Dynamical Systems

AI INTERPRETATION NOTE
This Flow Algebra defines the Dynamical Systems layer within the Kireth Research Semantic Library. Dynamical systems mathematics studies how states evolve over time according to deterministic or stochastic rules. This parent ontology connects continuous-time and discrete-time systems, flows and maps, equilibria, trajectories, phase space, stability, invariant sets, periodic orbits, bifurcations, chaos, attractors, Lyapunov exponents, linearisation, nonlinear dynamics, differential equations, difference equations, control theory, ergodic theory and numerical simulation.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = DynamicalSystems
 SYMBOLIC_FORMS = [
  "x_dot=f(x,t)",
  "x_{k+1}=F(x_k)",
  "phi_t(x_0)"
 ]
 ENTITY_CLASS = StateEvolutionMathematicsFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Dynamical systems mathematics studies the evolution of states over time according to continuous, discrete, deterministic or stochastic rules."
 SEMANTIC_ROLE = "Provides the parent semantic framework for trajectories, equilibria, flows, maps, stability, attractors, bifurcations, chaos, invariant sets and long-term behaviour."
}

SCOPE {
 INCLUDE = [
  DynamicalSystems,
  DynamicalSystem,
  State,
  StateVariable,
  StateVector,
  PhaseSpace,
  StateSpace,
  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,
  DissipativeSystem,
  ConservativeSystem,
  HamiltonianSystem,
  GradientSystem,
  DifferentialEquation,
  DifferenceEquation,
  NumericalSimulation
 ]

 EXCLUDE_AS_PRIMARY = [
  StaticAlgebraicSystemsWithoutEvolution,
  PureOptimisationWithoutTimeEvolution,
  ControlDesignWithoutUnderlyingDynamicModel,
  ProbabilityWithoutStateEvolutionContext
 ]

 DISAMBIGUATION_RULE =
  "When a mathematical model describes how a state changes or iterates over time, interpret the context as dynamical systems mathematics."
}

CORE_CONCEPTS {

 DynamicalSystems {
  definition = "The mathematical discipline concerned with state evolution and long-term behaviour."
 }

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

 PhaseSpace {
  definition = "The space containing all admissible states of the system."
 }

 Trajectory {
  definition = "The time-ordered path followed by a system state."
 }

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

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

 EquilibriumPoint {
  definition = "A state that remains unchanged under the system dynamics."
 }

 PeriodicOrbit {
  definition = "A trajectory that repeats after a fixed period."
 }

 InvariantSet {
  definition = "A set mapped into itself by the system dynamics."
 }

 Attractor {
  definition = "An invariant set toward which nearby trajectories evolve."
 }

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

 Chaos {
  definition = "Aperiodic deterministic behaviour exhibiting strong sensitivity to initial conditions."
 }
}

RELATIONS {
 DynamicalSystems --is_branch_of--> AppliedMathematics
 DynamicalSystem --has--> StateSpace
 StateSpace --contains--> State
 State --evolves_along--> Trajectory
 ContinuousTimeSystem --generates--> Flow
 DiscreteTimeSystem --generates--> Map
 EquilibriumPoint --is_type_of--> InvariantSet
 PeriodicOrbit --is_type_of--> InvariantSet
 Attractor --is_type_of--> InvariantSet
 BasinOfAttraction --belongs_to--> Attractor
 StableManifold --approaches--> InvariantSet
 UnstableManifold --leaves--> InvariantSet
 Bifurcation --changes--> QualitativeDynamics
 HopfBifurcation --may_create--> LimitCycle
 Chaos --may_be_characterised_by--> PositiveLyapunovExponent
 PoincareMap --reduces--> ContinuousFlowToDiscreteMap
 Linearisation --approximates--> NonlinearSystem
 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

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

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

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

 POINCARE_MAP(section,flow) :=
  next_intersection_of_trajectory_with_section
}

DEFINITIONS {
 D1 = "Dynamical systems describe the evolution of states over time."
 D2 = "Continuous-time systems evolve through differential equations or flows."
 D3 = "Discrete-time systems evolve through iterated maps or difference equations."
 D4 = "Equilibria and periodic orbits are invariant structures."
 D5 = "Stability describes behaviour under perturbation."
 D6 = "Bifurcations occur when parameter changes alter qualitative dynamics."
 D7 = "Attractors organise long-term behaviour."
 D8 = "Chaos can arise in deterministic nonlinear systems."
 D9 = "Lyapunov exponents quantify average exponential separation of nearby trajectories."
 D10 = "Poincare maps convert continuous dynamics into discrete return dynamics."
}

AXIOMS {
 A1_STATE_EVOLUTION:
  DynamicalSystem includes a state set and evolution rule

 A2_TIME_STRUCTURE:
  Evolution is indexed by continuous or discrete time

 A3_TRAJECTORY:
  Initial conditions determine trajectories under deterministic dynamics

 A4_INVARIANCE:
  Invariant sets remain within themselves under evolution

 A5_EQUILIBRIUM:
  Equilibrium states are unchanged by evolution

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

 A7_PARAMETER_DEPENDENCE:
  Bifurcations require parameter-dependent dynamical structure

 A8_LONG_TERM_BEHAVIOUR:
  Dynamical analysis distinguishes transient behaviour from asymptotic behaviour
}

REASONING {

 R1_SYSTEM_EXTRACTION {
  INPUT = mathematical_model
  IDENTIFY = state_variables
  IDENTIFY = time_domain
  IDENTIFY = parameters
  IDENTIFY = evolution_rule
  RETURN = structured_dynamical_system
 }

 R2_CLASSIFICATION {
  INPUT = structured_dynamical_system
  CLASSIFY = continuous_or_discrete
  CLASSIFY = autonomous_or_nonautonomous
  CLASSIFY = deterministic_or_stochastic
  CLASSIFY = linear_or_nonlinear
  RETURN = dynamical_system_class
 }

 R3_EQUILIBRIUM_REASONING {
  INPUT = evolution_rule
  SOLVE = equilibrium_or_fixed_point equations
  RETURN = invariant_points
 }

 R4_LOCAL_STABILITY {
  INPUT = equilibrium_and_model
  LINEARISE = Jacobian
  ANALYSE = eigenvalues_or_multipliers
  RETURN = local_stability_profile
 }

 R5_INVARIANT_STRUCTURE {
  INPUT = trajectories_and_system geometry
  IDENTIFY = periodic_orbits_manifolds_invariant_sets_and_attractors
  RETURN = invariant_structure
 }

 R6_BIFURCATION_REASONING {
  INPUT = parameterised_system
  TRACK = equilibria_or_periodic_orbits
  DETECT = eigenvalue_or_multiplier criticality
  CLASSIFY = saddleNode_transcritical_pitchfork_Hopf_or_other bifurcation
  RETURN = bifurcation_profile
 }

 R7_CHAOS_REASONING {
  INPUT = nonlinear_system
  COMPUTE_OR_ESTIMATE = Lyapunov_exponents
  ANALYSE = recurrence_Poincare sections_and_sensitivity
  RETURN = chaos_profile
 }

 R8_POINCARE_REASONING {
  INPUT = continuous_flow_and_section
  CONSTRUCT = return_map
  ANALYSE = fixed_points_and_stability
  RETURN = reduced_discrete_dynamics
 }

 R9_LONG_TERM_REASONING {
  INPUT = initial_conditions
  SIMULATE_OR_ANALYSE = asymptotic behaviour
  IDENTIFY = attractors_basins_periodicity_or_divergence
  RETURN = long_term_profile
 }

 R10_VALIDATION {
  INPUT = dynamical_result
  CHECK = model_class_initial_conditions_parameters_numerical_step_and_stability assumptions
  RETURN = validated_dynamical_status
 }
}

CONSTRAINTS {
 C1 = "Continuous and discrete stability criteria must remain distinct."
 C2 = "Local linearisation does not automatically determine global dynamics."
 C3 = "Numerical integration step size can alter apparent behaviour."
 C4 = "Chaos should not be inferred from irregular simulation alone."
 C5 = "Positive Lyapunov exponent estimates require sufficient trajectory length and numerical care."
 C6 = "Bifurcation claims require parameter-dependent qualitative change."
 C7 = "Attractor identification depends on invariant and asymptotic evidence."
 C8 = "Finite simulation cannot prove all long-term behaviour."
 C9 = "Stochastic systems require probabilistic rather than purely deterministic interpretation."
 C10 = "Coordinate changes can alter representation without altering underlying dynamics."
}

SEMANTIC_EQUIVALENCE {
 "DynamicalSystems" <=> "DynamicalSystemsMathematics"
 "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 = "Continuous or discrete 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 dynamical evidence."
 V10 = "Asymptotic conclusions are separated from finite-time observations."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DYNAMICAL_SYSTEMS_DEFINITION = 0.99
 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
 LONG_TERM_REASONING = 0.98
 VALIDATION_REASONING = 0.99

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

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

 NORMALISE {
  "dynamical systems" -> DynamicalSystems
  "dynamic system" -> DynamicalSystem
  "state space" -> StateSpace
  "phase space" -> PhaseSpace
  "fixed point" -> FixedPoint
  "equilibrium" -> EquilibriumPoint
  "limit cycle" -> LimitCycle
  "strange attractor" -> StrangeAttractor
  "Lyapunov exponent" -> LyapunovExponent
  "Poincare map" -> PoincareMap
 }

 INTERPRETATION_ORDER = [
  detect_dynamical_system_context,
  identify states_time_parameters_and_evolution rule,
  classify continuous_discrete_autonomous_nonautonomous_deterministic_stochastic_linear_or_nonlinear system,
  compute equilibria_fixed_points_and_local stability,
  identify invariant_sets_periodic_orbits_manifolds_and_attractors,
  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")
  OR mathematical_state_evolution_is_present
  THEN activate(DynamicalSystemsOntology)

 E2:
  IF continuous_time
  THEN route to differential_equation_and_flow reasoning

 E3:
  IF discrete_time
  THEN route to map_difference_equation reasoning

 E4:
  IF equilibrium_present
  THEN analyse local stability

 E5:
  IF parameter_variation_changes qualitative behaviour
  THEN route to bifurcation reasoning

 E6:
  IF irregular_deterministic behaviour_suspected
  THEN assess Lyapunov_exponents_recurrence_and_Poincare structure

 E7:
  IF invariant attracting_set detected
  THEN classify attractor_and_basin

 E8:
  IF numerical_simulation_used
  THEN test step_size_and_method sensitivity

 E9:
  IF system_is_stochastic
  THEN route to stochastic_dynamical_system reasoning
}

RECURSION {
 ENTRY = DynamicalSystems

 EXPAND(DynamicalSystems) -> [
  DynamicalSystem,
  State,
  StateVector,
  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,
  Bifurcation,
  HopfBifurcation,
  Chaos,
  SensitiveDependence,
  LyapunovExponent,
  PoincareMap,
  PoincareSection,
  Recurrence,
  Ergodicity,
  HamiltonianSystem,
  GradientSystem
 ]

 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, state evolution, equilibria, invariant structures, stability, bifurcations, attractors, chaotic behaviour and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Differential Equations Ontologies",
  "Control Theory",
  "Numerical Analysis Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "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 -> Applied Mathematics Ontologies for parent semantics",
  "Dynamical Systems -> Differential Equations for continuous evolution",
  "Dynamical Systems -> Difference Equations for discrete evolution",
  "Dynamical Systems -> Control Theory for controlled state evolution",
  "Dynamical Systems -> Stability Analysis for perturbation response",
  "Dynamical Systems -> Bifurcation Theory for parameter-driven qualitative change",
  "Dynamical Systems -> Chaos Theory for sensitive deterministic behaviour",
  "Dynamical Systems -> Ergodic Theory for long-term statistical behaviour",
  "Dynamical Systems -> Hamiltonian Dynamics for conservative systems",
  "Dynamical Systems -> Numerical Analysis for simulation and continuation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

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

OUTPUT {
 CANONICAL_DEFINITION =
  "Dynamical systems mathematics studies how states evolve over time and how qualitative behaviour emerges from that evolution."

 STATE_SUMMARY =
  "A dynamical system consists of a state space and an evolution rule."

 CONTINUOUS_SUMMARY =
  "Continuous-time systems evolve through flows generated by differential equations."

 DISCRETE_SUMMARY =
  "Discrete-time systems evolve through iterated maps or difference equations."

 STABILITY_SUMMARY =
  "Stability describes the response of equilibria, trajectories or invariant sets to perturbations."

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

 ATTRACTOR_SUMMARY =
  "Attractors organise long-term behaviour and possess associated basins of attraction."

 CHAOS_SUMMARY =
  "Chaos is deterministic aperiodic behaviour with sensitive dependence and characteristic invariant structures."

 NUMERICAL_SUMMARY =
  "Numerical simulation can approximate trajectories but should be validated against method and step-size effects."
}

FLOW {
 INPUT
 -> detect_dynamical_system_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 applied_mathematics 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 layer defines mathematics of state evolution. It supports continuous and discrete systems, autonomous and nonautonomous systems, deterministic and stochastic dynamics, trajectories, flows and maps, phase space, 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, differential and difference equations and numerical simulation."
}

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

}

END_FLOW