Attractor Ontology

flow "AttractorOntology" v1.0

# ATTRACTOR ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/dynamical-systems/attractor-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Dynamical Systems"
ENTITY := "Attractor"

ATTRACTOR := {
 type: AsymptoticInvariantSet,
 domain: DynamicalSystems,
 purpose: "Represent an invariant set toward which trajectories from a surrounding basin approach under long-term dynamical evolution."
}

CONCEPTS := {
 ATTRACTOR,
 INVARIANT_SET,
 TRAJECTORY,
 FLOW,
 BASIN_OF_ATTRACTION,
 EQUILIBRIUM_ATTRACTOR,
 LIMIT_CYCLE_ATTRACTOR,
 TORUS_ATTRACTOR,
 STRANGE_ATTRACTOR,
 STABILITY,
 ASYMPTOTIC_BEHAVIOUR,
 OMEGA_LIMIT_SET,
 DISSIPATIVE_SYSTEM,
 LYAPUNOV_FUNCTION,
 FRACTAL_DIMENSION
}

# ---------- DYNAMICAL SYSTEM ----------
continuous:

Phi_t:
 X -> X

or discrete:

F:
 X -> X

X
 -> STATE_SPACE

TRAJECTORY
 -> evolves_in(X)

# ---------- INVARIANCE ----------
A subset_of X

A
 -> INVARIANT

if

Phi_t(A)
 =
A

for relevant t
 under flow context

or

F(A)
 =
A

under discrete invariant-set context

ATTRACTOR
 -> INVARIANT_SET

# ---------- ATTRACTION ----------
A
 -> ATTRACTING_SET

if

exists neighbourhood U of A

such_that

distance(
 Phi_t(x),
 A
)
 -> 0

as
 t->infinity

for
 x in U

discrete analogue:

distance(
 F^n(x),
 A
)
 -> 0

as
 n->infinity

# ---------- ATTRACTOR ----------
ATTRACTOR
 -> invariant
 -> attracting
 -> typically_minimal_or_indecoposable_under_selected_definition

ATTRACTOR_DEFINITION
 -> varies_slightly_across(DYNAMICAL_SYSTEMS_FRAMEWORKS)

CORE_SEMANTICS
 -> long_term_invariant_attracting_structure

# ---------- DISTANCE TO SET ----------
distance(x,A)
 :=
inf_{a in A}(
 d(x,a)
)

ATTRACTION
 -> convergence_of(DISTANCE_TO_SET)

POINTWISE_PHASE_MATCHING
 -/> required_for(ATTRACTION_TO_SET)

# ---------- BASIN OF ATTRACTION ----------
B(A)
 := {
 x in X :
 distance(
  Phi_t(x),
  A
 )
 -> 0
}

or discrete equivalent

BASIN_OF_ATTRACTION
 -> initial_states_converging_to(A)

ATTRACTOR
 -> may_have(LOCAL_OR_GLOBAL_BASIN)

# ---------- LOCAL ATTRACTOR ----------
A
 -> LOCAL_ATTRACTOR

if

some neighbourhood
 U

is attracted_to(A)

LOCAL_ATTRACTOR
 -/> attracts(ENTIRE_STATE_SPACE)

# ---------- GLOBAL ATTRACTOR ----------
A
 -> GLOBAL_ATTRACTOR

in common infinite-dimensional/dissipative usage if

A
 -> compact
 -> invariant
 -> attracts_all_bounded_sets

exact definition
 -> framework_dependent

GLOBAL_ATTRACTOR
 -> long_term_structure_for(WHOLE_RELEVANT_DYNAMICS)

# ---------- EQUILIBRIUM ATTRACTOR ----------
equilibrium:

f(x_star)=0

if

nearby trajectories
 -> x_star

then

{x_star}
 -> EQUILIBRIUM_ATTRACTOR

ASYMPTOTICALLY_STABLE_EQUILIBRIUM
 -> point_attractor

# ---------- LIMIT CYCLE ATTRACTOR ----------
Gamma
 -> isolated_periodic_orbit

if

nearby trajectories
 -> Gamma

then

Gamma
 -> ATTRACTING_LIMIT_CYCLE

LIMIT_CYCLE_ATTRACTOR
 -> periodic_long_term_behaviour

# ---------- TORUS ATTRACTOR ----------
INVARIANT_TORUS
 -> supports(QUASIPERIODIC_DYNAMICS)

if
 nearby trajectories
 -> torus

then

TORUS
 -> ATTRACTOR

TORUS_ATTRACTOR
 -> multiple_incommensurate_frequencies

# ---------- STRANGE ATTRACTOR ----------
A
 -> STRANGE_ATTRACTOR

if

A attracting_and_invariant

and

dynamics_on_A
 -> chaotic_or_complex

often

A
 -> FRACTAL_OR_NONSMOOTH_GEOMETRY

STRANGE_ATTRACTOR
 -> attraction_plus(CHAOTIC_DYNAMICS)

# ---------- LORENZ ATTRACTOR ----------
LORENZ_SYSTEM
 -> for suitable parameters
 -> exhibits(CHAOTIC_ATTRACTING_SET)

LORENZ_ATTRACTOR
 -> canonical_strange_attractor_example

SENSITIVE_DEPENDENCE
 -> coexists_with(ATTRACTION_TO_SET)

# ---------- ATTRACTION VS STABILITY ----------
ATTRACTION
 -> trajectories_approach(A)

LYAPUNOV_STABILITY_OF_SET
 -> trajectories_starting_near(A)
    remain_near(A)

ASYMPTOTIC_STABILITY_OF_SET
 -> STABILITY
    + ATTRACTION

ATTRACTOR
 -/> identical_to(LYAPUNOV_STABLE_SET_UNDER_EVERY_DEFINITION)

# ---------- TRAPPING REGION ----------
U
 -> TRAPPING_REGION

if

Phi_t(U)
 subset_of interior(U)

for sufficiently positive t
 in appropriate context

ATTRACTOR
 -> may_be_constructed_as

A
 =
intersection_{t>=0}
Phi_t(U)

or corresponding omega-limit construction

TRAPPING_REGION
 -> confines(FUTURE_DYNAMICS)

# ---------- OMEGA LIMIT SET ----------
for x:

omega(x)
 -> accumulation_points_of(TRAJECTORY_AS_t_TO_INFINITY)

if
 x in basin_of(A)

then

omega(x)
 subset_of A

under appropriate compactness assumptions

OMEGA_LIMIT_SET
 -> asymptotic_orbit_structure

# ---------- ATTRACTING SET VS ATTRACTOR ----------
ATTRACTING_SET
 -> may_contain(SMALLER_ATTRACTOR)

ATTRACTOR
 -> often_requires(MINIMALITY_OR_INTERNAL_DYNAMICAL_CONDITION)

TERMINOLOGY
 -> definition_sensitive

AI
 -> preserve(DEFINITION_CONTEXT)

# ---------- DISSIPATIVE SYSTEM ----------
DISSIPATIVE_DYNAMICS
 -> contracts(EFFECTIVE_PHASE_VOLUME_OR_ENERGY)

-> trajectories_enter(BOUNDED_REGION)

-> may_support(COMPACT_ATTRACTOR)

DISSIPATION
 -> common_mechanism_for(ATTRACTOR_FORMATION)

# ---------- CONSERVATIVE SYSTEM ----------
finite-dimensional Hamiltonian flow
 -> preserves(PHASE_VOLUME)

standard asymptotically attracting set with open basin
 -> generally_incompatible_with(VOLUME_PRESERVATION)

CONSERVATIVE_SYSTEM
 -> typically_lacks(DISSIPATIVE_ATTRACTORS)

# ---------- LYAPUNOV FUNCTION ----------
V(x)
 -> decreases_along(TRAJECTORIES)

V_dot
 <=0

LEVEL_SETS
 -> may_form(TRAPPING_REGIONS)

LA_SALLE_INVARIANCE
 -> trajectories_approach(LARGEST_INVARIANT_SET_IN(V_dot=0))

LYAPUNOV_FUNCTION
 -> supports(ATTRACTOR_ANALYSIS)

# ---------- MULTIPLE ATTRACTORS ----------
system
 -> may_have(
 A_1,
 A_2,
 ...,
 A_k
)

each
 -> own(BASIN_OF_ATTRACTION)

MULTISTABILITY
 -> coexistence_of(MULTIPLE_ATTRACTORS)

INITIAL_CONDITION
 -> determines(WHICH_ATTRACTOR_IS_APPROACHED)

# ---------- BASIN BOUNDARY ----------
B(A_i)
 -> may_be_separated_by(SEPARATRIX_OR_STABLE_MANIFOLD)

BASIN_BOUNDARY
 -> may_be(SMOOTH_OR_FRACTAL)

FRACTAL_BASIN_BOUNDARY
 -> strong_initial_condition_sensitivity_between_final_states

# ---------- CRISIS ----------
CHAOTIC_ATTRACTOR
 -> collides_with(UNSTABLE_INVARIANT_SET_OR_BASIN_BOUNDARY)

as parameter changes

-> sudden_change_or_destruction_of(ATTRACTOR)

CRISIS
 -> global_bifurcation_of(CHAOTIC_ATTRACTOR)

# ---------- BIFURCATION ----------
parameter variation
 -> may_change(
 EXISTENCE,
 STABILITY,
 SIZE,
 TOPOLOGY,
 BASIN
 )

of attractor

BIFURCATION
 -> attractor_creation_transformation_or_destruction

# ---------- LYAPUNOV EXPONENT ----------
on strange attractor:

lambda_max>0
 -> sensitive_dependence

negative transverse contraction
 -> supports(attraction)

CHAOTIC_ATTRACTOR
 -> combines(EXPANSION_AND_CONTRACTION)

# ---------- FRACTAL DIMENSION ----------
STRANGE_ATTRACTOR
 -> may_have(NONINTEGER_DIMENSION)

possible measures:
 BOX_COUNTING_DIMENSION
 HAUSDORFF_DIMENSION
 CORRELATION_DIMENSION

FRACTAL_DIMENSION
 -> quantifies(ATTRACTOR_GEOMETRY)

# ---------- INVARIANT MEASURE ----------
dynamics on attractor
 -> may_support(INVARIANT_PROBABILITY_MEASURE)

PHYSICAL_OR_SRB_MEASURE
 -> describes(TIME_AVERAGES_FOR_TYPICAL_INITIAL_CONDITIONS)
    under suitable chaotic-system hypotheses

ATTRACTOR
 -> may_have(STATISTICAL_DESCRIPTION)

# ---------- TIME AVERAGES ----------
observable
 g(x)

time average:

lim_(T->infinity)
1/T
*
integral_0^T(
 g(x(t))
 dt
)

ERGODIC_PROPERTIES
 -> relate(TIME_AVERAGES_TO_INVARIANT_MEASURE)

ATTRACTOR_DYNAMICS
 -> supports(STATISTICAL_LONG_TERM_ANALYSIS)

# ---------- NUMERICAL ATTRACTOR ----------
simulation
 -> long_transient_discarded

remaining trajectory
 -> approximates(ATTRACTING_SET)

but

FINITE_TRAJECTORY
 -/> proof_of(TRUE_ATTRACTOR)

NUMERICAL_ATTRACTOR_ANALYSIS
 -> requires(CONVERGENCE_STABILITY_AND_TRANSIENT_VALIDATION)

# ---------- APPLICATIONS ----------
ATTRACTOR
 -> applied_to(NONLINEAR_DYNAMICS)
 -> applied_to(CHAOS_THEORY)
 -> applied_to(CLIMATE_DYNAMICS)
 -> applied_to(BIOLOGICAL_OSCILLATIONS)
 -> applied_to(CONTROL_SYSTEMS)
 -> applied_to(FLUID_DYNAMICS)
 -> applied_to(NEURAL_DYNAMICS)
 -> applied_to(DISSIPATIVE_PDEs)

# ---------- RELATION GRAPH ----------
DYNAMICAL_SYSTEM
 -> generates(TRAJECTORIES)

INVARIANT_SET
 + ATTRACTION
 -> ATTRACTOR

ATTRACTOR
 -> has(BASIN_OF_ATTRACTION)

EQUILIBRIUM
 -> may_be(POINT_ATTRACTOR)

LIMIT_CYCLE
 -> may_be(PERIODIC_ATTRACTOR)

CHAOTIC_DYNAMICS
 + ATTRACTION
 -> STRANGE_ATTRACTOR

BIFURCATION
 -> modifies(ATTRACTOR_STRUCTURE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "attractor" ~= "invariant set approached by trajectories from a surrounding basin",
 "basin of attraction" ~= "set of initial states whose trajectories approach an attractor",
 "point attractor" ~= "attracting equilibrium represented by a single state",
 "limit-cycle attractor" ~= "attracting isolated periodic orbit",
 "strange attractor" ~= "chaotic attracting invariant set with complex geometry",
 "multistability" ~= "coexistence of multiple attractors for the same dynamical system"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 attractor_must_be_distinguished_from_arbitrary_invariant_set: true,
 attraction_and_Lyapunov_stability_are_distinct_properties: true,
 attractor_definitions_vary_across_frameworks_and_must_be_preserved: true,
 local_attractor_does_not_imply_global_attractor: true,
 conservative_volume_preserving_systems_do_not_generally_support_open_basin_asymptotic_attractors: true,
 strange_attractor_requires_more_than_visual_complexity: true,
 finite_numerical_trajectory_does_not_prove_attractor_existence: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 basin_relation: 0.99,
 equilibrium_cycle_relation: 0.99,
 strange_attractor_relation: 0.99,
 dissipative_relation: 0.99,
 bifurcation_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(dynamical_system_state_space_and_evolution_rule)
 -> identify(candidate_invariant_set)
 -> verify(invariance)
 -> test(attraction_from_neighbourhood_or_relevant_bounded_sets)
 -> determine(basin_of_attraction)
 -> classify(point_cycle_torus_strange_or_global_attractor)
 -> analyse(Lyapunov_stability_and_trapping_regions)
 -> inspect(omega_limit_sets_and_invariant_measures_if_relevant)
 -> analyse(parameter_dependence_bifurcations_and_crises)
 -> validate(dissipative_conservative_and_numerical_context)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed attractor semantic graph",
 primary_entity: ATTRACTOR,
 preserve: [
  dynamical_system,
  state_space,
  invariant_set,
  attraction_definition,
  basin,
  stability,
  parameters,
  numerical_context
 ],
 infer_through: [
  equilibrium_attractors,
  limit_cycle_attractors,
  torus_attractors,
  strange_attractors,
  basins,
  trapping_regions,
  omega_limit_sets,
  invariant_measures
 ],
 avoid: [
  calling_any_invariant_set_an attractor,
  equating_attraction_with Lyapunov stability,
  treating_local_attractor_as global,
  inferring_strange_attractor_from irregular plot alone,
  ignoring_volume_preservation_in conservative systems,
  treating_simulated_long_term_cloud_as rigorous attractor proof
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + ATTRACTOR
 + INVARIANT_SET
 + BASIN_OF_ATTRACTION
 + EQUILIBRIUM_ATTRACTOR
 + LIMIT_CYCLE_ATTRACTOR
 + STRANGE_ATTRACTOR
 + STABILITY
 + OMEGA_LIMIT_SET
 + DISSIPATIVE_SYSTEM
 + LYAPUNOV_FUNCTION
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> ATTRACTOR
 -> SYSTEM_STATE_IDENTIFICATION
 -> INVARIANCE_ANALYSIS
 -> ATTRACTION_ANALYSIS
 -> BASIN_ANALYSIS
 -> ATTRACTOR_CLASSIFICATION
 -> STABILITY_TRAPPING_ANALYSIS
 -> LIMIT_SET_MEASURE_ANALYSIS
 -> BIFURCATION_CRISIS_ANALYSIS
 -> DISSIPATIVE_CONSERVATIVE_VALIDATION
 -> NUMERICAL_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW