaos-versus-randomness distinction.",
  "Preserve machine-readable dynamical-systems ontology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The Dynamical Systems Ontology formally represents state evolution, trajectories, equilibria, invariant sets, stability, bifurcations, attractors and chaotic behaviour."

 STATE_SUMMARY =
  "A dynamical system consists of a state space together with a continuous or discrete evolution rule."

 EQUILIBRIUM_SUMMARY =
  "Equilibria and fixed points are stationary invariant states of continuous and discrete systems."

 STABILITY_SUMMARY =
  "Stability describes how trajectories or invariant structures respond to perturbations."

 INVARIANT_SUMMARY =
  "Invariant sets include equilibria, periodic orbits, manifolds and attractors."

 BIFURCATION_SUMMARY =
  "Bifurcations are qualitative changes in system behaviour caused by parameter variation."

 CHAOS_SUMMARY =
  "Chaos is deterministic aperiodic behaviour characterised by sensitive dependence and nontrivial invariant structure."

 NUMERICAL_SUMMARY =
  "Numerical simulation approximates trajectories and should be validated for solver and step-size sensitivity."
}

FLOW {
 INPUT
 -> detect_dynamical_systems_ontology_context
 -> identify state_variables_phase_space_time_domain_parameters_initial_conditions_and_evolution rule
 -> classify continuous_discrete_autonomous_nonautonomous_deterministic_stochastic_linear_nonlinear_conservative_or_dissipative system
 -> compute equilibria_or_fixed points
 -> analyse local stability_via linearisation_or_Lyapunov methods
 -> identify invariant_sets_periodic orbits_manifolds_attractors_and_basins
 -> vary_parameters_and_detect bifurcations when relevant
 -> assess chaos_using Lyapunov_exponents_Poincare maps_recurrence_and_sensitive dependence
 -> if_numerical simulate trajectories_and_validate resolution
 -> route to dynamical_systems differential_equations difference_equations control stability nonlinear_analysis bifurcation chaos ergodic Hamiltonian stochastic_process numerical_analysis_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Dynamical Systems Ontology provides the formal semantic layer beneath dynamical systems mathematics. It represents state spaces, phase spaces, states, trajectories, flows and maps, continuous and discrete systems, autonomous and nonautonomous dynamics, deterministic and stochastic systems, equilibria and fixed points, periodic orbits and limit cycles, invariant sets and manifolds, attractors and basins, Lyapunov stability, bifurcations, chaos, sensitive dependence, Lyapunov exponents, Poincare maps and sections, recurrence, ergodicity, Hamiltonian and gradient systems, conservative and dissipative dynamics and numerical simulation."
}

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

}

END_FLOW
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