Chaos Theory Ontology

flow "ChaosTheoryOntology" v1.0

# CHAOS THEORY ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/dynamical-systems/chaos-theory-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Dynamical Systems"
ENTITY := "Chaos Theory"

CHAOS_THEORY := {
 type: NonlinearDynamicsFramework,
 domain: DynamicalSystems,
 purpose: "Study deterministic dynamical systems that exhibit complex aperiodic behaviour, sensitive dependence on initial conditions, stretching and folding, and related chaotic structures."
}

CONCEPTS := {
 CHAOS,
 DYNAMICAL_SYSTEM,
 NONLINEARITY,
 SENSITIVE_DEPENDENCE,
 INITIAL_CONDITION,
 LYAPUNOV_EXPONENT,
 STRANGE_ATTRACTOR,
 BIFURCATION,
 PERIOD_DOUBLING,
 POINCARE_MAP,
 SYMBOLIC_DYNAMICS,
 TOPOLOGICAL_MIXING,
 DENSE_PERIODIC_ORBITS,
 FRACTAL_DIMENSION,
 ENTROPY,
 LOGISTIC_MAP,
 LORENZ_SYSTEM
}

# ---------- DETERMINISTIC SYSTEM ----------
continuous:

dx/dt
 =
f(x)

or discrete:

x_(n+1)
 =
F(x_n)

CHAOTIC_SYSTEM
 -> may_be(DETERMINISTIC)

DETERMINISM
 -/> implies(LONG_TERM_PREDICTABILITY)

# ---------- SENSITIVE DEPENDENCE ----------
two nearby states:

x_0

and

x_0 + delta_x_0

evolve to

x(t)

and

x_delta(t)

SENSITIVE_DEPENDENCE
 -> small_initial_difference_can_grow_rapidly

approx locally:

||delta_x(t)||
 approx
exp(lambda*t)
*
||delta_x(0)||

# ---------- LYAPUNOV EXPONENT ----------
lambda
 :=
limsup_(t->infinity)
1/t
*
log(
 ||delta_x(t)||
 /
 ||delta_x(0)||
)

POSITIVE_MAXIMAL_LYAPUNOV_EXPONENT
 -> evidence_of(EXPONENTIAL_SENSITIVITY)

LYAPUNOV_TIME
 approx
1/lambda_max

when
 lambda_max>0

LYAPUNOV_TIME
 -> characteristic_predictability_horizon

# ---------- DISCRETE LYAPUNOV EXPONENT ----------
for one-dimensional map:

x_(n+1)
 =
F(x_n)

lambda
 :=
lim_(N->infinity)
1/N
*
sum_from_n_0_to_Nminus1(
 log(
  |F_prime(x_n)|
 )
)

when limit exists

lambda>0
 -> local_exponential_separation_on_average

# ---------- CHAOS DEFINITIONS ----------
CHAOS
 -> definition_depends_on(MATHEMATICAL_FRAMEWORK)

possible criteria include:

SENSITIVE_DEPENDENCE
TOPOLOGICAL_TRANSITIVITY
TOPOLOGICAL_MIXING
DENSE_PERIODIC_ORBITS
POSITIVE_ENTROPY
POSITIVE_LYAPUNOV_EXPONENT

NO_SINGLE_CRITERION
 -> universally_equivalent_in_all_settings

# ---------- DEVANEY CHAOS ----------
on suitable metric space:

DEVANEY_CHAOS
 -> TOPOLOGICALLY_TRANSITIVE
 -> DENSE_PERIODIC_POINTS
 -> SENSITIVE_DEPENDENCE
    under standard theorem relationships

DEVANEY_DEFINITION
 -> topological_chaos_framework

# ---------- TOPOLOGICAL TRANSITIVITY ----------
F
 -> TOPOLOGICALLY_TRANSITIVE

if for nonempty open sets
 U,V

exists n>=0

such_that

F^n(U)
 intersection V
 != empty_set

TRANSITIVITY
 -> trajectories_can_connect_regions_of_state_space

# ---------- TOPOLOGICAL MIXING ----------
F
 -> TOPOLOGICALLY_MIXING

if for nonempty open U,V

exists N

such_that

for all n>=N:

F^n(U)
 intersection V
 != empty_set

MIXING
 -> stronger_than(TRANSITIVITY)
    in standard contexts

# ---------- DENSE PERIODIC ORBITS ----------
PERIODIC_POINTS
 -> DENSE_IN(INVARIANT_SET)

means

every neighbourhood
 -> contains(PERIODIC_POINT)

CHAOTIC_SET
 -> may_contain(PERIODIC_ORBITS_OF_MANY_PERIODS)

# ---------- APERIODIC TRAJECTORY ----------
CHAOTIC_TRAJECTORY
 -> generally_nonperiodic

yet

CHAOTIC_INVARIANT_SET
 -> may contain(DENSE_PERIODIC_ORBITS)

APERIODICITY
 and
PERIODIC_ORBIT_STRUCTURE
 -> coexist

# ---------- STRETCHING AND FOLDING ----------
CHAOTIC_MECHANISM
 -> stretch(STATE_SPACE_REGIONS)
 -> fold(REGIONS_BACK)
 -> repeat

STRETCHING
 -> amplifies(SEPARATIONS)

FOLDING
 -> keeps(DYNAMICS_BOUNDED)

STRETCH_AND_FOLD
 -> canonical_geometric_chaos_mechanism

# ---------- HORSESHOE ----------
SMALE_HORSESHOE
 -> stretch
 -> contract
 -> fold

HORSESHOE_DYNAMICS
 -> conjugate_or_semiconjugate_to(SYMBOLIC_SHIFT)
    on invariant set

HORSESHOE
 -> rigorous_model_of(CHAOS)

# ---------- SYMBOLIC DYNAMICS ----------
partition state space

-> encode trajectory by symbols

sequence:

s_0,s_1,s_2,...

SHIFT_MAP
 -> advances(SYMBOL_SEQUENCE)

SYMBOLIC_DYNAMICS
 -> represents(COMPLEX_ORBITS_AS_SEQUENCES)

# ---------- BERNOULLI SHIFT ----------
binary sequence:

(...,s_-1,s_0,s_1,...)

SHIFT
 -> moves_index

BERNOULLI_SHIFT
 -> canonical_chaotic_symbolic_system

SYMBOLIC_ENTROPY
 -> measures(INFORMATION_PRODUCTION)

# ---------- LOGISTIC MAP ----------
F_r(x)
 :=
r*x*(1-x)

x in [0,1]

LOGISTIC_MAP
 -> nonlinear_discrete_system

as r increases:

FIXED_POINT
 -> PERIOD_TWO
 -> PERIOD_FOUR
 -> PERIOD_EIGHT
 -> CHAOTIC_REGIMES

PERIOD_DOUBLING_CASCADE
 -> route_to(CHAOS)

# ---------- FEIGENBAUM SCALING ----------
successive period-doubling parameter intervals

-> ratios approach
 FEIGENBAUM_DELTA

for broad universality class

FEIGENBAUM_CONSTANT
 -> universal_scaling_in(PERIOD_DOUBLING_ROUTE)

UNIVERSALITY
 -> depends_on(MAP_CLASS_AND_NONLINEARITY)

# ---------- LORENZ SYSTEM ----------
dx/dt
 =
sigma*(y-x)

dy/dt
 =
x*(rho-z)-y

dz/dt
 =
x*y-beta*z

LORENZ_SYSTEM
 -> nonlinear_three_dimensional_flow

for classical chaotic parameter regimes:

-> STRANGE_ATTRACTOR
-> SENSITIVE_DEPENDENCE
-> APERIODIC_TRAJECTORIES

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

if

A
 -> ATTRACTING_INVARIANT_SET

and

A
 -> exhibits(CHAOTIC_DYNAMICS_AND_COMPLEX_GEOMETRY)

often

FRACTAL_STRUCTURE
 -> present

STRANGE_ATTRACTOR
 -> combines(ATTRACTION_AND_CHAOS)

# ---------- FRACTAL DIMENSION ----------
chaotic invariant set
 -> may_have(NONINTEGER_DIMENSION)

possible measures:

BOX_COUNTING_DIMENSION
HAUSDORFF_DIMENSION
CORRELATION_DIMENSION

FRACTAL_DIMENSION
 -> quantifies(GEOMETRIC_COMPLEXITY)

# ---------- CORRELATION DIMENSION ----------
correlation sum:

C(r)
 -> probability_pair_distance_less_than(r)

for small r:

C(r)
 approx
r^D2

D2
 -> CORRELATION_DIMENSION

TIME_SERIES
 -> may_estimate(FRACTAL_DIMENSION)
    with substantial data and methodological caveats

# ---------- ENTROPY ----------
TOPOLOGICAL_ENTROPY
 -> growth_rate_of(DISTINGUISHABLE_ORBIT_COMPLEXITY)

METRIC_ENTROPY
 -> information_generation_rate_with_respect_to(INVARIANT_MEASURE)

POSITIVE_ENTROPY
 -> common_indicator_of(CHAOTIC_COMPLEXITY)

# ---------- PESIN RELATION ----------
under suitable smooth hyperbolic measure-theoretic hypotheses:

KS_ENTROPY
 =
sum_of_positive_Lyapunov_exponents
 weighted_by_multiplicity

PESIN_FORMULA
 -> links(INFORMATION_PRODUCTION_AND_INSTABILITY)

# ---------- BIFURCATION ----------
parameter change
 -> qualitative_change_in(DYNAMICS)

routes to chaos may include:

PERIOD_DOUBLING
INTERMITTENCY
QUASIPERIODIC_BREAKDOWN
CRISIS
HOMOCLINIC_TANGLES

CHAOS
 -> can_emerge_through(MULTIPLE_ROUTES)

# ---------- HOMOCLINIC TANGLE ----------
stable_manifold
 and
unstable_manifold

of saddle invariant object

-> intersect_transversely

repeated intersections
 -> complex_tangle

HOMOCLINIC_TANGLE
 -> mechanism_for(CHAOTIC_DYNAMICS)

# ---------- POINCARE MAP ----------
continuous flow
 -> sample_on(TRANSVERSE_SECTION)

return map
 -> DISCRETE_DYNAMICAL_SYSTEM

CHAOTIC_FLOW
 -> may_be_studied_via(CHAOTIC_POINCARE_MAP)

POINCARE_SECTION
 -> reduces(DIMENSION)

# ---------- RECURRENCE PLOT ----------
time series
 -> reconstruct_state_vectors

compare distances

-> RECURRENCE_MATRIX

RECURRENCE_PLOT
 -> visualises(RETURN_STRUCTURE)

RECURRENCE_QUANTIFICATION
 -> descriptive_tool_for(NONLINEAR_DYNAMICS)

# ---------- POWER SPECTRUM ----------
periodic signal
 -> discrete_spectral_lines

chaotic signal
 -> often_broadband_spectrum

BROADBAND_SPECTRUM
 -/> sufficient_proof_of(CHAOS)

SPECTRAL_ANALYSIS
 -> supporting_diagnostic_only

# ---------- CHAOS VS NOISE ----------
CHAOS
 -> deterministic_complexity

NOISE
 -> stochastic_variability

observed_irregularity
 -> may_arise_from(CHAOS_OR_NOISE_OR_BOTH)

DISTINGUISHING
 -> requires(MULTIPLE_DIAGNOSTICS_AND_MODEL_CONTEXT)

# ---------- PREDICTABILITY ----------
measurement uncertainty:

delta_0

growth:

delta(t)
 approx
delta_0*exp(lambda_max*t)

prediction horizon for tolerance Delta:

T_predict
 approx
1/lambda_max
*
log(
 Delta/delta_0
)

CHAOS
 -> limits(LONG_TERM_STATE_PREDICTION)

SHORT_TERM_PREDICTION
 -> may_remain_possible

# ---------- SHADOWING ----------
numerical pseudo-orbit
 -> may_be_close_to(TRUE_ORBIT)
    under(SHADOWING_HYPOTHESES)

CHAOTIC_NUMERICAL_SIMULATION
 -> sensitive_to(ROUNDING_AND_DISCRETISATION)

SHADOWING
 -> framework_for(INTERPRETING_NUMERICAL_CHAOTIC_TRAJECTORIES)

# ---------- APPLICATIONS ----------
CHAOS_THEORY
 -> applied_to(FLUID_DYNAMICS)
 -> applied_to(WEATHER_AND_CLIMATE)
 -> applied_to(CELESTIAL_MECHANICS)
 -> applied_to(ELECTRONIC_CIRCUITS)
 -> applied_to(BIOLOGICAL_SYSTEMS)
 -> applied_to(CHEMICAL_OSCILLATIONS)
 -> applied_to(CONTROL_SYSTEMS)
 -> applied_to(NONLINEAR_TIME_SERIES)

# ---------- RELATION GRAPH ----------
NONLINEAR_DYNAMICS
 -> may_generate(CHAOS)

SENSITIVE_DEPENDENCE
 -> quantified_by(LYAPUNOV_EXPONENT)

BIFURCATION
 -> may_lead_to(CHAOS)

CHAOTIC_ATTRACTOR
 -> may_have(FRACTAL_GEOMETRY)

POINCARE_MAP
 -> represents(CHAOTIC_FLOW_AS_DISCRETE_DYNAMICS)

SYMBOLIC_DYNAMICS
 -> encodes(CHAOTIC_ORBITS)

ENTROPY
 -> quantifies(ORBIT_COMPLEXITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "chaos theory" ~= "study of complex deterministic behaviour in nonlinear dynamical systems",
 "sensitive dependence" ~= "rapid divergence of trajectories from nearby initial conditions",
 "positive Lyapunov exponent" ~= "positive asymptotic exponential perturbation-growth rate",
 "strange attractor" ~= "attracting invariant set with chaotic dynamics and typically complex geometry",
 "period-doubling cascade" ~= "sequence of bifurcations in which stable orbit periods repeatedly double",
 "symbolic dynamics" ~= "encoding of dynamical trajectories by sequences of symbols"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 chaos_is_not_synonymous_with_randomness: true,
 irregular_time_series_alone_does_not_prove_chaos: true,
 positive_Lyapunov_exponent_is_not_universal_complete_definition_in_all_contexts: true,
 chaos_definitions_depend_on_topological_measure_theoretic_or_smooth_framework: true,
 finite_precision_simulation_has_limited_trajectory_predictability: true,
 fractal_dimension_and_entropy_estimators_require_data_and_methodological_validation: true,
 bifurcation_route_to_chaos_is_system_dependent: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 sensitivity_Lyapunov_relation: 0.99,
 logistic_period_doubling_relation: 0.99,
 strange_attractor_relation: 0.99,
 symbolic_entropy_relation: 0.98,
 numerical_predictability_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(dynamical_system_state_space_and_parameters)
 -> classify(continuous_or_discrete_nonlinear_dynamics)
 -> identify(invariant_set_or_long_term_trajectory)
 -> test(periodicity_recurrence_and_sensitivity)
 -> estimate_or_analyse(Lyapunov_exponents)
 -> inspect(Poincare_map_symbolic_dynamics_or_manifold_structure_if_relevant)
 -> analyse(bifurcation_route_and_parameter_dependence)
 -> quantify(fractal_dimension_or_entropy_if_supported)
 -> distinguish(deterministic_chaos_from_noise)
 -> assess(predictability_and_numerical_precision_limits)
 -> validate(definition_data_length_and_method_hypotheses)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed chaos-theory semantic graph",
 primary_entity: CHAOS_THEORY,
 preserve: [
  dynamical_system,
  state_space,
  evolution_rule,
  parameters,
  invariant_set,
  initial_conditions,
  sampling,
  numerical_precision
 ],
 infer_through: [
  sensitive_dependence,
  Lyapunov_exponents,
  bifurcations,
  period_doubling,
  strange_attractors,
  symbolic_dynamics,
  entropy,
  fractal_dimensions
 ],
 avoid: [
  labelling_any_irregular_signal_as chaos,
  equating_chaos_with stochastic randomness,
  relying_on_single_visual_diagnostic,
  treating_positive_Lyapunov_estimate_as infallible proof,
  ignoring_finite_data_and_noise_bias,
  claiming_long_term_numerical_trajectory_accuracy_beyond_predictability_horizon
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + CHAOS_THEORY
 + CHAOS
 + SENSITIVE_DEPENDENCE
 + LYAPUNOV_EXPONENT
 + STRANGE_ATTRACTOR
 + BIFURCATION
 + PERIOD_DOUBLING
 + POINCARE_MAP
 + SYMBOLIC_DYNAMICS
 + ENTROPY
 + LOGISTIC_MAP
 + LORENZ_SYSTEM
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> CHAOS_THEORY
 -> SYSTEM_PARAMETER_IDENTIFICATION
 -> DYNAMICS_CLASSIFICATION
 -> INVARIANT_LONG_TERM_ANALYSIS
 -> SENSITIVITY_ANALYSIS
 -> LYAPUNOV_ANALYSIS
 -> POINCARE_SYMBOLIC_MANIFOLD_ANALYSIS
 -> BIFURCATION_ANALYSIS
 -> FRACTAL_ENTROPY_ANALYSIS
 -> CHAOS_NOISE_DIFFERENTIATION
 -> PREDICTABILITY_PRECISION_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW