Dynamical Systems

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

flow "DynamicalSystemsOntology" v1.0

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

DYNAMICAL_SYSTEMS := {
 type: MathematicalEvolutionFramework,
 domain: CalculusAndAnalysis,
 purpose: "Study how states evolve over time according to deterministic or stochastic evolution rules, including equilibria, stability, periodic behaviour, bifurcations and chaos."
}

CONCEPTS := {
 DYNAMICAL_SYSTEM,
 STATE_SPACE,
 STATE,
 TIME,
 FLOW,
 MAP,
 TRAJECTORY,
 ORBIT,
 FIXED_POINT,
 EQUILIBRIUM,
 STABILITY,
 ATTRACTOR,
 PERIODIC_ORBIT,
 PHASE_SPACE,
 PHASE_PORTRAIT,
 BIFURCATION,
 CHAOS,
 LYAPUNOV_EXPONENT
}

# ---------- GENERAL SYSTEM ----------
DYNAMICAL_SYSTEM
 := {
  STATE_SPACE_X,
  TIME_SET_T,
  EVOLUTION_RULE
 }

STATE
 := x in X

EVOLUTION
 -> maps(INITIAL_STATE_TO_FUTURE_OR_PAST_STATE)

# ---------- CONTINUOUS TIME ----------
continuous-time system:

dx/dt
 =
f(x,t)

AUTONOMOUS_SYSTEM:

dx/dt
 =
f(x)

SOLUTION
 x(t)
 -> TRAJECTORY_IN(STATE_SPACE)

# ---------- FLOW ----------
for autonomous continuous system:

phi_t(x_0)
 -> state_at_time_t

FLOW_PROPERTY:

phi_0
 = identity

phi_(t+s)
 =
phi_t composed_with phi_s

when solutions exist uniquely

FLOW
 -> continuous_time_evolution_operator

# ---------- DISCRETE TIME ----------
discrete system:

x_(n+1)
 =
F(x_n)

iteration:

x_n
 =
F^n(x_0)

MAP
 -> discrete_time_evolution_rule

ORBIT
 := {
 x_0,
 x_1,
 x_2,
 ...
 }

# ---------- TRAJECTORY ----------
TRAJECTORY
 -> ordered_state_evolution

continuous:
 {x(t):t in T}

discrete:
 {x_n:n in integers_or_naturals}

TRAJECTORY
 -> determined_by(INITIAL_CONDITION_AND_EVOLUTION_RULE)
    under(DETERMINISTIC_UNIQUENESS)

# ---------- PHASE SPACE ----------
PHASE_SPACE
 -> space_of_all_possible_states

PHASE_POINT
 -> instantaneous_system_state

PHASE_PORTRAIT
 -> geometric_representation_of(TRAJECTORIES)

DYNAMICAL_SYSTEMS
 -> analyse(GEOMETRY_OF_STATE_EVOLUTION)

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

x_star
 -> EQUILIBRIUM

iff

f(x_star)
 = 0

then

x(t)
 = x_star

for equilibrium initial condition

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

x_star
 -> FIXED_POINT

iff

F(x_star)
 =
x_star

FIXED_POINT
 -> discrete_analogue_of(EQUILIBRIUM)

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

x
 =
x_star + xi

then approximately

d xi/dt
 =
J_f(x_star)
*
xi

J_f
 -> JACOBIAN_MATRIX

LINEARISATION
 -> local_first_order_dynamics

# ---------- EIGENVALUE STABILITY ----------
for continuous linearisation:

d xi/dt
 =
A xi

if all eigenvalues of A
 have negative real parts

-> ASYMPTOTICALLY_STABLE
   under(HYPERBOLIC_LINEARISATION_CONTEXT)

if any eigenvalue
 has positive real part

-> UNSTABLE

ZERO_REAL_PART_CASE
 -> requires(FURTHER_ANALYSIS)

# ---------- DISCRETE STABILITY ----------
for map linearisation:

xi_(n+1)
 =
A xi_n

if all eigenvalues satisfy

|lambda_i|<1

-> LOCALLY_ASYMPTOTICALLY_STABLE
   under appropriate nonlinear hypotheses

if any

|lambda_i|>1

-> UNSTABLE

UNIT_CIRCLE_CASE
 -> requires(FURTHER_ANALYSIS)

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

if

for every epsilon>0
exists delta>0

such_that

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

implies

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

for all future t

STABILITY
 -> nearby_states_remain_nearby

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

if

x_star
 -> LYAPUNOV_STABLE

and

nearby trajectories
 -> converge_to(x_star)

as
 t -> infinity

ASYMPTOTIC_STABILITY
 -> stability_plus_attraction

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

typically

V(x_star)=0

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

and

dV/dt
 <= 0

STABILITY
 -> inferred_without(EXPLICIT_SOLUTION)

if

dV/dt<0
 away_from_equilibrium

-> supports(ASYMPTOTIC_STABILITY)

under standard hypotheses

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

x(t+T)
 =
x(t)

for some
 T>0

PERIODIC_ORBIT
 -> closed_trajectory_in(PHASE_SPACE)

discrete:

F^p(x)
 =
x

PERIOD_p_ORBIT
 -> repeats_after(p_iterations)

# ---------- LIMIT CYCLE ----------
LIMIT_CYCLE
 -> isolated_periodic_orbit

STABLE_LIMIT_CYCLE
 -> attracts(NEARBY_TRAJECTORIES)

OSCILLATORY_SYSTEM
 -> may_have(LIMIT_CYCLE)

# ---------- INVARIANT SET ----------
S subset_of X

-> INVARIANT

if evolution of every state in S
 remains_in(S)

INVARIANT_SET
 -> preserved_by(DYNAMICS)

examples:
 EQUILIBRIA
 PERIODIC_ORBITS
 ATTRACTORS
 INVARIANT_MANIFOLDS

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

if
 A invariant

and
 nearby trajectories
 -> approach(A)

ATTRACTOR
 -> long_term_dynamical_structure

types:
 FIXED_POINT_ATTRACTOR
 LIMIT_CYCLE
 TORUS
 STRANGE_ATTRACTOR

# ---------- BASIN OF ATTRACTION ----------
B(A)
 := {
 x_0 :
 trajectory_from(x_0)
 approaches(A)
 }

BASIN_OF_ATTRACTION
 -> initial_conditions_leading_to(ATTRACTOR)

MULTIPLE_ATTRACTORS
 -> partition_state_space_into(BASINS_WITH_COMPLEX_BOUNDARIES)

# ---------- STABLE / UNSTABLE MANIFOLD ----------
for hyperbolic invariant object:

STABLE_MANIFOLD
 -> states_converging_to(object_forward_in_time)

UNSTABLE_MANIFOLD
 -> states_converging_to(object_backward_in_time)

MANIFOLDS
 -> organise(PHASE_SPACE_GEOMETRY)

# ---------- BIFURCATION ----------
parameter-dependent system:

dx/dt
 =
f(x,mu)

BIFURCATION
 -> qualitative_change_in(DYNAMICS_AS_PARAMETER_VARIES)

examples:
 SADDLE_NODE
 TRANSCRITICAL
 PITCHFORK
 HOPF
 PERIOD_DOUBLING

# ---------- SADDLE NODE ----------
normal form:

dx/dt
 =
mu - x^2

as mu changes:

number_and_stability_of_equilibria
 -> change

SADDLE_NODE_BIFURCATION
 -> creation_or_annihilation_of(FIXED_POINTS)

# ---------- HOPF ----------
pair_of_complex_conjugate_eigenvalues
 -> crosses(IMAGINARY_AXIS)

under suitable nondegeneracy conditions

EQUILIBRIUM
 -> may_exchange_stability_with(PERIODIC_ORBIT)

HOPF_BIFURCATION
 -> onset_of(OSCILLATIONS)

# ---------- CHAOS ----------
CHAOTIC_DYNAMICS
 -> deterministic
 -> aperiodic
 -> sensitive_to(INITIAL_CONDITIONS)

CHAOS
 -/> randomness

NONLINEAR_DYNAMICS
 -> may_generate(COMPLEX_UNPREDICTABLE_LONG_TERM_BEHAVIOUR)

# ---------- SENSITIVE DEPENDENCE ----------
nearby initial conditions:

||delta_x(0)||
 -> small

may evolve as

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

lambda>0
 -> exponential_separation

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

when limit exists appropriately

POSITIVE_MAXIMAL_LYAPUNOV_EXPONENT
 -> evidence_of(SENSITIVE_DEPENDENCE)

LYAPUNOV_EXPONENT
 -> quantitative_stability_or_instability_measure

# ---------- LOGISTIC MAP ----------
x_(n+1)
 =
r*x_n*(1-x_n)

LOGISTIC_MAP
 -> canonical_discrete_nonlinear_system

as r varies:
 FIXED_POINTS
 PERIOD_DOUBLING
 CHAOTIC_REGIMES

LOGISTIC_MAP
 -> example_of(BIFURCATION_ROUTE_TO_CHAOS)

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

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

dz/dt
 =
x*y-beta*z

LORENZ_SYSTEM
 -> nonlinear_continuous_dynamical_system

for suitable parameters:
 -> STRANGE_ATTRACTOR
 -> SENSITIVE_DEPENDENCE

# ---------- CONSERVATIVE SYSTEM ----------
CONSERVATIVE_DYNAMICS
 -> preserves(ENERGY_OR_OTHER_INVARIANTS)

HAMILTONIAN_SYSTEM
 -> phase_space_flow
 -> preserves(SYMPLECTIC_STRUCTURE)

LIOUVILLE_THEOREM
 -> preserves(PHASE_SPACE_VOLUME)

# ---------- DISSIPATIVE SYSTEM ----------
DISSIPATIVE_SYSTEM
 -> contracts_phase_space_volume_in_relevant_context
 -> loses_energy_or_effective_volume

DISSIPATION
 -> may_generate(ATTRACTORS)

CONSERVATIVE_AND_DISSIPATIVE
 -> distinct_dynamical_classes

# ---------- POINCARE MAP ----------
continuous periodic or recurrent flow

-> intersect_with(TRANSVERSE_SECTION)

successive intersections
 -> define(DISCRETE_MAP)

POINCARE_MAP
 -> reduces(CONTINUOUS_DYNAMICS_TO_DISCRETE_DYNAMICS)

PERIODIC_ORBIT
 -> fixed_point_of(POINCARE_MAP)

# ---------- POINCARE SECTION ----------
SECTION
 -> lower_dimensional_surface_transverse_to_flow

TRAJECTORY_INTERSECTIONS
 -> reveal(RECURRENCE_AND_CHAOS)

POINCARE_SECTION
 -> visualises(HIGH_DIMENSIONAL_DYNAMICS)

# ---------- APPLICATIONS ----------
DYNAMICAL_SYSTEMS
 -> applied_to(PHYSICS)
 -> applied_to(ENGINEERING)
 -> applied_to(BIOLOGY)
 -> applied_to(CHEMISTRY)
 -> applied_to(ECONOMICS)
 -> applied_to(CLIMATE_SCIENCE)
 -> applied_to(CONTROL_THEORY)
 -> applied_to(CELESTIAL_MECHANICS)
 -> applied_to(NEURAL_DYNAMICS)

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

EVOLUTION_RULE
 -> generates(TRAJECTORIES)

TRAJECTORY
 -> forms(ORBIT)

FIXED_POINT
 -> stationary_state_of(DISCRETE_MAP)

EQUILIBRIUM
 -> stationary_state_of(CONTINUOUS_FLOW)

LINEARISATION
 -> supports(LOCAL_STABILITY_ANALYSIS)

ATTRACTOR
 -> determines(LONG_TERM_BEHAVIOUR)

BIFURCATION
 -> changes(QUALITATIVE_DYNAMICS)

LYAPUNOV_EXPONENT
 -> measures(SENSITIVITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "dynamical system" ~= "state space equipped with a rule for time evolution",
 "trajectory" ~= "time-ordered path of a system state through state space",
 "equilibrium" ~= "continuous-time state unchanged by the dynamics",
 "fixed point" ~= "state unchanged by a discrete evolution map",
 "attractor" ~= "invariant set approached by nearby trajectories",
 "bifurcation" ~= "qualitative change in dynamics caused by parameter variation"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 continuous_and_discrete_time_systems_require_distinct_evolution_formulations: true,
 linear_stability_tests_require_local_and_hyperbolicity_context: true,
 zero_real_part_or_unit_modulus_eigenvalues_require_further_analysis: true,
 stability_and_attraction_are_distinct_properties: true,
 deterministic_chaos_is_not_equivalent_to_randomness: true,
 positive_Lyapunov_exponent_is_context_dependent_evidence_not_complete_universal_definition_of_chaos: true,
 numerical_simulation_must_not_be_confused_with_exact_long_term_dynamics: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 flow_map_relation: 0.99,
 equilibrium_stability_relation: 0.99,
 attractor_relation: 0.99,
 bifurcation_relation: 0.99,
 chaos_Lyapunov_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(state_space_time_set_and_evolution_rule)
 -> classify(continuous_discrete_autonomous_or_nonautonomous_system)
 -> identify(initial_conditions_and_trajectories)
 -> locate(equilibria_fixed_points_and_periodic_orbits)
 -> linearise(near_invariant_objects_if_valid)
 -> analyse(stability_and_Lyapunov_structure)
 -> identify(attractors_basins_and_invariant_manifolds)
 -> vary(parameters_for_bifurcation_analysis_if_requested)
 -> analyse(sensitivity_and_Lyapunov_exponents_if_chaotic_behavior_relevant)
 -> validate(local_global_and_numerical_assumptions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed dynamical-systems semantic graph",
 primary_entity: DYNAMICAL_SYSTEMS,
 preserve: [
  state_space,
  time_set,
  evolution_rule,
  parameters,
  initial_conditions,
  trajectory,
  invariant_sets,
  stability_context
 ],
 infer_through: [
  flows,
  maps,
  equilibria,
  fixed_points,
  periodic_orbits,
  attractors,
  bifurcations,
  Lyapunov_exponents,
  chaos
 ],
 avoid: [
  treating_continuous_and_discrete_systems_as identical,
  inferring_nonlinear_stability_from_inconclusive_linearisation,
  confusing_stability_with attraction,
  calling_complex_behavior_random_without analysis,
  claiming_chaos_from_visual_irregularity_alone,
  treating_finite_numerical_trajectory_as proof_of_asymptotic behavior
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + DYNAMICAL_SYSTEMS
 + STATE_SPACE
 + FLOW
 + MAP
 + TRAJECTORY
 + EQUILIBRIUM
 + STABILITY
 + ATTRACTOR
 + BIFURCATION
 + CHAOS
 + LYAPUNOV_EXPONENT
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> DYNAMICAL_SYSTEMS
 -> SYSTEM_CLASSIFICATION
 -> STATE_TIME_EVOLUTION_IDENTIFICATION
 -> TRAJECTORY_ANALYSIS
 -> INVARIANT_OBJECT_IDENTIFICATION
 -> LINEARISATION
 -> STABILITY_ANALYSIS
 -> ATTRACTOR_BASIN_ANALYSIS
 -> BIFURCATION_ANALYSIS
 -> CHAOS_SENSITIVITY_ANALYSIS
 -> ASSUMPTION_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW