Resonance Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Resonance Ontology within the Kireth Research Semantic Library. Resonance in dynamical systems describes amplified or otherwise strongly modified responses that occur when external forcing, parametric modulation, internal coupling or interacting modes align with characteristic frequencies or timescales of a system. This ontology supports linear resonance, nonlinear resonance, primary resonance, superharmonic and subharmonic resonance, internal resonance, parametric resonance, detuning, bandwidth, quality factor, resonance curves, jump phenomena, hysteresis, modal interaction, frequency response, stability of periodic responses and relationships with oscillation, nonlinear dynamics, bifurcation theory, control theory and signal processing.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Resonance
 SYMBOLIC_FORMS = [
  "omega approximately omega_n",
  "A(omega)=F0/m / sqrt((omega_n^2-omega^2)^2+(2*zeta*omega_n*omega)^2)",
  "n*omega_1 approximately m*omega_2"
 ]
 ENTITY_CLASS = FrequencyInteractionAmplificationFramework
 PRIMARY_CONTEXT = Oscillation
 DEFINITION = "Resonance is a dynamical phenomenon in which forcing, modulation or modal interaction aligns with characteristic system frequencies or timescales and produces a strongly enhanced or qualitatively altered response."
 SEMANTIC_ROLE = "Provides machine-readable semantics for forced, nonlinear, internal and parametric resonance, detuning, resonance bandwidth, quality factor, stability, bifurcation and modal interaction."
}

SCOPE {
 INCLUDE = [
  Resonance,
  LinearResonance,
  ForcedResonance,
  PrimaryResonance,
  NonlinearResonance,
  SuperharmonicResonance,
  SubharmonicResonance,
  InternalResonance,
  ParametricResonance,
  CombinationResonance,
  ModalResonance,
  NaturalFrequency,
  ForcingFrequency,
  ExcitationFrequency,
  Detuning,
  FrequencyMismatch,
  ResonanceFrequency,
  ResonancePeak,
  ResonanceCurve,
  FrequencyResponse,
  AmplitudeResponse,
  PhaseResponse,
  Bandwidth,
  HalfPowerBandwidth,
  QualityFactor,
  Damping,
  DampingRatio,
  EnergyTransfer,
  ModeCoupling,
  ModalInteraction,
  ModeShape,
  JumpPhenomenon,
  Hysteresis,
  BackboneCurve,
  FrequencyLocking,
  PhaseLocking,
  MathieuEquation,
  DuffingOscillator,
  VanDerPolOscillator,
  HarmonicBalance,
  MultipleScalesMethod,
  AveragingMethod,
  FloquetTheory,
  StabilityBoundary,
  ArnoldTongue,
  Bifurcation,
  PeriodicResponse
 ]

 EXCLUDE_AS_PRIMARY = [
  GenericLargeAmplitudeWithoutFrequencyInteraction,
  PureSpectralPeakWithoutDynamicalResponseContext,
  RandomExcitationWithoutResonantStructure,
  StaticAmplificationWithoutOscillatoryMechanism
 ]

 DISAMBIGUATION_RULE =
  "When system response is strongly enhanced or structurally altered because forcing, modulation or modal frequencies align with characteristic frequencies or rational frequency relationships, interpret the context as resonance."
}

CORE_CONCEPTS {

 Resonance {
  definition = "Strong dynamical response caused by frequency or timescale alignment."
 }

 PrimaryResonance {
  definition = "Resonance occurring when forcing frequency is close to a principal natural frequency."
 }

 NonlinearResonance {
  definition = "Resonance in a nonlinear system where amplitude-dependent frequency, multiple response branches or jump phenomena may occur."
 }

 SuperharmonicResonance {
  definition = "Resonance caused when an integer multiple of the forcing frequency aligns with a natural frequency."
 }

 SubharmonicResonance {
  definition = "Resonance in which the system responds strongly at a fractional multiple of the forcing frequency."
 }

 InternalResonance {
  definition = "Resonance between system modes whose natural frequencies satisfy an approximate low-order rational relation."
 }

 ParametricResonance {
  definition = "Resonance caused by periodic variation of a system parameter rather than direct additive forcing."
 }

 Detuning {
  definition = "The mismatch between forcing, modal or parametric excitation frequency and a resonant frequency condition."
 }

 QualityFactor {
  definition = "A measure of resonance sharpness related to stored versus dissipated energy."
 }

 ResonanceCurve {
  definition = "A relation between steady-state response amplitude and excitation frequency or detuning."
 }
}

RELATIONS {
 Resonance --is_child_of--> Oscillation
 ForcedResonance --is_type_of--> Resonance
 NonlinearResonance --is_type_of--> Resonance
 InternalResonance --is_type_of--> Resonance
 ParametricResonance --is_type_of--> Resonance
 NaturalFrequency --interacts_with--> ForcingFrequency
 Detuning --measures--> FrequencyMismatch
 Damping --reduces--> ResonancePeak
 Damping --increases--> ResonanceBandwidth
 QualityFactor --inversely_relates_to--> Damping
 InternalResonance --requires--> ModalFrequencyRelation
 ParametricResonance --depends_on--> ParameterModulation
 ModalInteraction --enables--> EnergyTransfer
 NonlinearResonance --may_exhibit--> JumpPhenomenon
 NonlinearResonance --may_exhibit--> Hysteresis
 FloquetTheory --analyses--> ParametricResonanceStability
 Bifurcation --may_bound--> ResonantResponseBranch
}

OPERATORS {
 LINEAR_FORCED_OSCILLATOR(x,zeta,omega_n,F0,m,omega) :=
  x''+2*zeta*omega_n*x'+omega_n^2*x=(F0/m)*cos(omega*t)

 LINEAR_AMPLITUDE_RESPONSE(F0,m,omega_n,zeta,omega) :=
  A=(F0/m)/sqrt((omega_n^2-omega^2)^2+(2*zeta*omega_n*omega)^2)

 PHASE_RESPONSE(omega_n,zeta,omega) :=
  phi=atan2(2*zeta*omega_n*omega,omega_n^2-omega^2)

 DETUNING(omega,omega_r) :=
  sigma=omega-omega_r

 NORMALISED_FREQUENCY_RATIO(omega,omega_n) :=
  r=omega/omega_n

 QUALITY_FACTOR(zeta) :=
  Q approximately 1/(2*zeta)
  for lightly_damped_linear_systems

 HALF_POWER_BANDWIDTH(f1,f2) :=
  Delta_f=f2-f1

 INTERNAL_RESONANCE_CONDITION(omega_i,omega_j,m,n) :=
  m*omega_i approximately n*omega_j

 PARAMETRIC_RESONANCE_CONDITION(omega_p,omega_n) :=
  omega_p approximately 2*omega_n/k
  for appropriate integer k

 SUPERHARMONIC_CONDITION(omega,omega_n,n) :=
  n*omega approximately omega_n

 SUBHARMONIC_CONDITION(omega,omega_n,n) :=
  omega approximately n*omega_n
}

DEFINITIONS {
 D1 = "Resonance depends on interaction between excitation and intrinsic system timescales."
 D2 = "In linear systems, damping controls resonance peak sharpness."
 D3 = "Primary resonance occurs near a principal natural frequency."
 D4 = "Nonlinear resonance can produce amplitude-dependent frequency response."
 D5 = "Internal resonance transfers energy between coupled modes."
 D6 = "Parametric resonance arises from periodic parameter variation."
 D7 = "Detuning measures departure from exact resonance."
 D8 = "Jump phenomena arise from multiple stable and unstable nonlinear response branches."
 D9 = "Quality factor characterises resonance sharpness in lightly damped systems."
 D10 = "Resonant periodic responses require their own stability analysis."
}

AXIOMS {
 A1_FREQUENCY_INTERACTION:
  Resonance requires interaction between characteristic and excitation frequencies or timescales

 A2_RESPONSE_ENHANCEMENT:
  Resonance produces amplified or qualitatively altered response relative to nearby nonresonant conditions

 A3_DAMPING:
  Dissipation limits resonance amplitude in ordinary stable linear systems

 A4_NONLINEAR_SHIFT:
  Nonlinearity can shift resonant frequencies as response amplitude changes

 A5_INTERNAL_COUPLING:
  InternalResonance requires coupling between modes satisfying an approximate rational frequency relationship

 A6_PARAMETRIC_MODULATION:
  ParametricResonance requires time-varying system parameters

 A7_STABILITY:
  Large resonant amplitude alone does not establish stability of the resonant response

 A8_BIFURCATION:
  Nonlinear resonant branches can appear disappear or exchange stability through bifurcation
}

REASONING {

 R1_RESONANCE_CONTEXT {
  INPUT = oscillatory_system
  IDENTIFY = natural_frequencies
  IDENTIFY = forcing_or_modulation frequencies
  IDENTIFY = damping
  IDENTIFY = nonlinearities_and_coupling
  RETURN = resonance_context
 }

 R2_LINEAR_RESONANCE {
  INPUT = linear_forced_system
  COMPUTE = frequency_response
  IDENTIFY = resonance_peak_bandwidth_phase
  RETURN = linear_resonance_profile
 }

 R3_DETUNING_ANALYSIS {
  INPUT = excitation_and_resonant frequencies
  COMPUTE = detuning
  RETURN = proximity_to_resonance
 }

 R4_NONLINEAR_RESONANCE {
  INPUT = nonlinear_oscillator
  SELECT = harmonic_balance_multiple_scales_averaging_or_continuation
  COMPUTE = amplitude_frequency branches
  IDENTIFY = folds_jumps_and_hysteresis
  RETURN = nonlinear_resonance_profile
 }

 R5_INTERNAL_RESONANCE {
  INPUT = coupled_modes
  CHECK = low_order_rational frequency relation
  ANALYSE = modal_energy_transfer_and_phase locking
  RETURN = internal_resonance_profile
 }

 R6_PARAMETRIC_RESONANCE {
  INPUT = periodically_modulated_parameter
  DERIVE = Mathieu_or_equivalent form
  ANALYSE = Floquet multipliers_and_instability tongues
  RETURN = parametric_resonance_profile
 }

 R7_STABILITY_ANALYSIS {
  INPUT = resonant_periodic_response
  COMPUTE = Floquet_multipliers_or_Poincare derivative
  RETURN = resonant_response_stability
 }

 R8_BIFURCATION_REASONING {
  INPUT = resonance_branch_and_parameter
  TRACK = response_branches
  DETECT = folds_Hopf_periodDoubling_or_other transitions
  RETURN = resonance_bifurcation_profile
 }

 R9_ENERGY_TRANSFER {
  INPUT = coupled_or_forced system
  COMPUTE = modal_or_input_energy flow
  RETURN = resonance_energy profile
 }

 R10_VALIDATION {
  INPUT = resonance_claim
  CHECK = frequency_relation_damping_branch_structure_stability_transients_and_numerical resolution
  RETURN = validated_resonance_status
 }
}

CONSTRAINTS {
 C1 = "Large amplitude alone does not establish resonance."
 C2 = "Natural and forcing frequencies must not be conflated."
 C3 = "Linear resonance formulas may fail for strongly nonlinear systems."
 C4 = "Damping changes resonance peak location and bandwidth."
 C5 = "Internal resonance requires modal coupling as well as frequency commensurability."
 C6 = "Parametric resonance differs from direct forced resonance."
 C7 = "Jump phenomena require multiple nonlinear response branches."
 C8 = "Finite frequency sweeps can miss unstable resonant branches."
 C9 = "Numerical integration step size can distort resonant amplitude and phase."
 C10 = "Transient amplification must be distinguished from steady resonant response."
}

SEMANTIC_EQUIVALENCE {
 "Resonance" <=> "DynamicalResonance"
 "PrimaryResonance" <=> "FundamentalForcedResonance"
 "InternalResonance" <=> "ModalInternalResonance"
 "ParametricResonance" <=> "ParameterExcitedResonance"
 "Detuning" <=> "FrequencyDetuning"
 "QualityFactor" <=> "ResonanceQFactor"
}

VALIDATION {
 V1 = "Natural frequency or modal frequencies are identified."
 V2 = "Excitation or modulation frequency is explicit."
 V3 = "Damping is represented."
 V4 = "Transient and steady-state responses are distinguished."
 V5 = "Nonlinear branch structure is checked where relevant."
 V6 = "Internal resonance includes a coupling mechanism."
 V7 = "Parametric resonance includes parameter modulation."
 V8 = "Periodic-response stability is checked."
 V9 = "Frequency sweep or continuation resolution is documented."
 V10 = "Numerical results are checked for time-step and solver sensitivity."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 RESONANCE_DEFINITION = 0.99
 LINEAR_RESONANCE_REASONING = 0.99
 DETUNING_REASONING = 0.99
 NONLINEAR_RESONANCE_REASONING = 0.98
 INTERNAL_RESONANCE_REASONING = 0.98
 PARAMETRIC_RESONANCE_REASONING = 0.98
 STABILITY_REASONING = 0.98
 BIFURCATION_REASONING = 0.98
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when natural frequencies, damping, excitation type, coupling, branch structure, modulation frequency, stability or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ResonanceReasoning,
  LinearResonanceReasoning,
  ForcedResonanceReasoning,
  NonlinearResonanceReasoning,
  InternalResonanceReasoning,
  ParametricResonanceReasoning,
  DetuningReasoning,
  FrequencyResponseReasoning,
  QualityFactorReasoning,
  HarmonicBalanceReasoning,
  MultipleScalesReasoning,
  AveragingReasoning,
  FloquetReasoning,
  BifurcationReasoning,
  ModalInteractionReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "resonance" -> Resonance
  "forced resonance" -> ForcedResonance
  "primary resonance" -> PrimaryResonance
  "nonlinear resonance" -> NonlinearResonance
  "internal resonance" -> InternalResonance
  "parametric resonance" -> ParametricResonance
  "detuning" -> Detuning
  "quality factor" -> QualityFactor
  "Q factor" -> QualityFactor
  "resonance curve" -> ResonanceCurve
 }

 INTERPRETATION_ORDER = [
  detect_resonance_context,
  identify natural_modal_forcing_and_modulation frequencies,
  identify damping_nonlinearity_and_coupling,
  classify forced_primary_superharmonic_subharmonic_internal_parametric_or_combination resonance,
  compute amplitude_phase_bandwidth_detuning_or_quality factor,
  if_nonlinear derive response branches_and_stability,
  if_parametric apply Floquet_and_instability tongue analysis,
  if_internal analyse modal_energy_transfer,
  validate against transients_numerical artefacts_and_branch omissions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("resonance")
  OR frequency_alignment_changes_dynamical_response
  THEN activate(ResonanceOntology)

 E2:
  IF forcing_frequency_near_natural_frequency
  THEN consider PrimaryResonance

 E3:
  IF integer_multiple_of_forcing_near_natural_frequency
  THEN consider SuperharmonicResonance

 E4:
  IF response_frequency_is_fraction_of_forcing
  THEN consider SubharmonicResonance

 E5:
  IF modal_frequencies_have_low_order_rational relation
  AND modes_are_coupled
  THEN consider InternalResonance

 E6:
  IF parameter_modulation_present
  THEN consider ParametricResonance

 E7:
  IF nonlinear_response_has_folds
  THEN analyse jump_and_hysteresis behaviour

 E8:
  IF resonant_periodic_response
  THEN assess Floquet_or_Poincare stability

 E9:
  IF branch_tracking_required
  THEN use numerical_continuation
}

RECURSION {
 ENTRY = Resonance

 EXPAND(Resonance) -> [
  LinearResonance,
  ForcedResonance,
  PrimaryResonance,
  NonlinearResonance,
  SuperharmonicResonance,
  SubharmonicResonance,
  InternalResonance,
  ParametricResonance,
  CombinationResonance,
  ModalResonance,
  NaturalFrequency,
  ForcingFrequency,
  Detuning,
  ResonanceFrequency,
  ResonancePeak,
  ResonanceCurve,
  FrequencyResponse,
  AmplitudeResponse,
  PhaseResponse,
  Bandwidth,
  HalfPowerBandwidth,
  QualityFactor,
  Damping,
  EnergyTransfer,
  ModeCoupling,
  ModalInteraction,
  JumpPhenomenon,
  Hysteresis,
  BackboneCurve,
  FrequencyLocking,
  PhaseLocking,
  MathieuEquation,
  DuffingOscillator,
  HarmonicBalance,
  MultipleScalesMethod,
  AveragingMethod,
  FloquetTheory,
  ArnoldTongue,
  Bifurcation,
  PeriodicResponse
 ]

 EXPAND(ResonanceClasses) -> [
  PrimaryResonance,
  SuperharmonicResonance,
  SubharmonicResonance,
  InternalResonance,
  ParametricResonance,
  CombinationResonance
 ]

 EXPAND(AnalysisMethods) -> [
  FrequencyResponse,
  HarmonicBalance,
  MultipleScalesMethod,
  AveragingMethod,
  FloquetTheory,
  NumericalContinuation
 ]

 STOP_CONDITION =
  "Stop when resonance class, relevant frequencies, detuning, damping, response amplitude and phase, branch structure, stability and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Dynamical Systems"

 IMMEDIATE_CONTEXTS = [
  "Oscillation Ontology",
  "Nonlinear Dynamics Ontology",
  "Stability Ontology",
  "Bifurcation Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Dynamical Systems Ontology",
  "Oscillation Ontology",
  "Nonlinear Dynamics Ontology",
  "Stability Ontology",
  "Bifurcation Ontology",
  "Chaos Theory Ontology",
  "Ordinary Differential Equation Ontology",
  "Control Theory Ontologies",
  "Signal Processing Ontology",
  "Fourier Analysis Ontology",
  "Perturbation Method Ontologies",
  "Numerical Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Resonance -> Oscillation Ontology for frequency-amplitude-phase semantics",
  "Resonance -> Nonlinear Dynamics for amplitude-dependent resonant branches",
  "Resonance -> Stability for resonant periodic-response stability",
  "Resonance -> Bifurcation for folds, Hopf and period-doubling of response branches",
  "Resonance -> Chaos Theory for resonance-driven routes to chaos",
  "Resonance -> ODE Ontology for governing oscillator equations",
  "Resonance -> Control Theory for resonance suppression and shaping",
  "Resonance -> Signal Processing and Fourier Analysis for spectral interpretation",
  "Resonance -> Perturbation Methods for multiple-scales and averaging analysis",
  "Resonance -> Numerical Analysis for continuation and simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible resonance classes, interaction mechanisms or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve frequency-interaction semantics.",
  "Preserve direct-forcing versus parametric distinction.",
  "Preserve linear-versus-nonlinear resonance distinction.",
  "Preserve internal-resonance coupling requirements.",
  "Preserve detuning semantics.",
  "Preserve branch-stability semantics.",
  "Preserve transient-versus-steady-state distinction.",
  "Preserve machine-readable resonance semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Resonance is a dynamical response produced when forcing, modulation or modal interactions align with characteristic frequencies or timescales."

 LINEAR_SUMMARY =
  "Linear resonance produces a frequency-response peak whose amplitude and bandwidth depend strongly on damping."

 NONLINEAR_SUMMARY =
  "Nonlinear resonance can shift with amplitude and generate multiple branches, jumps and hysteresis."

 INTERNAL_SUMMARY =
  "Internal resonance transfers energy between coupled modes with approximately rationally related frequencies."

 PARAMETRIC_SUMMARY =
  "Parametric resonance is driven by periodic variation of a system parameter and is commonly analysed using Floquet theory."

 DETUNING_SUMMARY =
  "Detuning measures departure from exact resonance and strongly influences amplitude and phase response."

 STABILITY_SUMMARY =
  "Resonant periodic solutions require stability analysis independently of response amplitude."

 VALIDATION_SUMMARY =
  "Resonance claims should distinguish steady response from transients and should account for damping, nonlinearity, coupling and numerical sensitivity."
}

FLOW {
 INPUT
 -> detect_resonance_context
 -> identify natural_modal_forcing_modulation frequencies_damping_nonlinearity_and_coupling
 -> classify linear_forced_primary_superharmonic_subharmonic_internal_parametric_or_combination resonance
 -> compute detuning_frequency_response_amplitude_phase_bandwidth_and_quality factor
 -> if_nonlinear derive amplitude_frequency_branches_backbone_curve_jumps_and_hysteresis
 -> if_internal analyse modal_coupling_phase locking_and_energy transfer
 -> if_parametric derive instability tongues_and_Floquet stability
 -> assess bifurcations_and_periodic response stability
 -> route to oscillation nonlinear_dynamics stability bifurcation chaos ODE control signal_processing Fourier perturbation_methods_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Resonance Ontology defines strong dynamical responses caused by frequency or timescale alignment. It supports linear and forced resonance, primary, superharmonic, subharmonic, internal, parametric and combination resonance, natural and forcing frequencies, detuning, resonance peaks and curves, bandwidth, quality factor, damping, modal interaction, energy transfer, jump phenomena, hysteresis, backbone curves, phase and frequency locking, harmonic balance, multiple scales, averaging, Floquet theory, Arnold tongues, bifurcation and stability of resonant periodic responses."
}

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

}

END_FLOW