Stability Ontology

flow "StabilityOntology" v1.0

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

STABILITY := {
 type: DynamicalBehaviorProperty,
 domain: DynamicalSystems,
 purpose: "Characterise how trajectories respond to small perturbations of states, initial conditions or system parameters, especially near equilibria, fixed points and invariant sets."
}

CONCEPTS := {
 STABILITY,
 LYAPUNOV_STABILITY,
 ASYMPTOTIC_STABILITY,
 EXPONENTIAL_STABILITY,
 INSTABILITY,
 EQUILIBRIUM,
 FIXED_POINT,
 PERTURBATION,
 LYAPUNOV_FUNCTION,
 LINEARISATION,
 EIGENVALUE,
 BASIN_OF_ATTRACTION,
 ORBITAL_STABILITY,
 STRUCTURAL_STABILITY,
 INPUT_TO_STATE_STABILITY
}

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

equilibrium
 x_star

satisfies

f(x_star)
 =
0

STABILITY
 -> behaviour_of(TRAJECTORIES_NEAR_x_star)

# ---------- LYAPUNOV STABILITY ----------
x_star
 -> LYAPUNOV_STABLE

iff

for every
 epsilon>0

exists
 delta>0

such_that

||x(0)-x_star||<delta

implies

||x(t)-x_star||<epsilon

for all
 t>=0

LYAPUNOV_STABILITY
 -> nearby_trajectories_remain_nearby

# ---------- INSTABILITY ----------
x_star
 -> UNSTABLE

if
 LYAPUNOV_STABILITY_CONDITION
 -> fails

INSTABILITY
 -> arbitrarily_small_perturbations_can_leave_specified_neighbourhood

# ---------- ATTRACTIVITY ----------
x_star
 -> LOCALLY_ATTRACTIVE

if

exists delta>0

such_that

||x(0)-x_star||<delta

implies

x(t)
 -> x_star

as
 t->infinity

ATTRACTIVITY
 -/> same_as(STABILITY)

# ---------- ASYMPTOTIC STABILITY ----------
x_star
 -> ASYMPTOTICALLY_STABLE

iff

x_star
 -> LYAPUNOV_STABLE

and

x_star
 -> LOCALLY_ATTRACTIVE

ASYMPTOTIC_STABILITY
 -> stability_plus_convergence

# ---------- GLOBAL ASYMPTOTIC STABILITY ----------
x_star
 -> GLOBALLY_ASYMPTOTICALLY_STABLE

if

x_star
 -> LYAPUNOV_STABLE

and

for all admissible initial states:

x(t)
 -> x_star

GLOBAL_STABILITY
 -> entire_relevant_state_space_in(BASIN_OF_ATTRACTION)

# ---------- EXPONENTIAL STABILITY ----------
x_star
 -> EXPONENTIALLY_STABLE

if

exists
 M>=1
 alpha>0
 delta>0

such_that

||x(t)-x_star||
 <=
M
*
exp(-alpha*t)
*
||x(0)-x_star||

for initial states within delta

EXPONENTIAL_STABILITY
 -> quantitative_decay_rate

EXPONENTIAL_STABILITY
 -> implies(ASYMPTOTIC_STABILITY)

# ---------- DISCRETE SYSTEM ----------
x_(n+1)
 =
F(x_n)

fixed point:

F(x_star)
 =
x_star

LYAPUNOV_STABILITY
 -> analogous_epsilon_delta_condition_over(n>=0)

ASYMPTOTIC_STABILITY
 -> x_n approaches x_star

# ---------- LINEAR CONTINUOUS SYSTEM ----------
dx/dt
 =
A*x

origin
 -> equilibrium

if all eigenvalues of A satisfy

Re(lambda_i)<0

then

origin
 -> EXPONENTIALLY_STABLE

if any

Re(lambda_i)>0

then

origin
 -> UNSTABLE

IMAGINARY_AXIS_CASE
 -> requires(JORDAN_AND_ADDITIONAL_ANALYSIS)

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

if

spectral_radius(A)<1

then

origin
 -> EXPONENTIALLY_STABLE

if

spectral_radius(A)>1

then

origin
 -> UNSTABLE

UNIT_CIRCLE_CASE
 -> requires(MULTIPLICITY_AND_JORDAN_ANALYSIS)

# ---------- NONLINEAR LINEARISATION ----------
nonlinear:

dx/dt
 =
f(x)

Jacobian:

A
 :=
Df(x_star)

if all eigenvalues
 Re(lambda)<0

then

x_star
 -> LOCALLY_ASYMPTOTICALLY_STABLE

if any eigenvalue
 Re(lambda)>0

then

x_star
 -> UNSTABLE

if eigenvalues on imaginary axis

LINEARISATION
 -> INCONCLUSIVE_IN_GENERAL

# ---------- HYPERBOLIC EQUILIBRIUM ----------
x_star
 -> HYPERBOLIC

if

Df(x_star)
 -> has_no_eigenvalues_with_zero_real_part

HYPERBOLICITY
 -> permits(ROBUST_LOCAL_LINEAR_CLASSIFICATION)

# ---------- LYAPUNOV FUNCTION ----------
V(x)
 -> scalar_function

requirements near x_star:

V(x_star)
 =
0

V(x)
 >
0
for x != x_star

time derivative:

V_dot(x)
 :=
grad(V)
 dot
f(x)

if

V_dot(x)
 <=0

then

x_star
 -> STABLE
 under(LYAPUNOV_THEOREM_HYPOTHESES)

# ---------- STRICT LYAPUNOV DECREASE ----------
if

V_dot(x)
 <0

for
 x != x_star

then

x_star
 -> ASYMPTOTICALLY_STABLE

under suitable local conditions

LYAPUNOV_METHOD
 -> does_not_require(EXPLICIT_TRAJECTORY_SOLUTION)

# ---------- RADIAL UNBOUNDEDNESS ----------
if

V(x)
 -> infinity

as
 ||x|| -> infinity

and

V_dot
 -> negative_definite_globally

then

GLOBAL_ASYMPTOTIC_STABILITY
 -> may_be_established

RADIAL_UNBOUNDEDNESS
 -> supports(GLOBAL_RESULT)

# ---------- LASALLE INVARIANCE ----------
if

V_dot
 <=0

then trajectories
 -> approach(LARGEST_INVARIANT_SET_WITHIN(V_dot=0))

LASALLE_INVARIANCE_PRINCIPLE
 -> strengthens(NONSESTRICT_LYAPUNOV_ANALYSIS)

# ---------- BASIN OF ATTRACTION ----------
B(x_star)
 := {
 x_0 :
 trajectory_from(x_0)
 -> x_star
}

BASIN_OF_ATTRACTION
 -> region_of(CONVERGENT_INITIAL_CONDITIONS)

LOCAL_STABILITY
 -/> automatically(GLOBAL_BASIN)

# ---------- ORBITAL STABILITY ----------
for periodic orbit
 Gamma

ORBITAL_STABILITY
 -> trajectories_starting_near(Gamma)
    remain_near(Gamma)

phase_shift_along_orbit
 -> permitted

ORBITAL_STABILITY
 -/> pointwise_state_stability

# ---------- FLOQUET STABILITY ----------
for periodic solution:

linearised_variational_system
 -> MONODROMY_MATRIX

eigenvalues
 -> FLOQUET_MULTIPLIERS

periodic orbit stability
 -> determined_by(MULTIPLIER_MAGNITUDES)
    under appropriate hypotheses

neutral multiplier
 -> associated_with(TIME_TRANSLATION_IN_AUTONOMOUS_SYSTEM)

# ---------- STRUCTURAL STABILITY ----------
SYSTEM
 -> STRUCTURALLY_STABLE

if

small perturbation_of(vector_field_or_map)

-> produces(QUALITATIVELY_EQUIVALENT_DYNAMICS)

STRUCTURAL_STABILITY
 -> robustness_of(DYNAMICAL_STRUCTURE)

STRUCTURAL_STABILITY
 -/> same_as(TRAJECTORY_STABILITY)

# ---------- PARAMETRIC STABILITY ----------
system depends on
 parameter mu

STABILITY_REGION
 := {
 mu :
 desired_stability_condition_holds
}

PARAMETER_CHANGE
 -> may_cross(STABILITY_BOUNDARY)

BIFURCATION
 -> often_associated_with(STABILITY_CHANGE)

# ---------- INPUT-TO-STATE STABILITY ----------
controlled system:

dx/dt
 =
f(x,u)

INPUT_TO_STATE_STABILITY
 -> bounds(STATE_RESPONSE_BY_INITIAL_STATE_AND_INPUT_MAGNITUDE)

ISS
 -> robustness_to(EXTERNAL_INPUTS)

CONTROL_THEORY
 -> extends(STABILITY_BEYOND_UNFORCED_SYSTEMS)

# ---------- BIBO STABILITY ----------
input-output system

-> BOUNDED_INPUT_BOUNDED_OUTPUT_STABLE

if

bounded input
 -> bounded output

BIBO_STABILITY
 -> input_output_property

BIBO
 -/> identical_to(INTERNAL_STATE_STABILITY)

# ---------- NUMERICAL STABILITY DISTINCTION ----------
NUMERICAL_STABILITY
 -> property_of(NUMERICAL_ALGORITHM)

DYNAMICAL_STABILITY
 -> property_of(MATHEMATICAL_SYSTEM)

NUMERICAL_STABILITY
 -/> same_as(DYNAMICAL_STABILITY)

simulation
 -> must_preserve_or_resolve(RELEVANT_DYNAMICAL_STABILITY)

# ---------- PERTURBATION ----------
PERTURBATION
 -> change_in(
 INITIAL_STATE,
 STATE,
 PARAMETER,
 INPUT,
 MODEL
)

STABILITY_ANALYSIS
 -> determines(RESPONSE_TO_PERTURBATION)

# ---------- APPLICATIONS ----------
STABILITY
 -> applied_to(DYNAMICAL_SYSTEMS)
 -> applied_to(CONTROL_THEORY)
 -> applied_to(MECHANICS)
 -> applied_to(ELECTRICAL_SYSTEMS)
 -> applied_to(BIOLOGICAL_MODELS)
 -> applied_to(ECONOMIC_MODELS)
 -> applied_to(FLUID_DYNAMICS)
 -> applied_to(POWER_SYSTEMS)
 -> applied_to(ROBOTICS)

# ---------- RELATION GRAPH ----------
EQUILIBRIUM
 -> candidate_for(STABILITY_ANALYSIS)

PERTURBATION
 -> tests(STABILITY)

LINEARISATION
 -> supports(LOCAL_STABILITY_CLASSIFICATION)

EIGENVALUES
 -> determine(LINEAR_STABILITY)

LYAPUNOV_FUNCTION
 -> certifies(STABILITY_WITHOUT_EXPLICIT_SOLUTION)

ATTRACTIVITY
 + STABILITY
 -> ASYMPTOTIC_STABILITY

BASIN_OF_ATTRACTION
 -> defines(REGION_OF_CONVERGENCE_TO_ATTRACTOR)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "stability" ~= "persistence of nearby dynamical behaviour under small perturbations",
 "Lyapunov stability" ~= "property that trajectories starting sufficiently near remain near",
 "asymptotic stability" ~= "Lyapunov stability combined with convergence to the equilibrium",
 "exponential stability" ~= "stability with exponentially decaying perturbation magnitude",
 "basin of attraction" ~= "set of initial states whose trajectories converge to an attractor",
 "structural stability" ~= "robustness of qualitative dynamics under small system perturbations"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 stability_and_attractivity_are_distinct: true,
 asymptotic_stability_requires_both_stability_and_attraction: true,
 linearisation_can_be_inconclusive_for_nonhyperbolic_equilibria: true,
 continuous_and_discrete_spectral_stability_conditions_are_distinct: true,
 orbital_stability_differs_from_pointwise_equilibrium_stability: true,
 structural_stability_differs_from_Lyapunov_stability: true,
 numerical_stability_must_not_be_confused_with_dynamical_stability: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 Lyapunov_relation: 0.99,
 asymptotic_relation: 0.99,
 linearisation_relation: 0.99,
 basin_relation: 0.99,
 structural_control_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(system_equilibrium_fixed_point_or_invariant_object)
 -> identify(continuous_discrete_periodic_or_controlled_context)
 -> define(perturbation_and_stability_notions)
 -> linearise(if_local_and_valid)
 -> inspect(eigenvalues_or_Floquet_multipliers)
 -> construct_or_test(Lyapunov_function_if_available)
 -> analyse(attractivity_and_basin)
 -> apply(LaSalle_if_nonstrict_Lyapunov_derivative)
 -> distinguish(orbital_structural_input_output_or_numerical_stability)
 -> validate(local_global_and_hyperbolicity_hypotheses)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed dynamical-stability semantic graph",
 primary_entity: STABILITY,
 preserve: [
  system,
  invariant_object,
  perturbation_type,
  time_type,
  local_or_global_scope,
  norm_or_distance,
  parameters,
  inputs
 ],
 infer_through: [
  Lyapunov_stability,
  asymptotic_stability,
  exponential_stability,
  linearisation,
  eigenvalues,
  Lyapunov_functions,
  basins_of_attraction,
  orbital_stability,
  structural_stability
 ],
 avoid: [
  equating_attraction_with stability,
  using_continuous_eigenvalue_criterion_for discrete maps,
  concluding_from_inconclusive_nonhyperbolic_linearisation,
  confusing_orbital_and_point stability,
  confusing_structural_and_Lyapunov stability,
  confusing_numerical_algorithm_stability_with physical dynamics
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + STABILITY
 + LYAPUNOV_STABILITY
 + ASYMPTOTIC_STABILITY
 + EXPONENTIAL_STABILITY
 + EQUILIBRIUM
 + PERTURBATION
 + LYAPUNOV_FUNCTION
 + LINEARISATION
 + BASIN_OF_ATTRACTION
 + STRUCTURAL_STABILITY
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> STABILITY
 -> SYSTEM_OBJECT_IDENTIFICATION
 -> CONTEXT_CLASSIFICATION
 -> PERTURBATION_DEFINITION
 -> LINEARISATION_ANALYSIS
 -> SPECTRAL_ANALYSIS
 -> LYAPUNOV_FUNCTION_ANALYSIS
 -> ATTRACTIVITY_BASIN_ANALYSIS
 -> LASALLE_ANALYSIS
 -> STABILITY_TYPE_DIFFERENTIATION
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW