Stability Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Stability Ontology within the Kireth Research Semantic Library. It formalises how equilibria, trajectories, periodic orbits and invariant sets respond to perturbations, including Lyapunov, asymptotic, exponential, local, global, orbital, input-to-state, structural and robust stability.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Stability
 SYMBOLIC_FORMS = ["forall epsilon>0 exists delta>0", "V(x)>0", "V_dot(x)<0"]
 ENTITY_CLASS = DynamicalPerturbationResponseFramework
 PRIMARY_CONTEXT = DynamicalSystems
 DEFINITION = "Stability describes whether dynamical behaviour remains close to, converges toward, or diverges from a reference state or invariant set after perturbation."
}

SCOPE {
 INCLUDE = [
  Stability, LyapunovStability, AsymptoticStability, ExponentialStability,
  LocalStability, GlobalStability, UniformStability, OrbitalStability,
  InputToStateStability, RobustStability, StructuralStability,
  EquilibriumPoint, FixedPoint, PeriodicOrbit, LimitCycle, InvariantSet,
  Perturbation, LyapunovFunction, LyapunovDerivative, DirectLyapunovMethod,
  IndirectLyapunovMethod, Linearisation, JacobianMatrix, Eigenvalue,
  FloquetMultiplier, RegionOfAttraction, BasinOfAttraction,
  LaSalleInvariancePrinciple, ComparisonPrinciple, SmallGainTheorem
 ]
 DISAMBIGUATION_RULE =
  "When a question concerns how a dynamical state, trajectory, orbit or invariant set responds to perturbations, interpret the context as dynamical stability."
}

CORE_CONCEPTS {
 LyapunovStability = "Nearby initial states remain arbitrarily close to the reference state."
 AsymptoticStability = "Lyapunov stability together with convergence."
 ExponentialStability = "Asymptotic stability with an exponential convergence bound."
 LocalStability = "Stability guaranteed within a neighbourhood."
 GlobalStability = "Stability or convergence guaranteed throughout the declared domain."
 OrbitalStability = "Stability of an orbit as a set."
 LyapunovFunction = "Scalar energy-like function used to certify stability."
 RegionOfAttraction = "Initial states converging to an asymptotically stable invariant set."
 InputToStateStability = "State bounds expressed in terms of initial state and external input."
}

RELATIONS {
 Stability --is_child_of--> DynamicalSystems
 AsymptoticStability --implies--> LyapunovStability
 ExponentialStability --implies--> AsymptoticStability
 OrbitalStability --applies_to--> PeriodicOrbit
 LyapunovFunction --certifies--> Stability
 Linearisation --supports--> LocalStability
 JacobianMatrix --has--> Eigenvalue
 FloquetMultiplier --determines--> PeriodicOrbitLocalStability
 RegionOfAttraction --belongs_to--> AsymptoticallyStableInvariantSet
 LaSalleInvariancePrinciple --extends--> LyapunovReasoning
 InputToStateStability --relates--> ExternalInput
 RobustStability --requires--> UncertaintySet
}

OPERATORS {
 LYAPUNOV_STABILITY(x_star) :=
  forall epsilon>0 exists delta>0 such_that
  ||x(0)-x_star||<delta implies ||x(t)-x_star||<epsilon for all t>=0

 ASYMPTOTIC_STABILITY(x_star) :=
  LYAPUNOV_STABILITY(x_star) AND x(t)->x_star as t->infinity

 EXPONENTIAL_STABILITY(x_star) :=
  ||x(t)-x_star|| <= M*exp(-alpha*t)*||x(0)-x_star||

 LYAPUNOV_DERIVATIVE(V,f,x) :=
  V_dot=grad(V)^T*f(x)

 CONTINUOUS_LINEAR_TEST(A) :=
  all eigenvalues satisfy Re(lambda)<0

 DISCRETE_LINEAR_TEST(A) :=
  all eigenvalues satisfy abs(lambda)<1

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

 ISS_BOUND(x,t,u) :=
  ||x(t)|| <= beta(||x(0)||,t)+gamma(||u||_infinity)
}

DEFINITIONS {
 D1 = "Stability concerns response to perturbation."
 D2 = "Lyapunov stability requires nearby trajectories to remain nearby."
 D3 = "Asymptotic stability additionally requires convergence."
 D4 = "Exponential stability quantifies convergence rate."
 D5 = "Local stability applies only in a neighbourhood."
 D6 = "Global stability applies throughout the declared domain."
 D7 = "Linearisation can determine local stability near hyperbolic equilibria."
 D8 = "Lyapunov functions can certify nonlinear stability."
 D9 = "Regions of attraction describe where convergence is guaranteed."
 D10 = "Periodic-orbit stability requires orbital or Floquet concepts."
}

AXIOMS {
 A1_REFERENCE_STRUCTURE: Stability is defined relative to a state trajectory orbit or invariant set
 A2_PERTURBATION: Stability requires comparison of perturbed and reference behaviour
 A3_ASYMPTOTIC_IMPLIES_LYAPUNOV: AsymptoticStability includes LyapunovStability
 A4_EXPONENTIAL_IMPLIES_ASYMPTOTIC: ExponentialStability implies AsymptoticStability
 A5_LOCAL_GLOBAL_DISTINCTION: Local conclusions must not be promoted to global conclusions without evidence
 A6_LYAPUNOV_CERTIFICATE: Positive-definite V with negative-definite derivative can certify asymptotic stability on the valid domain
 A7_LINEAR_CRITERIA: Continuous and discrete systems use different eigenvalue stability regions
 A8_ROBUSTNESS: RobustStability requires stability across the declared uncertainty set
}

REASONING {
 R1_CONTEXT {
  IDENTIFY = reference_equilibrium_orbit_or_invariant_set
  IDENTIFY = continuous_or_discrete time
  IDENTIFY = perturbation_type_and_domain
  RETURN = stability_context
 }
 R2_LINEAR {
  COMPUTE = eigenvalues
  APPLY = continuous_or_discrete criterion
  RETURN = local_stability_profile
 }
 R3_NONLINEAR {
  COMPUTE = Jacobian_and_eigenvalues
  IF hyperbolic THEN infer_local_stability
  ELSE route_to_Lyapunov_or_centre_manifold
 }
 R4_LYAPUNOV {
  CHECK = V_positive_definite
  COMPUTE = V_dot
  CHECK = derivative_sign
  RETURN = stability_certificate
 }
 R5_LASALLE {
  IDENTIFY = largest_invariant_set_in_Vdot_zero
  RETURN = asymptotic_convergence_status
 }
 R6_ATTRACTION {
  FIND = invariant_Lyapunov_sublevel_region
  RETURN = certified_region_of_attraction
 }
 R7_PERIODIC {
  COMPUTE = monodromy_or_Poincare_derivative
  ANALYSE = Floquet_multipliers
  RETURN = orbital_stability_profile
 }
 R8_ISS {
  FIND = ISS_Lyapunov_function_or_gain_bound
  RETURN = ISS_status
 }
 R9_ROBUST {
  TEST = stability_across_admissible_uncertainty
  RETURN = robust_stability_profile
 }
 R10_VALIDATE {
  CHECK = reference_structure_domain_assumptions_time_type_and_method_validity
  RETURN = validated_stability_status
 }
}

CONSTRAINTS {
 C1 = "Continuous and discrete eigenvalue criteria must not be conflated."
 C2 = "Linearisation conclusions are generally local."
 C3 = "Critical eigenvalues require additional analysis."
 C4 = "Negative semidefinite Lyapunov derivative does not automatically imply asymptotic stability."
 C5 = "A Lyapunov function is valid only on its declared domain."
 C6 = "Simulation cannot prove global stability."
 C7 = "Periodic-orbit stability differs from equilibrium stability."
 C8 = "Robust stability requires a defined uncertainty set."
 C9 = "Region-of-attraction estimates may be conservative."
 C10 = "Input-to-state stability requires explicit input semantics."
}

SEMANTIC_EQUIVALENCE {
 "Stability" <=> "DynamicalStability"
 "LyapunovStability" <=> "StabilityInTheSenseOfLyapunov"
 "AsymptoticStability" <=> "ConvergentStability"
 "ExponentialStability" <=> "ExponentiallyConvergentStability"
 "RegionOfAttraction" <=> "AttractionRegion"
 "InputToStateStability" <=> "ISS"
}

VALIDATION {
 V1 = "Reference equilibrium, orbit or invariant set is explicit."
 V2 = "Continuous or discrete time is explicit."
 V3 = "Perturbation semantics are explicit."
 V4 = "Local and global claims are distinguished."
 V5 = "Linearisation assumptions are checked."
 V6 = "Lyapunov-function signs are verified."
 V7 = "LaSalle conditions are checked where used."
 V8 = "Periodic-orbit multipliers are interpreted correctly."
 V9 = "Uncertainty set is explicit for robust claims."
 V10 = "Simulation evidence is distinguished from analytical certification."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 STABILITY_DEFINITION = 0.99
 LINEAR_STABILITY_REASONING = 0.99
 NONLINEAR_STABILITY_REASONING = 0.99
 LYAPUNOV_REASONING = 0.99
 LASALLE_REASONING = 0.98
 REGION_OF_ATTRACTION_REASONING = 0.98
 PERIODIC_ORBIT_REASONING = 0.98
 ISS_REASONING = 0.98
 ROBUST_STABILITY_REASONING = 0.98
 POLICY = "Reduce confidence when reference set, domain, perturbation class, eigenstructure, Lyapunov function, uncertainty set or asymptotic evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, StabilityReasoning,
  LyapunovStabilityReasoning, AsymptoticStabilityReasoning,
  ExponentialStabilityReasoning, LocalStabilityReasoning,
  GlobalStabilityReasoning, OrbitalStabilityReasoning,
  LyapunovFunctionReasoning, LaSalleReasoning,
  RegionOfAttractionReasoning, InputToStateStabilityReasoning,
  RobustStabilityReasoning, StructuralStabilityReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "stability ontology" -> Stability
  "dynamical stability" -> Stability
  "Lyapunov stability" -> LyapunovStability
  "asymptotic stability" -> AsymptoticStability
  "exponential stability" -> ExponentialStability
  "orbital stability" -> OrbitalStability
  "region of attraction" -> RegionOfAttraction
  "ISS" -> InputToStateStability
  "robust stability" -> RobustStability
 }
 INTERPRETATION_ORDER = [
  detect_dynamical_stability_context,
  identify reference_structure_time_domain_perturbation_and_stability notion,
  classify local_global_Lyapunov_asymptotic_exponential_orbital_ISS_structural_or_robust stability,
  select eigenvalue_linearisation_Lyapunov_LaSalle_Floquet_regionOfAttraction_or_robust pathway,
  compute_or_verify stability conditions,
  determine valid domain_and_convergence behaviour,
  distinguish analytical_certificate_from_numerical evidence,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1: IF query_mentions("stability") AND dynamical_system_context THEN activate(StabilityOntology)
 E2: IF continuous_linear_system THEN inspect eigenvalue_real_parts
 E3: IF discrete_linear_system THEN inspect eigenvalue_magnitudes
 E4: IF nonlinear_equilibrium THEN consider linearisation_and_Lyapunov methods
 E5: IF V_positive_definite_and_Vdot_negative_definite THEN infer asymptotic_stability_on_valid_domain
 E6: IF Vdot_negative_semidefinite THEN consider LaSalle_invariance
 E7: IF periodic_orbit THEN analyse orbital_stability_and_Floquet multipliers
 E8: IF external_inputs_present THEN consider input_to_state_stability
 E9: IF uncertainty_present THEN consider robust_stability
}

RECURSION {
 ENTRY = Stability
 EXPAND(Stability) -> [
  LyapunovStability, AsymptoticStability, ExponentialStability,
  LocalStability, GlobalStability, OrbitalStability, InputToStateStability,
  RobustStability, StructuralStability, EquilibriumPoint, FixedPoint,
  PeriodicOrbit, LimitCycle, InvariantSet, Perturbation, LyapunovFunction,
  DirectLyapunovMethod, IndirectLyapunovMethod, Linearisation,
  JacobianMatrix, Eigenvalue, FloquetMultiplier, RegionOfAttraction,
  BasinOfAttraction, LaSalleInvariancePrinciple, SmallGainTheorem
 ]
 STOP_CONDITION =
  "Stop when reference structure, stability notion, perturbation class, method, valid domain, convergence behaviour and certification status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Dynamical Systems"
 IMMEDIATE_CONTEXTS = [
  "Dynamical Systems Ontology",
  "Phase Space Ontology",
  "Ordinary Differential Equation Ontology",
  "Stability Analysis Ontology"
 ]
 RELATED_ONTOLOGIES = [
  "Dynamical Systems Ontology",
  "Phase Space Ontology",
  "Ordinary Differential Equation Ontology",
  "Partial Differential Equation Ontology",
  "Stability Analysis Ontology",
  "Control Theory Ontologies",
  "Robust Control Ontology",
  "Nonlinear Analysis Ontologies",
  "Bifurcation Theory Ontologies",
  "Chaos Theory Ontologies",
  "Numerical Analysis Ontologies"
 ]
 ROUTE_RULES = [
  "Stability -> Dynamical Systems Ontology for core evolution semantics",
  "Stability -> Phase Space for invariant geometry and attraction regions",
  "Stability -> ODE Ontology for finite-dimensional continuous dynamics",
  "Stability -> PDE Ontology for distributed-parameter stability",
  "Stability -> Control Theory for closed-loop stability",
  "Stability -> Robust Control for uncertainty-aware stability",
  "Stability -> Bifurcation Theory for loss-of-stability transitions",
  "Stability -> Numerical Analysis for computational assessment"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible stability notions, certificates or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve perturbation-response semantics.",
  "Preserve Lyapunov-versus-asymptotic distinction.",
  "Preserve local-versus-global distinction.",
  "Preserve continuous-versus-discrete criteria.",
  "Preserve equilibrium-versus-orbital stability distinction.",
  "Preserve region-of-attraction semantics.",
  "Preserve robust and input-to-state distinctions.",
  "Preserve machine-readable stability semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Stability describes whether a dynamical state, trajectory, orbit or invariant set remains close to or attracts nearby behaviour after perturbation."
 LYAPUNOV_SUMMARY =
  "Lyapunov stability requires sufficiently small perturbations to remain sufficiently small."
 ASYMPTOTIC_SUMMARY =
  "Asymptotic stability adds convergence to the reference structure."
 EXPONENTIAL_SUMMARY =
  "Exponential stability provides a quantitative exponential convergence bound."
 LOCAL_GLOBAL_SUMMARY =
  "Local stability applies near the reference structure, while global stability applies throughout the declared domain."
 ORBITAL_SUMMARY =
  "Orbital stability concerns perturbations of periodic orbits rather than a fixed phase point."
 REGION_SUMMARY =
  "A region of attraction identifies initial states guaranteed to converge to an asymptotically stable invariant set."
 ROBUST_SUMMARY =
  "Robust stability requires stability across the declared uncertainty family."
}

FLOW {
 INPUT
 -> detect_dynamical_stability_context
 -> identify reference_equilibrium_fixedPoint_periodicOrbit_or_invariantSet
 -> identify continuous_discrete_time_perturbation_class_and_domain
 -> classify Lyapunov_asymptotic_exponential_local_global_orbital_ISS_structural_or_robust stability
 -> select eigenvalue_linearisation_Lyapunov_LaSalle_Floquet_regionOfAttraction_smallGain_or_robust pathway
 -> compute_or_verify stability conditions
 -> determine valid domain_and_convergence behaviour
 -> distinguish local_global_nominal_robust_and_analytical_numerical evidence
 -> route to dynamical_systems phase_space ODE PDE control robust_control nonlinear_analysis bifurcation chaos_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Stability Ontology defines perturbation response in dynamical systems. It supports equilibrium and fixed-point stability, Lyapunov, asymptotic and exponential stability, local and global stability, orbital stability, input-to-state stability, robust and structural stability, Lyapunov functions, direct and indirect methods, linearisation, eigenvalue and Floquet criteria, LaSalle invariance, regions and basins of attraction, comparison principles and small-gain reasoning."
}

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

}

END_FLOW