Nonlinear Differential Equation Ontology

flow "NonlinearDifferentialEquationOntology" v1.0

# ============================================================
# NONLINEAR DIFFERENTIAL EQUATION ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/differential-equations/nonlinear-differential-equation-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Differential Equations"
ENTITY := "Nonlinear Differential Equation"

# ---------- CORE ENTITY ----------
NONLINEAR_DIFFERENTIAL_EQUATION := {
    type: DifferentialEquation,
    domain: DifferentialEquations,
    purpose: "Represent a differential equation in which the unknown function, its derivatives, or combinations of them occur nonlinearly."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    NONLINEAR_DIFFERENTIAL_EQUATION,
    UNKNOWN_FUNCTION,
    DERIVATIVE,
    NONLINEARITY,
    NONLINEAR_ODE,
    NONLINEAR_PDE,
    AUTONOMOUS_SYSTEM,
    EQUILIBRIUM,
    LINEARISATION,
    JACOBIAN,
    STABILITY,
    BIFURCATION,
    CHAOS,
    NUMERICAL_SOLUTION
}

# ---------- GENERAL STRUCTURE ----------
NONLINEAR_DIFFERENTIAL_EQUATION
    := F(
        independent_variables,
        unknown_function,
        derivatives
    ) = 0

NONLINEARITY
    -> occurs_when(LINEAR_OPERATOR_STRUCTURE_FAILS)

# ---------- SOURCES OF NONLINEARITY ----------
NONLINEARITY
    -> may_arise_from(PRODUCTS_OF_UNKNOWN_AND_DERIVATIVES)
    -> may_arise_from(POWERS_OF_UNKNOWN_OR_DERIVATIVES)
    -> may_arise_from(NONLINEAR_FUNCTIONS_OF_UNKNOWN)
    -> may_arise_from(NONLINEAR_COEFFICIENT_DEPENDENCE_ON_UNKNOWN)

examples:
    y*y_prime
    y^2
    (y_prime)^2
    sin(y)
    exp(y)

# ---------- NONLINEAR ODE ----------
NONLINEAR_ODE
    -> nonlinear_differential_equation_with(ONE_INDEPENDENT_VARIABLE)

example:

y_prime
    = y^2 - x

example:

y_double_prime
    + sin(y)
    = 0

NONLINEAR_ODE
    -> may_require(SPECIAL_ANALYTICAL_OR_NUMERICAL_METHODS)

# ---------- NONLINEAR PDE ----------
NONLINEAR_PDE
    -> nonlinear_differential_equation_with(PARTIAL_DERIVATIVES)

examples:

BURGERS_EQUATION:
    partial_u/partial_t
    + u*partial_u/partial_x
    =
    nu*partial_squared_u/partial_x_squared

NONLINEAR_WAVE_EQUATION:
    partial_squared_u/partial_t_squared
    - c^2*LAPLACIAN(u)
    + g(u)
    = 0

NONLINEAR_PDE
    -> may_form(SHOCKS)
    -> may_form(SOLITONS)
    -> may_form(PATTERNS)

# ---------- SUPERPOSITION FAILURE ----------
if
    y1 and y2
        solve(NONLINEAR_EQUATION)
then

c1*y1 + c2*y2
    -/> generally_solution

SUPERPOSITION_PRINCIPLE
    -> generally_not_available

NONLINEAR_SOLUTION_SET
    -/> generally_VECTOR_SPACE

# ---------- AUTONOMOUS SYSTEM ----------
AUTONOMOUS_NONLINEAR_SYSTEM
    := x_prime = F(x)

F
    -> NONLINEAR_VECTOR_FIELD

PHASE_SPACE
    -> domain_of(STATE_VECTOR)

TRAJECTORY
    -> solution_curve_in(PHASE_SPACE)

# ---------- EQUILIBRIUM ----------
EQUILIBRIUM_POINT
    := x_star
       such_that
       F(x_star) = 0

NONLINEAR_SYSTEM
    -> may_have(MULTIPLE_EQUILIBRIA)

EQUILIBRIA
    -> may_have(DIFFERENT_STABILITY_TYPES)

# ---------- LINEARISATION ----------
near equilibrium x_star:

F(x)
    ~= F(x_star)
       + J_F(x_star)*(x-x_star)

because
    F(x_star) = 0

x_prime
    ~= J_F(x_star)*(x-x_star)

LINEARISATION
    -> approximates(NONLINEAR_DYNAMICS_LOCALLY)

# ---------- JACOBIAN ----------
JACOBIAN(F)
    := matrix(
        partial_F_i/partial_x_j
    )

JACOBIAN_AT_EQUILIBRIUM
    -> determines(LOCAL_LINEAR_SYSTEM)

EIGENVALUES
    -> support(LOCAL_STABILITY_ANALYSIS)
       under(APPROPRIATE_HYPERBOLIC_CONDITIONS)

# ---------- STABILITY ----------
STABILITY
    -> describes(RESPONSE_TO_SMALL_PERTURBATIONS)

LYAPUNOV_STABILITY
    -> nonlinear_stability_framework

ASYMPTOTIC_STABILITY
    -> trajectories_approach(EQUILIBRIUM)

INSTABILITY
    -> perturbations_may_grow_or_depart

LINEARISATION
    -> may_be_inconclusive_for(NONHYPERBOLIC_EQUILIBRIA)

# ---------- LYAPUNOV FUNCTION ----------
LYAPUNOV_FUNCTION
    := V(x)

desired_properties_may_include:
    V(x_star) = 0
    V(x) > 0 near x_star
    dV/dt <= 0

LYAPUNOV_METHOD
    -> studies(STABILITY_WITHOUT_EXPLICIT_SOLUTION)

# ---------- BIFURCATION ----------
BIFURCATION
    -> qualitative_change_in(DYNAMICAL_BEHAVIOUR)
       as(PARAMETER_VARIES)

NONLINEAR_SYSTEM
    -> may_exhibit(SADDLE_NODE_BIFURCATION)
    -> may_exhibit(TRANSCRITICAL_BIFURCATION)
    -> may_exhibit(PITCHFORK_BIFURCATION)
    -> may_exhibit(HOPF_BIFURCATION)

BIFURCATION_POINT
    -> parameter_value_where(QUALITATIVE_STRUCTURE_CHANGES)

# ---------- PERIODIC SOLUTIONS ----------
NONLINEAR_SYSTEM
    -> may_have(LIMIT_CYCLE)

LIMIT_CYCLE
    := isolated_closed_trajectory

LIMIT_CYCLE
    -> may_be(STABLE)
    -> may_be(UNSTABLE)

PERIODIC_BEHAVIOUR
    -> cannot_generally_be_understood_by_SUPERPOSITION)

# ---------- CHAOS ----------
NONLINEAR_DYNAMICAL_SYSTEM
    -> may_exhibit(CHAOTIC_DYNAMICS)

CHAOS
    -> may_include(SENSITIVE_DEPENDENCE_ON_INITIAL_CONDITIONS)
    -> may_include(APERIODIC_LONG_TERM_BEHAVIOUR)
    -> may_include(STRUCTURED_ATTRACTORS)

NONLINEARITY
    -> necessary_but_not_sufficient_for(CHAOS_IN_STANDARD_DETERMINISTIC_CONTEXTS)

# ---------- EXACT / IMPLICIT SOLUTIONS ----------
NONLINEAR_EQUATION
    -> may_have(EXACT_SOLUTION)
    -> may_have(IMPLICIT_SOLUTION)
    -> may_require(SPECIAL_FUNCTIONS)
    -> may_have(NO_CLOSED_FORM_SOLUTION)

ANALYTICAL_SOLVABILITY
    -> depends_on(SPECIFIC_STRUCTURE)

# ---------- SPECIAL METHODS ----------
METHODS := {
    SEPARATION_OF_VARIABLES,
    SUBSTITUTION,
    PHASE_PLANE_ANALYSIS,
    PERTURBATION_METHODS,
    ASYMPTOTIC_METHODS,
    VARIATIONAL_METHODS,
    CHARACTERISTICS,
    TRANSFORMATIONS
}

METHOD_SELECTION
    -> depends_on(NONLINEAR_STRUCTURE)

# ---------- PERTURBATION ----------
if
    small_parameter epsilon
then

solution
    -> may_be_expanded_as(ASYMPTOTIC_SERIES)

PERTURBATION_METHOD
    -> approximates(NONLINEAR_SOLUTION)

REGULAR_PERTURBATION
    -/> sufficient_for(SINGULARLY_PERTURBED_PROBLEMS)

# ---------- NUMERICAL METHODS ----------
NUMERICAL_METHODS := {
    RUNGE_KUTTA_METHODS,
    NEWTON_ITERATION,
    SHOOTING_METHOD,
    FINITE_DIFFERENCE_METHOD,
    FINITE_ELEMENT_METHOD,
    FINITE_VOLUME_METHOD
}

NONLINEAR_DISCRETISATION
    -> may_produce(NONLINEAR_ALGEBRAIC_SYSTEM)

NEWTON_METHOD
    -> may_solve(DISCRETE_NONLINEAR_SYSTEM)

# ---------- INITIAL CONDITIONS ----------
INITIAL_VALUE_PROBLEM
    := NONLINEAR_DIFFERENTIAL_EQUATION
       + INITIAL_DATA

EXISTENCE_AND_UNIQUENESS
    -> depend_on(REGULARITY_AND_LIPSCHITZ_TYPE_CONDITIONS)

NONLINEARITY
    -/> automatically_implies(NONUNIQUENESS)

# ---------- BOUNDARY VALUE PROBLEM ----------
BOUNDARY_VALUE_PROBLEM
    := NONLINEAR_DIFFERENTIAL_EQUATION
       + BOUNDARY_DATA

NONLINEAR_BVP
    -> may_have(NO_SOLUTION)
    -> may_have(ONE_SOLUTION)
    -> may_have(MULTIPLE_SOLUTIONS)

SOLUTION_MULTIPLICITY
    -> depends_on(PROBLEM_STRUCTURE)

# ---------- CONSERVATION LAWS ----------
NONLINEAR_CONSERVATION_LAW
    := partial_u/partial_t
       + DIVERGENCE(F(u))
       = 0

CLASSICAL_SOLUTION
    -> may_develop(DISCONTINUITY)

WEAK_SOLUTION
    -> extends(SOLUTION_AFTER_SHOCK_FORMATION)

ENTROPY_CONDITION
    -> may_select(PHYSICALLY_RELEVANT_WEAK_SOLUTION)

# ---------- APPLICATIONS ----------
NONLINEAR_DIFFERENTIAL_EQUATION
    -> applied_to(FLUID_DYNAMICS)
    -> applied_to(POPULATION_DYNAMICS)
    -> applied_to(CHEMICAL_REACTIONS)
    -> applied_to(NONLINEAR_OSCILLATIONS)
    -> applied_to(GENERAL_RELATIVITY)
    -> applied_to(PLASMA_PHYSICS)
    -> applied_to(BIOLOGICAL_SYSTEMS)
    -> applied_to(CONTROL_THEORY)

# ---------- RELATION GRAPH ----------
NONLINEAR_TERMS
    -> define(NONLINEAR_DIFFERENTIAL_EQUATION)

NONLINEAR_SYSTEM
    -> has(EQUILIBRIA)

JACOBIAN
    -> linearises(NONLINEAR_SYSTEM)

LINEARISATION
    -> supports(LOCAL_STABILITY_ANALYSIS)

PARAMETER_CHANGE
    -> may_trigger(BIFURCATION)

NONLINEAR_DYNAMICS
    -> may_generate(LIMIT_CYCLES)
    -> may_generate(CHAOS)

NUMERICAL_DISCRETISATION
    -> produces(NONLINEAR_ALGEBRAIC_SYSTEM)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "nonlinear differential equation" ~= "differential equation not linear in the unknown function and its derivatives",
    "nonlinear ODE" ~= "ordinary differential equation with nonlinear dependence on the unknown or derivatives",
    "nonlinear PDE" ~= "partial differential equation with nonlinear dependence on the unknown or derivatives",
    "linearisation" ~= "local first-order linear approximation of nonlinear dynamics",
    "bifurcation" ~= "qualitative change in solution or dynamical structure as a parameter varies",
    "limit cycle" ~= "isolated periodic orbit of a dynamical system"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    nonlinear_equation_does_not_obey_general_superposition: true,
    nonlinear_does_not_mean_unsolvable: true,
    nonlinear_does_not_automatically_mean_chaotic: true,
    linearisation_is_local_approximation: true,
    linearisation_may_be_inconclusive_at_nonhyperbolic_equilibria: true,
    existence_uniqueness_require_problem_specific_hypotheses: true,
    numerical_solution_requires_error_and_stability_analysis: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    superposition_failure: 0.99,
    linearisation_relation: 0.99,
    stability_relation: 0.98,
    bifurcation_relation: 0.98,
    numerical_method_relation: 0.98,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(differential_equation)
    -> identify(nonlinear_terms)
    -> classify(ode_or_pde)
    -> determine(order_and_domain)
    -> identify(initial_or_boundary_conditions)
    -> detect(equilibria_if_dynamical_system)
    -> construct(local_linearisation_if_useful)
    -> analyse(stability_bifurcation_or_phase_structure_if_requested)
    -> select(analytical_asymptotic_or_numerical_method)
    -> solve_or_approximate()
    -> verify(solution_or_residual)
    -> validate(existence_uniqueness_and_method_conditions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed nonlinear-differential-equation semantic graph",
    primary_entity: NONLINEAR_DIFFERENTIAL_EQUATION,
    preserve: [
        unknown_function,
        independent_variables,
        nonlinear_terms,
        derivative_order,
        parameters,
        initial_conditions,
        boundary_conditions,
        domain
    ],
    infer_through: [
        nonlinear_odes,
        nonlinear_pdes,
        equilibria,
        jacobians,
        linearisations,
        stability,
        bifurcations,
        numerical_methods
    ],
    avoid: [
        applying_superposition_to nonlinear solutions,
        assuming_nonlinear means chaotic,
        treating_linearisation_as global model,
        assuming_unique_solution_without hypotheses,
        ignoring_multiple_solution branches,
        treating_numerical approximation as exact solution
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + NONLINEAR_DIFFERENTIAL_EQUATION
    + NONLINEARITY
    + NONLINEAR_ODE
    + NONLINEAR_PDE
    + EQUILIBRIUM
    + LINEARISATION
    + BIFURCATION
    + CHAOS
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> NONLINEAR_DIFFERENTIAL_EQUATION
    -> EQUATION_IDENTIFICATION
    -> NONLINEARITY_ANALYSIS
    -> ODE_PDE_CLASSIFICATION
    -> ORDER_DOMAIN_ANALYSIS
    -> CONDITION_ANALYSIS
    -> EQUILIBRIUM_ANALYSIS
    -> LINEARISATION_ANALYSIS
    -> STABILITY_BIFURCATION_ANALYSIS
    -> METHOD_SELECTION
    -> SOLUTION_OR_APPROXIMATION
    -> SOLUTION_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW