Dynamical Systems Ontology

flow "DynamicalSystemsOntology2" v1.0

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

DYNAMICAL_SYSTEM := {
 type: StateEvolutionSystem,
 domain: DynamicalSystems,
 purpose: "Represent mathematical systems whose states evolve through time according to continuous flows, discrete maps or other evolution laws."
}

CONCEPTS := {
 DYNAMICAL_SYSTEM,
 STATE_SPACE,
 STATE,
 TIME_SET,
 EVOLUTION_RULE,
 FLOW,
 DISCRETE_MAP,
 TRAJECTORY,
 ORBIT,
 EQUILIBRIUM,
 FIXED_POINT,
 PERIODIC_ORBIT,
 INVARIANT_SET,
 STABILITY,
 ATTRACTOR,
 BASIN_OF_ATTRACTION,
 BIFURCATION,
 CHAOS,
 LYAPUNOV_EXPONENT
}

# ---------- ABSTRACT SYSTEM ----------
DYNAMICAL_SYSTEM
 := (X,T,Phi)

X
 -> STATE_SPACE

T
 -> TIME_SET

Phi
 -> EVOLUTION_OPERATOR

Phi:
 T x X -> X

subject_to
 appropriate_composition_law

# ---------- CONTINUOUS SEMIGROUP / FLOW ----------
for forward-time system:

Phi_0
 = identity

Phi_(t+s)
 =
Phi_t composed_with Phi_s

for
 t,s >= 0

if time evolution invertible for all real t:

Phi
 -> FLOW

FLOW
 -> group_action_of(TIME_ADDITION)

# ---------- DISCRETE SYSTEM ----------
T
 := integers_or_naturals

x_(n+1)
 =
F(x_n)

then

x_n
 =
F^n(x_0)

DISCRETE_MAP
 -> generates(ORBIT)

# ---------- CONTINUOUS ODE SYSTEM ----------
dx/dt
 =
f(x,t)

if autonomous:

dx/dt
 =
f(x)

ODE
 -> generates(LOCAL_OR_GLOBAL_FLOW)
    under(EXISTENCE_AND_UNIQUENESS_HYPOTHESES)

# ---------- STATE ----------
x(t)
 -> STATE_AT_TIME(t)

INITIAL_STATE
 := x_0

EVOLUTION
 -> maps(INITIAL_STATE_TO_TRAJECTORY)

# ---------- TRAJECTORY ----------
TRAJECTORY(x_0)
 :=
{
 Phi_t(x_0) :
 t in T
}

continuous trajectory
 -> curve_in(STATE_SPACE)

discrete trajectory
 -> sequence_of(STATES)

# ---------- ORBIT ----------
ORBIT
 -> set_or_ordered_collection_of(STATES_REACHED_BY_EVOLUTION)

FORWARD_ORBIT
 := {
 Phi_t(x_0):
 t>=0
}

FULL_ORBIT
 -> may_include(PAST_AND_FUTURE)

# ---------- EQUILIBRIUM ----------
continuous autonomous:

x_star
 -> EQUILIBRIUM

iff

f(x_star)=0

equivalently

Phi_t(x_star)
 =
x_star

for all t

# ---------- FIXED POINT ----------
discrete:

x_star
 -> FIXED_POINT

iff

F(x_star)
 =
x_star

FIXED_POINT
 -> invariant_singleton

# ---------- PERIODIC ORBIT ----------
continuous:

exists T>0

such_that

Phi_T(x)
 =
x

PERIODIC_POINT
 -> generates(PERIODIC_ORBIT)

discrete:

F^p(x)
 =
x

for integer p>0

MINIMAL_PERIOD
 -> smallest_positive_return_time_or_iteration

# ---------- INVARIANT SET ----------
A subset_of X

A
 -> FORWARD_INVARIANT

if

Phi_t(A)
 subset_of A

for all forward t

A
 -> INVARIANT

if

Phi_t(A)
 =
A

under appropriate invertible-flow context

INVARIANCE
 -> preserved_under(EVOLUTION)

# ---------- LINEAR SYSTEM ----------
continuous:

dx/dt
 =
A*x

solution:

x(t)
 =
exp(A*t)
*
x_0

MATRIX_EXPONENTIAL
 -> continuous_linear_flow

# ---------- DISCRETE LINEAR SYSTEM ----------
x_(n+1)
 =
A*x_n

therefore

x_n
 =
A^n*x_0

SPECTRUM_OF_A
 -> determines(MAJOR_STABILITY_PROPERTIES)

# ---------- LINEARISATION ----------
nonlinear autonomous system:

dx/dt
 =
f(x)

near equilibrium x_star:

xi
 :=
x-x_star

d xi/dt
 approx
Df(x_star)
*
xi

JACOBIAN
 -> LOCAL_LINEAR_MODEL

# ---------- CONTINUOUS LOCAL STABILITY ----------
if every eigenvalue of
 Df(x_star)

has

Re(lambda)<0

then

x_star
 -> LOCALLY_ASYMPTOTICALLY_STABLE

under standard smoothness and hyperbolicity assumptions

if any

Re(lambda)>0

then

x_star
 -> UNSTABLE

# ---------- DISCRETE LOCAL STABILITY ----------
for fixed point:

DF(x_star)

if every eigenvalue satisfies

|lambda|<1

then

x_star
 -> LOCALLY_ASYMPTOTICALLY_STABLE

if any

|lambda|>1

then

x_star
 -> UNSTABLE

BOUNDARY_SPECTRUM
 -> requires(NONLINEAR_OR_HIGHER_ORDER_ANALYSIS)

# ---------- LYAPUNOV STABILITY ----------
x_star
 -> STABLE

if

for every epsilon>0
exists delta>0

such_that

distance(x_0,x_star)<delta

implies

distance(Phi_t(x_0),x_star)<epsilon

for all future t

# ---------- ATTRACTIVITY ----------
x_star
 -> ATTRACTIVE

if

nearby initial states
 -> converge_to(x_star)

as
 t -> infinity

STABILITY
 and
ATTRACTIVITY
 -> ASYMPTOTIC_STABILITY

# ---------- LYAPUNOV FUNCTION ----------
V:X->R

V(x_star)=0

V(x)>0
 nearby for x != x_star

and

dV/dt
 <=0

-> supports(STABILITY)

if

dV/dt<0
 away_from_equilibrium

-> supports(ASYMPTOTIC_STABILITY)

under standard theorem hypotheses

# ---------- ATTRACTOR ----------
A
 -> ATTRACTOR

if

A invariant

and

exists neighbourhood U

such_that

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

for x in U

ATTRACTOR
 -> long_term_state_structure

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

BASIN_OF_ATTRACTION
 -> initial_states_attracted_to(A)

# ---------- LIMIT SET ----------
OMEGA_LIMIT_SET(x)
 -> accumulation_points_of(FORWARD_ORBIT)

ALPHA_LIMIT_SET(x)
 -> accumulation_points_of(BACKWARD_ORBIT)

LIMIT_SET
 -> describes(ASYMPTOTIC_BEHAVIOUR)

# ---------- RECURRENCE ----------
state
 -> RECURRENT

if trajectory
 returns_arbitrarily_close_to(INITIAL_STATE)

RECURRENCE
 -> repeated_near_return

PERIODICITY
 -> strong_form_of(RECURRENCE)

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

dx/dt
 =
f(x,mu)

BIFURCATION
 -> qualitative_change_of(INVARIANT_OBJECTS_OR_STABILITY)

as
 mu crosses(CRITICAL_VALUE)

# ---------- COMMON BIFURCATIONS ----------
SADDLE_NODE
 -> equilibria_created_or_destroyed

TRANSCRITICAL
 -> equilibria_exchange_stability

PITCHFORK
 -> symmetry_related_branch_change

HOPF
 -> equilibrium_periodic_orbit_transition

PERIOD_DOUBLING
 -> periodic_orbit_changes_period

# ---------- CHAOS ----------
CHAOS
 -> complex_deterministic_dynamics

often associated with:
 SENSITIVE_DEPENDENCE
 TOPOLOGICAL_MIXING
 DENSE_PERIODIC_ORBITS
 POSITIVE_LYAPUNOV_EXPONENT

exact_definition
 -> framework_dependent

CHAOS
 -/> RANDOMNESS

# ---------- LYAPUNOV EXPONENT ----------
perturbation
 delta_x(t)

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

MAXIMAL_LYAPUNOV_EXPONENT
 -> exponential_separation_rate

lambda_max>0
 -> evidence_of(SENSITIVE_DEPENDENCE)

# ---------- STABLE MANIFOLD ----------
for hyperbolic invariant object A:

W_s(A)
 := states_approaching(A)_forward_in_time

W_u(A)
 := states_approaching(A)_backward_in_time

STABLE_AND_UNSTABLE_MANIFOLDS
 -> organise(GLOBAL_PHASE_SPACE)

# ---------- POINCARE MAP ----------
continuous flow
 + transverse_section

-> return_map

P:
 section -> section

POINCARE_MAP
 -> discrete_representation_of(CONTINUOUS_RECURRENCE)

PERIODIC_ORBIT
 -> FIXED_OR_PERIODIC_POINT_OF(POINCARE_MAP)

# ---------- CONSERVATIVE SYSTEM ----------
CONSERVATIVE_SYSTEM
 -> preserves(INVARIANT_QUANTITY_OR_MEASURE)

HAMILTONIAN_SYSTEM
 -> preserves(SYMPLECTIC_STRUCTURE)

LIOUVILLE_THEOREM
 -> phase_volume_preservation

# ---------- DISSIPATIVE SYSTEM ----------
DISSIPATIVE_SYSTEM
 -> contracts_effective_phase_volume
    in common contexts

DISSIPATION
 -> enables(ATTRACTING_SETS)

CONSERVATIVE
 and
DISSIPATIVE
 -> distinct_structural_classes

# ---------- APPLICATIONS ----------
DYNAMICAL_SYSTEM
 -> applied_to(ODEs)
 -> applied_to(DIFFERENCE_EQUATIONS)
 -> applied_to(CONTROL_THEORY)
 -> applied_to(MECHANICS)
 -> applied_to(BIOLOGY)
 -> applied_to(ECONOMICS)
 -> applied_to(CLIMATE)
 -> applied_to(NEURAL_SYSTEMS)
 -> applied_to(CHAOS_THEORY)

# ---------- RELATION GRAPH ----------
STATE_SPACE
 -> contains(STATES)

EVOLUTION_RULE
 -> generates(TRAJECTORY)

TRAJECTORY
 -> defines(ORBIT)

EQUILIBRIUM
 -> invariant_under(FLOW)

FIXED_POINT
 -> invariant_under(DISCRETE_MAP)

STABILITY
 -> characterises(RESPONSE_TO_PERTURBATION)

ATTRACTOR
 -> governs(LONG_TERM_BEHAVIOUR)

BIFURCATION
 -> changes(DYNAMICAL_STRUCTURE)

LYAPUNOV_EXPONENT
 -> quantifies(PERTURBATION_GROWTH)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "dynamical system" ~= "state space with a time-evolution rule",
 "flow" ~= "continuous-time evolution family satisfying a composition law",
 "orbit" ~= "states reached from an initial condition under repeated evolution",
 "equilibrium" ~= "continuous-time state unchanged by evolution",
 "fixed point" ~= "discrete-time state unchanged by the map",
 "attractor" ~= "invariant set approached by nearby trajectories"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 state_space_time_set_and_evolution_rule_must_be_identified: true,
 continuous_and_discrete_systems_have_distinct_stability_spectra: true,
 linearisation_is_local_and_can_be_inconclusive_at_nonhyperbolic_points: true,
 stability_and_attractivity_are_distinct: true,
 invariant_forward_invariant_and_attracting_sets_are_distinct: true,
 chaos_definition_depends_on_mathematical_framework: true,
 numerical_trajectory_does_not_by_itself_prove_asymptotic_dynamics: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 evolution_relation: 0.99,
 equilibrium_fixed_point_relation: 0.99,
 stability_relation: 0.99,
 attractor_relation: 0.99,
 bifurcation_chaos_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(state_space_time_set_and_evolution_rule)
 -> classify(continuous_or_discrete_dynamics)
 -> identify(initial_state_and_trajectory)
 -> locate(equilibria_fixed_points_periodic_orbits_and_invariant_sets)
 -> linearise(near_candidate_invariant_objects_if_valid)
 -> analyse(stability_attractivity_and_Lyapunov_functions)
 -> determine(attractors_basins_and_limit_sets)
 -> analyse(parameter_dependence_and_bifurcations_if_requested)
 -> evaluate(sensitivity_and_Lyapunov_exponents_if_relevant)
 -> validate(local_global_existence_and_numerical_assumptions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed dynamical-system semantic graph",
 primary_entity: DYNAMICAL_SYSTEM,
 preserve: [
  state_space,
  time_set,
  evolution_rule,
  initial_state,
  parameters,
  trajectory,
  invariant_sets,
  stability_context
 ],
 infer_through: [
  flows,
  discrete_maps,
  equilibria,
  fixed_points,
  periodic_orbits,
  invariant_sets,
  attractors,
  bifurcations,
  Lyapunov_exponents
 ],
 avoid: [
  mixing_continuous_and_discrete_stability_conditions,
  inferring_global_behavior_from_local_linearisation,
  confusing_attractor_with any invariant set,
  equating_chaos_with randomness,
  claiming_stability_from simulation alone,
  ignoring_existence_and_uniqueness_of_evolution
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + DYNAMICAL_SYSTEM
 + STATE_SPACE
 + EVOLUTION_RULE
 + FLOW
 + DISCRETE_MAP
 + TRAJECTORY
 + EQUILIBRIUM
 + FIXED_POINT
 + STABILITY
 + ATTRACTOR
 + BIFURCATION
 + LYAPUNOV_EXPONENT
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> DYNAMICAL_SYSTEM
 -> STATE_TIME_EVOLUTION_IDENTIFICATION
 -> SYSTEM_CLASSIFICATION
 -> TRAJECTORY_ORBIT_ANALYSIS
 -> INVARIANT_OBJECT_ANALYSIS
 -> LINEARISATION
 -> STABILITY_ATTRACTIVITY_ANALYSIS
 -> ATTRACTOR_LIMIT_SET_ANALYSIS
 -> BIFURCATION_ANALYSIS
 -> CHAOS_LYAPUNOV_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW