Fixed Point Ontology

flow "FixedPointOntology" v1.0

# FIXED POINT ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/dynamical-systems/fixed-point-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Dynamical Systems"
ENTITY := "Fixed Point"

FIXED_POINT := {
 type: InvariantStateEntity,
 domain: DynamicalSystems,
 purpose: "Represent a state that is unchanged by a mapping or discrete evolution rule and provide a foundation for equilibrium, stability, iteration and fixed-point theorem analysis."
}

CONCEPTS := {
 FIXED_POINT,
 MAP,
 STATE_SPACE,
 ITERATION,
 ORBIT,
 EQUILIBRIUM,
 PERIODIC_POINT,
 STABILITY,
 ATTRACTING_FIXED_POINT,
 REPELLING_FIXED_POINT,
 NEUTRAL_FIXED_POINT,
 DERIVATIVE,
 JACOBIAN,
 CONTRACTION_MAPPING,
 BANACH_FIXED_POINT_THEOREM,
 BASIN_OF_ATTRACTION
}

# ---------- DEFINITION ----------
let

F:
 X -> X

x_star in X

x_star
 -> FIXED_POINT_OF(F)

iff

F(x_star)
 =
x_star

FIXED_POINT
 -> invariant_under(SINGLE_MAP_APPLICATION)

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

if

x_0
 =
x_star

then

x_n
 =
x_star

for all n

FIXED_POINT
 -> CONSTANT_ORBIT

# ---------- ORBIT ----------
ORBIT(x_0)
 :=
{
 x_0,
 F(x_0),
 F^2(x_0),
 ...
 }

if
 x_0=x_star

then

ORBIT(x_star)
 =
{x_star}

FIXED_POINT
 -> period_one_orbit

# ---------- PERIODIC POINT RELATION ----------
x
 -> PERIODIC_POINT_OF_PERIOD_p

if

F^p(x)
 =
x

FIXED_POINT
 -> PERIODIC_POINT_WITH_PERIOD_ONE

PERIODIC_POINT
 -/> necessarily(FIXED_POINT_OF_F)

# ---------- EQUILIBRIUM RELATION ----------
continuous system:

dx/dt
 =
f(x)

equilibrium:

f(x_star)
 =
0

time-h flow map:

Phi_h(x_star)
 =
x_star

EQUILIBRIUM
 -> fixed_point_of(FLOW_MAP)

FIXED_POINT
 -> discrete_time_analogue_of(EQUILIBRIUM)

# ---------- SCALAR LOCAL STABILITY ----------
for

F:R->R

fixed point x_star

linearise:

x_(n+1)-x_star
 approx
F_prime(x_star)
*
(
 x_n-x_star
)

MULTIPLIER
 :=
F_prime(x_star)

# ---------- ATTRACTING FIXED POINT ----------
if

|F_prime(x_star)|
 < 1

then

x_star
 -> LOCALLY_ATTRACTING

under standard differentiability hypotheses

nearby iterates
 -> converge_to(x_star)

ATTRACTING_FIXED_POINT
 -> local_attractor

# ---------- REPELLING FIXED POINT ----------
if

|F_prime(x_star)|
 > 1

then

x_star
 -> LOCALLY_REPELLING

nearby perturbations
 -> grow_under_iteration

# ---------- NEUTRAL / NONHYPERBOLIC ----------
if

|F_prime(x_star)|
 = 1

then

LINEAR_TEST
 -> INCONCLUSIVE_IN_GENERAL

HIGHER_ORDER_TERMS
 -> determine(LOCAL_BEHAVIOUR)

NEUTRAL_FIXED_POINT
 -> requires(FURTHER_ANALYSIS)

# ---------- MULTIDIMENSIONAL MAP ----------
F:
 R^n -> R^n

fixed point:

F(x_star)
 =
x_star

linearisation:

xi_(n+1)
 approx
J_F(x_star)
*
xi_n

where

xi_n
 :=
x_n-x_star

# ---------- SPECTRAL STABILITY ----------
if every eigenvalue lambda_i of

J_F(x_star)

satisfies

|lambda_i|<1

then

x_star
 -> LOCALLY_ASYMPTOTICALLY_STABLE

if any

|lambda_i|>1

then

x_star
 -> UNSTABLE

UNIT_CIRCLE_EIGENVALUES
 -> require(NONLINEAR_ANALYSIS)

# ---------- HYPERBOLIC FIXED POINT ----------
x_star
 -> HYPERBOLIC_FIXED_POINT

if

no eigenvalue of J_F(x_star)
lies_on(UNIT_CIRCLE)

HYPERBOLICITY
 -> robust_local_classification

# ---------- STABLE MANIFOLD ----------
for hyperbolic fixed point:

W_s(x_star)
 := {
 x :
 F^n(x)
 -> x_star
 as n->infinity
}

W_u(x_star)
 -> states_approaching(x_star)_under_backward_iteration
    when defined

STABLE_UNSTABLE_MANIFOLDS
 -> organise(LOCAL_AND_GLOBAL_DYNAMICS)

# ---------- BASIN OF ATTRACTION ----------
for attracting fixed point:

B(x_star)
 := {
 x_0 :
 F^n(x_0)
 -> x_star
}

BASIN_OF_ATTRACTION
 -> set_of(CONVERGENT_INITIAL_STATES)

LOCAL_ATTRACTION
 -/> automatically(GLOBAL_ATTRACTION)

# ---------- FIXED POINT ITERATION ----------
solve equation:

G(x)
 =
0

rewrite as

x
 =
F(x)

then iterate:

x_(n+1)
 =
F(x_n)

FIXED_POINT_ITERATION
 -> numerical_root_solving_framework

CONVERGENCE
 -> depends_on(MAP_AND_INITIAL_GUESS)

# ---------- CONTRACTION ----------
metric space
 (X,d)

F:X->X

F
 -> CONTRACTION

if

exists
 0<=q<1

such_that

d(
 F(x),
 F(y)
)
<=
q*d(x,y)

for all x,y

CONTRACTION
 -> distances_shrink

# ---------- BANACH FIXED POINT THEOREM ----------
if

(X,d)
 -> COMPLETE_METRIC_SPACE

and

F:X->X
 -> CONTRACTION

then

exists unique
 x_star in X

such_that

F(x_star)
 =
x_star

and

for every x_0 in X:

F^n(x_0)
 -> x_star

BANACH_FIXED_POINT_THEOREM
 -> existence_uniqueness_and_constructive_convergence

# ---------- BANACH ERROR BOUND ----------
for contraction factor q:

d(
 x_n,
 x_star
)
<=
q^n
/
(1-q)
*
d(
 x_1,
 x_0
)

one standard a priori bound

CONTRACTION_CONSTANT
 -> controls(CONVERGENCE_RATE)

# ---------- BROUWER FIXED POINT ----------
if

K
 -> nonempty_compact_convex_subset_of(R^n)

and

F:K->K
 -> CONTINUOUS

then

F
 -> has_at_least_one_fixed_point

BROUWER_FIXED_POINT_THEOREM
 -> existence_without(CONSTRUCTIVE_ITERATION_OR_UNIQUENESS)

# ---------- SCHAUDER RELATION ----------
in suitable infinite-dimensional settings:

continuous compact mapping
 on appropriate closed convex set

-> may_have(FIXED_POINT)

SCHAUDER_FIXED_POINT_THEOREM
 -> extends(EXISTENCE_THEORY)

FIXED_POINT_THEOREMS
 -> have(DIFFERENT_HYPOTHESES_AND_CONCLUSIONS)

# ---------- NEWTON METHOD ----------
root equation:

g(x)=0

Newton map:

F_N(x)
 :=
x
-
g(x)/g_prime(x)

root
 x_star

-> fixed_point_of(F_N)

NEWTON_ITERATION
 -> fixed_point_iteration_with(SPECIAL_MAP)

under standard local hypotheses:
 -> QUADRATIC_CONVERGENCE

# ---------- LOGISTIC MAP ----------
F(x)
 =
r*x*(1-x)

fixed points solve:

x
 =
r*x*(1-x)

therefore

x_star
 =
0

or

x_star
 =
1-1/r

when defined

stability
 -> determined_by(|F_prime(x_star)|)

LOGISTIC_MAP
 -> canonical_fixed_point_bifurcation_example

# ---------- BIFURCATION ----------
parameterised map:

F_mu

fixed point branch:

F_mu(x_star(mu))
 =
x_star(mu)

as mu varies:

multiplier
 -> may_cross(UNIT_CIRCLE)

FIXED_POINT_STABILITY
 -> may_change

BIFURCATION
 -> qualitative_change_near(FIXED_POINT)

# ---------- PERIOD DOUBLING ----------
if real multiplier crosses

-1

under suitable nondegeneracy conditions

FIXED_POINT
 -> may_lose_stability

and
 PERIOD_TWO_ORBIT
 -> may_emerge

PERIOD_DOUBLING
 -> common_route_to(CHAOS)

# ---------- SADDLE NODE MAP BIFURCATION ----------
multiplier crosses
 +1

under suitable conditions

FIXED_POINTS
 -> created_or_annihilated

SADDLE_NODE_BIFURCATION
 -> fixed_point_pair_change

# ---------- INVARIANT SET RELATION ----------
{x_star}
 -> INVARIANT_SET

FIXED_POINT
 -> simplest_nonempty_invariant_set

ATTRACTOR
 -> may_be(FIXED_POINT)

REPELLER
 -> may_be(FIXED_POINT)

# ---------- APPLICATIONS ----------
FIXED_POINT
 -> applied_to(DYNAMICAL_SYSTEMS)
 -> applied_to(NUMERICAL_ANALYSIS)
 -> applied_to(NONLINEAR_EQUATIONS)
 -> applied_to(DIFFERENTIAL_EQUATIONS)
 -> applied_to(ECONOMICS)
 -> applied_to(GAME_THEORY)
 -> applied_to(OPTIMISATION)
 -> applied_to(FUNCTIONAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
MAP
 -> acts_on(STATE_SPACE)

FIXED_POINT
 -> unchanged_by(MAP)

FIXED_POINT
 -> generates(CONSTANT_ORBIT)

DERIVATIVE_OR_JACOBIAN
 -> determines(LOCAL_LINEAR_STABILITY)

CONTRACTION
 -> guarantees(UNIQUE_FIXED_POINT)

BASIN_OF_ATTRACTION
 -> contains(STATES_CONVERGING_TO_FIXED_POINT)

PARAMETER_CHANGE
 -> may_trigger(FIXED_POINT_BIFURCATION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "fixed point" ~= "point unchanged by a mapping",
 "period-one orbit" ~= "fixed-point orbit under discrete iteration",
 "attracting fixed point" ~= "fixed point approached by nearby iterates",
 "repelling fixed point" ~= "fixed point from which nearby iterates move away",
 "fixed-point iteration" ~= "iteration of a map to approximate a solution satisfying x=F(x)",
 "contraction fixed point" ~= "unique fixed point guaranteed by the Banach contraction theorem"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 fixed_point_is_defined_relative_to_specific_map: true,
 fixed_point_and_equilibrium_are_related_but_context_dependent: true,
 scalar_derivative_test_is_inconclusive_when_absolute_multiplier_equals_one: true,
 multidimensional_stability_depends_on_Jacobian_spectrum: true,
 local_attraction_does_not_imply_global_attraction: true,
 Banach_Brouwer_and_Schauder_theorems_have_distinct_hypotheses_and_conclusions: true,
 numerical_fixed_point_iteration_requires_convergence_analysis: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 orbit_relation: 0.99,
 stability_relation: 0.99,
 contraction_relation: 0.99,
 theorem_relation: 0.99,
 bifurcation_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(map_state_space_and_candidate_point)
 -> solve_or_test(F(x)=x)
 -> classify(fixed_periodic_or_equilibrium_relation)
 -> compute(derivative_or_Jacobian)
 -> analyse(local_spectral_stability)
 -> identify(basin_stable_or_unstable_manifolds_if_relevant)
 -> test(contraction_hypotheses_if_iteration_or_uniqueness_requested)
 -> select(Banach_Brouwer_Schauder_or_other_fixed_point_theorem_if_valid)
 -> analyse(parameter_dependence_and_bifurcation_if_present)
 -> validate(local_global_and_completeness_hypotheses)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed fixed-point semantic graph",
 primary_entity: FIXED_POINT,
 preserve: [
  map,
  state_space,
  candidate_point,
  derivative_or_Jacobian,
  parameter,
  metric,
  iteration,
  stability_context
 ],
 infer_through: [
  constant_orbits,
  periodic_points,
  equilibria,
  attracting_fixed_points,
  repelling_fixed_points,
  contractions,
  fixed_point_theorems,
  basins_of_attraction,
  bifurcations
 ],
 avoid: [
  calling_point_fixed_without specifying map,
  confusing_fixed_point_with arbitrary_periodic_point,
  applying_scalar_derivative_test_to multidimensional map,
  concluding_stability_when_multiplier_on unit circle,
  assuming_Brouwer_gives uniqueness_or constructive convergence,
  assuming_fixed_point_iteration_converges_without conditions
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + FIXED_POINT
 + MAP
 + ITERATION
 + ORBIT
 + EQUILIBRIUM
 + STABILITY
 + ATTRACTING_FIXED_POINT
 + REPELLING_FIXED_POINT
 + CONTRACTION_MAPPING
 + BANACH_FIXED_POINT_THEOREM
 + BASIN_OF_ATTRACTION
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> FIXED_POINT
 -> MAP_STATE_IDENTIFICATION
 -> FIXED_POINT_EQUATION_ANALYSIS
 -> ORBIT_RELATION_ANALYSIS
 -> DERIVATIVE_JACOBIAN_ANALYSIS
 -> LOCAL_STABILITY_ANALYSIS
 -> BASIN_MANIFOLD_ANALYSIS
 -> CONTRACTION_ANALYSIS
 -> FIXED_POINT_THEOREM_SELECTION
 -> BIFURCATION_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW