Limit Cycle Ontology

flow "LimitCycleOntology" v1.0

# LIMIT CYCLE ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/dynamical-systems/limit-cycle-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Dynamical Systems"
ENTITY := "Limit Cycle"

LIMIT_CYCLE := {
 type: IsolatedPeriodicOrbit,
 domain: DynamicalSystems,
 purpose: "Represent an isolated periodic orbit of a continuous dynamical system and describe its stability, attraction, repulsion, bifurcation and oscillatory behaviour."
}

CONCEPTS := {
 LIMIT_CYCLE,
 PERIODIC_ORBIT,
 PHASE_SPACE,
 TRAJECTORY,
 PERIOD,
 FREQUENCY,
 ISOLATED_ORBIT,
 STABILITY,
 ATTRACTING_LIMIT_CYCLE,
 REPELLING_LIMIT_CYCLE,
 SEMISTABLE_LIMIT_CYCLE,
 BASIN_OF_ATTRACTION,
 POINCARE_MAP,
 FLOQUET_MULTIPLIER,
 HOPF_BIFURCATION,
 OSCILLATION
}

# ---------- CONTINUOUS SYSTEM ----------
autonomous system:

dx/dt
 =
f(x)

trajectory:

x(t)
 -> PHASE_SPACE

PERIODIC_SOLUTION
 if

exists
 T>0

such_that

x(t+T)
 =
x(t)

for all t

# ---------- PERIODIC ORBIT ----------
Gamma
 :=
{
 x(t):
 0<=t<T
}

Gamma
 -> PERIODIC_ORBIT

T
 -> PERIOD

minimal_positive_T
 -> FUNDAMENTAL_PERIOD

frequency:

omega
 :=
2*pi/T

# ---------- LIMIT CYCLE ----------
Gamma
 -> LIMIT_CYCLE

iff

Gamma
 -> PERIODIC_ORBIT

and

Gamma
 -> ISOLATED_AMONG_PERIODIC_ORBITS

ISOLATION
 -> distinguishes(LIMIT_CYCLE_FROM_CONTINUOUS_FAMILY_OF_PERIODIC_ORBITS)

# ---------- ISOLATED ORBIT ----------
exists neighbourhood U of Gamma

such_that

U
 -> contains_no_other_periodic_orbit_sufficiently_close_to(Gamma)

LIMIT_CYCLE
 -> isolated_closed_trajectory

# ---------- PHASE PORTRAIT ----------
LIMIT_CYCLE
 -> closed_curve_in(PHASE_SPACE)

nearby trajectories
 -> may_spiral_toward
 -> may_spiral_away
 -> may_approach_from_one_side_only

PHASE_PORTRAIT
 -> reveals(LIMIT_CYCLE_STABILITY)

# ---------- ATTRACTING LIMIT CYCLE ----------
Gamma
 -> ATTRACTING

if

exists neighbourhood U

such_that

distance(
 Phi_t(x_0),
 Gamma
)
 -> 0

as
 t->infinity

for x_0 in U

ATTRACTING_LIMIT_CYCLE
 -> stable_self_sustained_oscillation
    under appropriate stability conditions

# ---------- REPELLING LIMIT CYCLE ----------
Gamma
 -> REPELLING

if nearby trajectories
 -> move_away_from(Gamma)
    in forward time

REPELLING_LIMIT_CYCLE
 -> attracting_under_backward_time
    where backward flow exists

# ---------- SEMISTABLE LIMIT CYCLE ----------
Gamma
 -> SEMISTABLE

if

attracting_from_one_side

and

repelling_from_other_side

SEMISTABILITY
 -> one_sided_stability_structure

# ---------- ORBITAL STABILITY ----------
Gamma
 -> ORBITALLY_STABLE

if trajectories starting sufficiently close

remain close to
 the orbit as a set

PHASE_SHIFT
 -> allowed

ORBITAL_STABILITY
 -/> pointwise_synchronisation_to_same_phase

# ---------- ASYMPTOTIC ORBITAL STABILITY ----------
Gamma
 -> ASYMPTOTICALLY_ORBITALLY_STABLE

if

Gamma orbitally_stable

and

nearby trajectories
 -> approach(Gamma)

possibly with phase shift

ATTRACTING_LIMIT_CYCLE
 -> commonly analysed_via(ASYMPTOTIC_ORBITAL_STABILITY)

# ---------- POINCARE SECTION ----------
choose
 Sigma
 -> TRANSVERSE_SECTION_TO(Gamma)

flow
 -> successive_returns_to(Sigma)

POINCARE_MAP:

P:
 Sigma -> Sigma

periodic orbit intersection
 x_star

satisfies

P(x_star)
 =
x_star

LIMIT_CYCLE
 -> FIXED_POINT_OF(POINCARE_RETURN_MAP)

# ---------- POINCARE STABILITY ----------
DP(x_star)
 -> derivative_or_Jacobian_of_return_map

if transverse eigenvalues satisfy

|lambda_i|<1

then

Gamma
 -> LOCALLY_ASYMPTOTICALLY_ORBITALLY_STABLE

if any

|lambda_i|>1

then

Gamma
 -> UNSTABLE

POINCARE_MAP
 -> reduces(PERIODIC_ORBIT_STABILITY_TO_FIXED_POINT_STABILITY)

# ---------- FLOQUET THEORY ----------
linearisation along periodic orbit:

delta_dot
 =
A(t)*delta

where

A(t+T)
 =
A(t)

fundamental matrix over one period:

M
 -> MONODROMY_MATRIX

eigenvalues of M
 -> FLOQUET_MULTIPLIERS

# ---------- NEUTRAL MULTIPLIER ----------
for autonomous continuous system:

one Floquet multiplier
 =
1

associated_with
 perturbation_along(ORBIT_DIRECTION)

TRANSVERSE_MULTIPLIERS
 -> determine(ORBITAL_STABILITY)

# ---------- FLOQUET EXPONENT ----------
multiplier:

rho_i

Floquet exponent:

mu_i
 :=
(1/T)
*
log(rho_i)

subject_to
 logarithm_branch

Re(mu_i)<0
 for transverse modes

-> transverse_decay

# ---------- BASIN OF ATTRACTION ----------
for attracting Gamma:

B(Gamma)
 := {
 x_0 :
 distance(
  Phi_t(x_0),
  Gamma
 )
 -> 0
}

BASIN_OF_ATTRACTION
 -> initial_states_converging_to(LIMIT_CYCLE)

# ---------- AMPLITUDE / PHASE ----------
near stable limit cycle:

perturbation
 -> decomposed_into(
 TRANSVERSE_AMPLITUDE_COMPONENT,
 TANGENTIAL_PHASE_COMPONENT
)

TRANSVERSE_COMPONENT
 -> decays_if_stable

PHASE_COMPONENT
 -> may_persist_as(PHASE_SHIFT)

# ---------- HOPF BIFURCATION ----------
parameterised system:

dx/dt
 =
f(x,mu)

equilibrium
 -> changes_stability

complex conjugate eigenvalues
 -> cross(IMAGINARY_AXIS)

under nondegeneracy conditions

LIMIT_CYCLE
 -> may_be_created_or_destroyed

HOPF_BIFURCATION
 -> principal_limit_cycle_generation_mechanism

# ---------- SUPERCRITICAL HOPF ----------
in standard normal-form context:

SUPERCRITICAL_HOPF
 -> small_stable_limit_cycle_emerges
    on one side of critical parameter

equilibrium
 -> loses_stability

exact orientation
 -> depends_on(PARAMETER_CONVENTION)

# ---------- SUBCRITICAL HOPF ----------
SUBCRITICAL_HOPF
 -> unstable_limit_cycle_involved_near_transition

may produce
 -> abrupt_stability_change

exact branch orientation
 -> convention_and_system_dependent

# ---------- SADDLE-NODE OF CYCLES ----------
two limit cycles
 -> collide

one stable
one unstable
 commonly

-> annihilate_or_appear

SADDLE_NODE_OF_LIMIT_CYCLES
 -> periodic_orbit_bifurcation

# ---------- PLANAR SYSTEM ----------
for two-dimensional autonomous system:

dx/dt
 =
f(x,y)

dy/dt
 =
g(x,y)

LIMIT_CYCLE
 -> closed_isolated_orbit_in_plane

PLANAR_DYNAMICS
 -> permits(POINCARE_BENDIXSON_ANALYSIS)

# ---------- POINCARE-BENDIXSON ----------
for suitable planar flow:

bounded_forward_trajectory

with compact omega-limit set

and no equilibria in omega-limit set

-> omega-limit_set_is(PERIODIC_ORBIT)

under theorem hypotheses

POINCARE_BENDIXSON
 -> supports(EXISTENCE_OF_PERIODIC_BEHAVIOUR_IN_2D)

# ---------- BENDIXSON-DULAC ----------
if suitable Dulac function B exists

such_that

div(
 B*f_vector
)

has one sign
and is not identically zero
on simply connected region

then

no periodic orbit lies entirely in region

BENDIXSON_DULAC
 -> nonexistence_test_for(LIMIT_CYCLES)

# ---------- VAN DER POL ----------
system:

x_double_dot
 -
mu*(1-x^2)*x_dot
 +
x
 =
0

for
 mu>0

-> has stable_limit_cycle
   under standard Van_der_Pol_dynamics

VAN_DER_POL_OSCILLATOR
 -> canonical_self_sustained_oscillator

# ---------- SELF-SUSTAINED OSCILLATION ----------
ATTRACTING_LIMIT_CYCLE
 -> oscillation_amplitude_selected_by(NONLINEAR_DYNAMICS)

initial conditions within basin
 -> approach(SAME_PERIODIC_ORBIT)

SELF_SUSTAINED_OSCILLATION
 -> independent_of(SMALL_INITIAL_AMPLITUDE_AFTER_TRANSIENT)
    in basin context

# ---------- CONSERVATIVE SYSTEM DISTINCTION ----------
Hamiltonian planar systems
 -> may_have(CONTINUOUS_FAMILIES_OF_PERIODIC_ORBITS)

such periodic orbits
 -/> necessarily(LIMIT_CYCLES)

because

LIMIT_CYCLE
 -> requires(ISOLATION)

DISSIPATIVE_NONLINEAR_SYSTEMS
 -> commonly_support(ATTRACTING_LIMIT_CYCLES)

# ---------- MULTIPLE LIMIT CYCLES ----------
system
 -> may_have(MULTIPLE_LIMIT_CYCLES)

each
 -> may_have(DIFFERENT_STABILITY)

BASIN_BOUNDARIES
 -> separate(ATTRACTING_CYCLES_OR_OTHER_ATTRACTORS)

# ---------- APPLICATIONS ----------
LIMIT_CYCLE
 -> applied_to(NONLINEAR_OSCILLATIONS)
 -> applied_to(BIOLOGICAL_RHYTHMS)
 -> applied_to(NEURAL_OSCILLATORS)
 -> applied_to(ELECTRICAL_CIRCUITS)
 -> applied_to(CONTROL_SYSTEMS)
 -> applied_to(CHEMICAL_OSCILLATIONS)
 -> applied_to(MECHANICAL_SYSTEMS)
 -> applied_to(FLUID_DYNAMICS)

# ---------- RELATION GRAPH ----------
PERIODIC_SOLUTION
 -> traces(PERIODIC_ORBIT)

PERIODIC_ORBIT
 + ISOLATION
 -> LIMIT_CYCLE

LIMIT_CYCLE
 -> fixed_point_of(POINCARE_MAP)

POINCARE_MAP
 -> determines(TRANSVERSE_STABILITY)

FLOQUET_MULTIPLIERS
 -> quantify(PERIODIC_ORBIT_STABILITY)

HOPF_BIFURCATION
 -> may_create(LIMIT_CYCLE)

ATTRACTING_LIMIT_CYCLE
 -> has(BASIN_OF_ATTRACTION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "limit cycle" ~= "isolated periodic orbit of a continuous dynamical system",
 "attracting limit cycle" ~= "isolated periodic orbit approached by nearby trajectories",
 "repelling limit cycle" ~= "isolated periodic orbit from which nearby trajectories diverge forward in time",
 "orbital stability" ~= "stability relative to the periodic orbit as a set, allowing phase displacement",
 "Poincare map fixed point" ~= "section intersection representing a periodic orbit",
 "Floquet multiplier" ~= "eigenvalue describing linear perturbation evolution over one period"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 limit_cycle_requires_periodicity_and_isolation: true,
 periodic_orbit_is_not_automatically_limit_cycle: true,
 orbital_stability_differs_from_pointwise_phase_stability: true,
 autonomous_periodic_orbit_has_neutral_tangential_direction: true,
 Hopf_bifurcation_requires_nontrivial_hypotheses_beyond_eigenvalue_crossing: true,
 conservative_families_of_periodic_orbits_are_not_limit_cycles_when_nonisolated: true,
 numerical_closed_trajectory_alone_does_not_prove_limit_cycle_existence: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 periodic_isolation_relation: 0.99,
 Poincare_relation: 0.99,
 Floquet_relation: 0.99,
 Hopf_relation: 0.99,
 planar_system_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(continuous_dynamical_system_and_candidate_periodic_orbit)
 -> verify(periodicity_and_minimal_period)
 -> test(isolation_from_nearby_periodic_orbits)
 -> construct(Poincare_section_and_return_map_if_possible)
 -> compute(return_map_derivative_or_Floquet_multipliers)
 -> classify(attracting_repelling_semistable_or_neutral_behavior)
 -> determine(basin_of_attraction_if_attracting)
 -> analyse(parameter_dependence_and_Hopf_or_cycle_bifurcation)
 -> apply(planar_existence_or_nonexistence_theorems_if_relevant)
 -> validate(autonomous_transverse_and_numerical_assumptions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed limit-cycle semantic graph",
 primary_entity: LIMIT_CYCLE,
 preserve: [
  dynamical_system,
  periodic_orbit,
  period,
  isolation,
  phase_space,
  Poincare_section,
  Floquet_multipliers,
  parameter
 ],
 infer_through: [
  orbital_stability,
  attracting_cycles,
  repelling_cycles,
  basins_of_attraction,
  Poincare_maps,
  Floquet_theory,
  Hopf_bifurcations,
  planar_limit_cycle_theorems
 ],
 avoid: [
  calling_every_periodic_orbit_a limit cycle,
  ignoring_isolation_requirement,
  treating_phase_shift_as instability,
  using_all_Floquet_multipliers_without separating neutral tangent multiplier,
  claiming_Hopf_cycle_without nondegeneracy analysis,
  inferring_limit_cycle_from finite simulation alone
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(limit_cycle_context)
 -> compare(existing_relations)
 -> preserve(period_isolation_stability_section_and_parameter_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + LIMIT_CYCLE
 + PERIODIC_ORBIT
 + PERIOD
 + ISOLATED_ORBIT
 + STABILITY
 + ATTRACTING_LIMIT_CYCLE
 + POINCARE_MAP
 + FLOQUET_MULTIPLIER
 + HOPF_BIFURCATION
 + BASIN_OF_ATTRACTION
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> LIMIT_CYCLE
 -> SYSTEM_ORBIT_IDENTIFICATION
 -> PERIODICITY_ANALYSIS
 -> ISOLATION_ANALYSIS
 -> POINCARE_MAP_ANALYSIS
 -> FLOQUET_STABILITY_ANALYSIS
 -> CYCLE_CLASSIFICATION
 -> BASIN_ANALYSIS
 -> BIFURCATION_ANALYSIS
 -> PLANAR_THEOREM_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW