Bifurcation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Bifurcation Ontology within the Kireth Research Semantic Library. Bifurcation theory studies qualitative changes in dynamical behaviour as parameters vary, including saddle-node, transcritical, pitchfork, Hopf, period-doubling, Neimark-Sacker and global bifurcations.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Bifurcation
 SYMBOLIC_FORMS = ["x_dot=f(x,mu)", "F(x,mu)=0", "Re(lambda(mu_c))=0"]
 ENTITY_CLASS = ParameterDrivenQualitativeDynamicsFramework
 PRIMARY_CONTEXT = DynamicalSystems
 DEFINITION = "A bifurcation is a qualitative change in the structure or stability of a dynamical system caused by variation of one or more parameters."
}

SCOPE {
 INCLUDE = [Bifurcation, BifurcationTheory, Parameter, CriticalParameter, BifurcationPoint, EquilibriumBranch, PeriodicOrbitBranch, LocalBifurcation, GlobalBifurcation, Codimension, SaddleNodeBifurcation, TranscriticalBifurcation, PitchforkBifurcation, HopfBifurcation, PeriodDoublingBifurcation, NeimarkSackerBifurcation, HomoclinicBifurcation, HeteroclinicBifurcation, BorderCollisionBifurcation, NormalForm, CentreManifold, EigenvalueCrossing, FloquetMultiplier, Continuation, NumericalContinuation, BifurcationDiagram, Hysteresis, Multistability, RouteToChaos]
 DISAMBIGUATION_RULE = "When parameter variation causes a qualitative change in equilibria, periodic orbits, stability, invariant sets or long-term dynamics, interpret the context as bifurcation theory."
}

CORE_CONCEPTS {
 Bifurcation = "Parameter-driven qualitative change in dynamical structure."
 CriticalParameter = "Parameter value at which qualitative dynamics change."
 SaddleNodeBifurcation = "Creation or destruction of two colliding invariant solutions."
 TranscriticalBifurcation = "Intersecting branches exchange stability."
 PitchforkBifurcation = "Symmetry-associated branch splitting or merging."
 HopfBifurcation = "Complex eigenvalue crossing associated with periodic behaviour."
 PeriodDoublingBifurcation = "Periodic orbit loses stability to an orbit of twice the period."
 NeimarkSackerBifurcation = "Discrete complex multiplier crossing associated with an invariant circle."
 NormalForm = "Simplified local representation retaining essential bifurcation structure."
}

RELATIONS {
 Bifurcation --is_child_of--> DynamicalSystems
 Bifurcation --depends_on--> Parameter
 CriticalParameter --defines--> BifurcationPoint
 SaddleNodeBifurcation --is_type_of--> LocalBifurcation
 TranscriticalBifurcation --is_type_of--> LocalBifurcation
 PitchforkBifurcation --is_type_of--> LocalBifurcation
 HopfBifurcation --is_type_of--> LocalBifurcation
 HomoclinicBifurcation --is_type_of--> GlobalBifurcation
 HeteroclinicBifurcation --is_type_of--> GlobalBifurcation
 CentreManifold --supports--> LocalBifurcationReduction
 NormalForm --classifies--> LocalBifurcation
 Continuation --tracks--> EquilibriumBranch
 Continuation --tracks--> PeriodicOrbitBranch
 PeriodDoublingBifurcation --may_contribute_to--> RouteToChaos
}

OPERATORS {
 PARAMETERISED_FLOW(f,x,mu) := x_dot=f(x,mu)
 EQUILIBRIUM_BRANCH(f,x,mu) := f(x,mu)=0
 SADDLE_NODE_NORMAL_FORM(x,mu) := x_dot=mu-x^2
 TRANSCRITICAL_NORMAL_FORM(x,mu) := x_dot=mu*x-x^2
 PITCHFORK_NORMAL_FORM(x,mu) := x_dot=mu*x-x^3
 HOPF_NORMAL_FORM(r,mu,beta) := r_dot=mu*r-beta*r^3
 CONTINUOUS_CRITICALITY(lambda,mu_c) := Re(lambda(mu_c))=0
 DISCRETE_CRITICALITY(lambda,mu_c) := abs(lambda(mu_c))=1
 PERIOD_DOUBLING_CONDITION(mu) := Floquet_multiplier_crosses_minus_one
 NEIMARK_SACKER_CONDITION(mu) := complex_multiplier_pair_crosses_unit_circle
}

DEFINITIONS {
 D1 = "Bifurcations are qualitative rather than merely quantitative parameter responses."
 D2 = "Local bifurcations concern dynamics near a critical invariant object."
 D3 = "Global bifurcations involve large-scale invariant structures."
 D4 = "Codimension counts independent parameter conditions generically required."
 D5 = "Saddle-node bifurcations create or destroy pairs of equilibria."
 D6 = "Transcritical bifurcations exchange stability between branches."
 D7 = "Pitchfork bifurcations commonly arise in symmetric systems."
 D8 = "Hopf bifurcations create or destroy periodic behaviour."
 D9 = "Period doubling is an important route toward chaos."
 D10 = "Numerical continuation tracks invariant-solution branches through parameter space."
}

AXIOMS {
 A1_PARAMETER_DEPENDENCE: Bifurcation requires a parameterised dynamical system
 A2_QUALITATIVE_CHANGE: Bifurcation requires qualitative dynamical change
 A3_CRITICALITY: Local bifurcations occur at critical state-parameter configurations
 A4_STABILITY_RELATION: Many bifurcations involve stability change
 A5_NORMAL_FORM: Generic local bifurcations admit reduced normal forms under suitable assumptions
 A6_CENTRE_MANIFOLD: Nonhyperbolic local dynamics can often be reduced to a centre manifold
 A7_CONTINUOUS_DISCRETE_DISTINCTION: Continuous eigenvalue and discrete multiplier crossings use different criteria
 A8_GLOBAL_STRUCTURE: Global bifurcations cannot generally be inferred from local eigenvalues alone
}

REASONING {
 R1_CONTEXT { IDENTIFY = parameters_invariant_objects_and_stability }
 R2_BRANCH_TRACKING { CONTINUE = equilibrium_or_periodic_branch; RECORD = stability }
 R3_CRITICAL_DETECTION { DETECT = zero_eigenvalue_imaginary_crossing_minusOne_or_unitCircle_crossing }
 R4_LOCAL_CLASSIFICATION { CLASSIFY = saddleNode_transcritical_pitchfork_Hopf_periodDoubling_or_NeimarkSacker }
 R5_HOPF { CHECK = complex_pair_crossing_and_transversality; CLASSIFY = supercritical_or_subcritical }
 R6_GLOBAL { DETECT = homoclinic_heteroclinic_collision_or_crisis }
 R7_HYSTERESIS { IDENTIFY = overlapping_stable_branches_and_folds }
 R8_CHAOS_ROUTE { TRACK = periodDoubling_intermittency_torus_breakdown_or_crisis }
 R9_CONTINUATION { APPLY = predictor_corrector_or_pseudoArclength }
 R10_VALIDATE { CHECK = criticality_transversality_branch_structure_and_resolution }
}

CONSTRAINTS {
 C1 = "Quantitative parameter change alone is not necessarily a bifurcation."
 C2 = "Local classification requires nondegeneracy conditions."
 C3 = "Critical eigenvalues alone may be insufficient without transversality."
 C4 = "Pitchfork classification generally depends on symmetry or equivalent structure."
 C5 = "Global bifurcations cannot be established from local linearisation alone."
 C6 = "Continuation step size can miss narrow bifurcations."
 C7 = "Branch stability must be distinguished from branch existence."
 C8 = "Subcritical and supercritical bifurcations have different hysteresis implications."
 C9 = "Finite parameter scans can miss unstable branches."
 C10 = "Routes to chaos require dynamical evidence beyond irregular trajectories."
}

SEMANTIC_EQUIVALENCE {
 "Bifurcation" <=> "DynamicalBifurcation"
 "SaddleNodeBifurcation" <=> "FoldBifurcation"
 "PeriodDoublingBifurcation" <=> "FlipBifurcation"
 "NeimarkSackerBifurcation" <=> "TorusBifurcation"
 "CriticalParameter" <=> "BifurcationParameterValue"
}

VALIDATION {
 V1 = "Control parameter is explicit."
 V2 = "Invariant branch is identified."
 V3 = "Stability is tracked along the branch."
 V4 = "Critical eigenvalue or multiplier behaviour is verified."
 V5 = "Nondegeneracy and transversality assumptions are checked."
 V6 = "Local and global bifurcations are distinguished."
 V7 = "Supercritical and subcritical cases are distinguished."
 V8 = "Unstable branches are included where possible."
 V9 = "Continuation resolution is documented."
 V10 = "Classification is supported by structural evidence."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 BIFURCATION_DEFINITION = 0.99
 BRANCH_TRACKING_REASONING = 0.98
 CRITICAL_POINT_REASONING = 0.99
 LOCAL_CLASSIFICATION_REASONING = 0.98
 GLOBAL_BIFURCATION_REASONING = 0.97
 VALIDATION_REASONING = 0.99
 POLICY = "Reduce confidence when parameterisation, branch structure, eigenvalues, multipliers, symmetry, transversality, continuation resolution or global geometry are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [SemanticSearch, OntologyGrounding, BifurcationReasoning, SaddleNodeReasoning, TranscriticalReasoning, PitchforkReasoning, HopfReasoning, PeriodDoublingReasoning, NeimarkSackerReasoning, HomoclinicReasoning, HeteroclinicReasoning, NormalFormReasoning, CentreManifoldReasoning, ContinuationReasoning, BifurcationDiagramReasoning, HysteresisReasoning, MultistabilityReasoning, RouteToChaosReasoning, KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference]
 NORMALISE = ["bifurcation"->Bifurcation, "fold bifurcation"->SaddleNodeBifurcation, "period doubling"->PeriodDoublingBifurcation, "flip bifurcation"->PeriodDoublingBifurcation, "Neimark-Sacker"->NeimarkSackerBifurcation]
 INTERPRETATION_ORDER = [detect_parameterised_context, identify_branches_and_stability, track_branches, detect_criticality, classify_bifurcation, reduce_by_normal_form_when_needed, analyse_hysteresis_or_chaos_route, validate_conditions]
}

EXECUTION_RULES {
 E1: IF query_mentions_bifurcation OR parameter_variation_changes_qualitative_dynamics THEN activate(BifurcationOntology)
 E2: IF simple_zero_eigenvalue_and_fold_conditions THEN consider_SaddleNodeBifurcation
 E3: IF branches_cross_and_exchange_stability THEN consider_TranscriticalBifurcation
 E4: IF symmetry_and_branch_splitting THEN consider_PitchforkBifurcation
 E5: IF complex_pair_crosses_imaginary_axis THEN consider_HopfBifurcation
 E6: IF periodic_multiplier_crosses_minus_one THEN consider_PeriodDoublingBifurcation
 E7: IF discrete_complex_pair_crosses_unit_circle THEN consider_NeimarkSackerBifurcation
 E8: IF homoclinic_or_heteroclinic_connection_changes THEN classify_GlobalBifurcation
}

RECURSION {
 ENTRY = Bifurcation
 EXPAND(Bifurcation) -> [Parameter, CriticalParameter, BifurcationPoint, EquilibriumBranch, PeriodicOrbitBranch, LocalBifurcation, GlobalBifurcation, Codimension, SaddleNodeBifurcation, TranscriticalBifurcation, PitchforkBifurcation, HopfBifurcation, PeriodDoublingBifurcation, NeimarkSackerBifurcation, HomoclinicBifurcation, HeteroclinicBifurcation, NormalForm, CentreManifold, FloquetMultiplier, Continuation, NumericalContinuation, BifurcationDiagram, Hysteresis, Multistability, RouteToChaos]
 STOP_CONDITION = "Stop when parameterisation, invariant branches, stability, criticality, bifurcation type, normal form, continuation and global structure are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Dynamical Systems"
 RELATED_ONTOLOGIES = ["Dynamical Systems Ontology", "Stability Ontology", "Phase Space Ontology", "Chaos Theory Ontology", "Ordinary Differential Equation Ontology", "Nonlinear Analysis Ontologies", "Centre Manifold Ontologies", "Normal Form Ontologies", "Numerical Analysis Ontologies"]
 ROUTE_RULES = [
  "Bifurcation -> Dynamical Systems Ontology for state-evolution semantics",
  "Bifurcation -> Stability for stability-loss conditions",
  "Bifurcation -> Phase Space for invariant geometry",
  "Bifurcation -> Chaos Theory for routes to chaos",
  "Bifurcation -> ODE Ontology for continuous-time models",
  "Bifurcation -> Numerical Analysis for continuation and detection"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible bifurcation classes, continuation methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [preserve_parameter_driven_semantics, preserve_local_global_distinction, preserve_continuous_discrete_criticality, preserve_normal_form_semantics, preserve_supercritical_subcritical_distinction, preserve_branch_stability, preserve_route_to_chaos, preserve_machine_readable_semantics]
}

OUTPUT {
 CANONICAL_DEFINITION = "A bifurcation is a qualitative change in dynamical structure or stability caused by parameter variation."
 LOCAL_SUMMARY = "Local bifurcations include saddle-node, transcritical, pitchfork, Hopf, period-doubling and Neimark-Sacker types."
 GLOBAL_SUMMARY = "Global bifurcations involve large-scale invariant structures such as homoclinic and heteroclinic connections."
 CONTINUATION_SUMMARY = "Numerical continuation tracks solution branches and detects critical points as parameters vary."
 CHAOS_SUMMARY = "Sequences of bifurcations can provide routes to chaotic dynamics."
}

FLOW {
 INPUT
 -> detect_parameterised_dynamical_context
 -> identify control_parameters_equilibria_periodic_orbits_and_stability
 -> continue invariant_solution_branches
 -> detect critical_eigenvalue_or_multiplier_conditions
 -> classify local_or_global_bifurcation
 -> apply centre_manifold_and_normal_form_reduction_when_required
 -> determine supercritical_subcritical_hysteresis_and_multistability structure
 -> analyse routes_to_chaos_when_relevant
 -> validate nondegeneracy_transversality_and_numerical_resolution
 -> route to dynamical_systems stability phase_space chaos ODE nonlinear_analysis centre_manifold normal_form_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Bifurcation Ontology defines parameter-driven qualitative changes in dynamical systems. It supports critical parameters and bifurcation points, equilibrium and periodic-orbit branches, local and global bifurcations, saddle-node, transcritical, pitchfork, Hopf, period-doubling, Neimark-Sacker, homoclinic and heteroclinic bifurcations, normal forms, centre manifolds, eigenvalue and Floquet criticality, numerical continuation, bifurcation diagrams, hysteresis, multistability and routes to chaos."
}

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

}

END_FLOW