Lean math dynamical

Lean 4 Nonlinear Dynamics & Stability

Guide to formalizing dynamical systems, stability theory, and bifurcation in Lean 4.

Routing

• USE FOR: nonlinear dynamical systems, Lyapunov stability, bifurcation theory, catastrophe theory, control theory, phase-portrait analysis, attractor classification, and any deterministic system with state evolution over time in Lean 4.
• DO NOT USE FOR: stochastic dynamics (delegate to @lean-math-stochastic); pure analysis / topology (delegate to @lean-math-analysis); optimization-only control problems (delegate to @lean-math-optimization); writing one specific proof (delegate to @lean-proof).
• TRIGGERS: Lyapunov, stability, bifurcation, cusp catastrophe, attractor, phase portrait, dynamical system, control theory, equilibrium.

Workflow

1. Classify the system (Part 1) — discrete-time, continuous-time, gradient, conservative, controlled.
2. Pick the appropriate technique below (Lyapunov, contraction, bifurcation-normal-form, catastrophe-classification).
3. Handoff to @lean-proof for the concrete proof; if it bottoms out in a derivative / measure step, handoff to @lean-math-analysis.

Recovery & STOP

• STOP if the proof depends on a manifold or smooth-structure API not in Mathlib at the current pin — escalate to @lean-research.
• STOP if a stochastic perturbation enters the model — re-route to @lean-math-stochastic; this skill covers only deterministic dynamics.

Handoffs

• Predecessors: agent:gateway, skill:lean-proof (mid-proof stability goal), skill:lean-research (catastrophe-theory result survey).
• Successors: skill:lean-proof (apply the dynamical pattern), skill:lean-proof-review (audit Lyapunov candidate), skill:lean-math-analysis (continuous-derivative or contraction reduction).

Detailed reference

Full encyclopaedia content (Parts 1 through 6) lives in references/lean4-math-dynamical.md. Load that file when authoring; the SKILL.md only carries the dispatch contract and the high-frequency pitfalls / recipes (kept inline below).

Part	Topic	Covers
Part 1	Dynamical Systems Taxonomy	discrete-time, continuous-time, gradient, conservative, controlled
Part 2	Lyapunov Stability Theory	candidate construction, positive-definiteness, LaSalle
Part 3	Bifurcation and Catastrophe Theory	cusp / fold / pitchfork normal forms
Part 4	Phase Space Analysis	phase portraits, equilibria classification
Part 5	Control Theory Connections	Lyapunov-control, control-Lyapunov functions
Part 6	Nonlinear Methods Toolbox	linearisation, normal-form reduction, numerical-continuation hints

Part 7 — Research Council Integration

Consolidated into the single canonical routing matrix: references/research-council-skill-map.md (see the "Dynamical" section). When dispatching a question to a council member, cite that table rather than restating the rows here.

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Part 8 — IVT Sign-Change Pattern

Extracted to single canonical reference: references/lean4-ivt-patterns.md. That file owns the canonical incantation, the asymmetric_three_roots_ivt walk-through, and the polynomial-continuity prerequisite.

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Part 9 — Common Pitfalls (Stability & Bifurcation)

Pitfall	Symptom	Recovery
Lyapunov candidate not positive-definite	V 0 = 0 proved, but V x > 0 for x ≠ 0 fails	Add a quadratic-form witness (x^T Q x with Q PSD); check posDef_iff_eigenvalues_pos family
Contraction map without [CompleteSpace α]	ContractingWith.fixedPoint won't apply	Add [CompleteSpace α] instance or restrict to a closed subset and use IsCompact.completeSpace
IVT sign-error in bifurcation diagram	Existential ∃ c, f c = 0 won't close	Re-check sign of f a and f b; see references/lean4-ivt-patterns.md for the canonical asymmetric_three_roots_ivt walk-through
Catastrophe normal form drifted from repository canon	cusp / fold polynomial signs don't match expectations	Compare against Mathlib.Analysis.SpecialFunctions.Pow.Real; there is no canonical catastrophe API in Mathlib — keep local definitions in the host repository's catastrophe namespace
Discrete- vs continuous-time confusion	Tendsto with the wrong filter	Discrete: atTop on ℕ; continuous: atTop on ℝ (often with a measure-preserving step)
Spurious equilibrium from simp overreach	f x = x "proved" by simplifying both sides to 0	Disable simp for the candidate equilibrium proof; use linear_combination or explicit substitution

Cross-reference: repository stability tactics

• Host-repository Lyapunov modules — quadratic-form Lyapunov constructions for local cusp + phase-portrait models.
• Host-repository catastrophe modules — cusp-form polynomial bifurcation (the 3-real-roots case).
• references/lean4-ivt-patterns.md — asymmetric_three_roots_ivt walk-through (canonical entry-point for sign-change existence).
• references/lean4-contraction-catalog.md — catalog of contraction-mapping templates (uses ContractingWith / LipschitzWith).

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Part 10 — Banach Fixed-Point Recipe

The most common "I need a fixed point" pattern in this corpus:

-- Given a self-map f : α → α and a contraction constant K < 1:
example {α : Type*} [MetricSpace α] [CompleteSpace α]
    (f : α → α) {K : NNReal} (hK : K < 1)
    (hC : ContractingWith K f) :
    ∃! x, f x = x :=
  ⟨hC.fixedPoint, hC.fixedPoint_isFixedPt,
   fun y hy => hC.fixedPoint_unique hy⟩

Project-relevant adaptations:

• Sub-Banach setting: IsCompact.completeSpace lets you restrict to a closed ball when global completeness is unwieldy.
• Iteration bounds: ContractingWith.aux_dist_le gives dist (f^[n] x) (fixedPoint) ≤ K^n * dist x (fixedPoint) / (1 - K).
• Existence-only (no uniqueness): Schauder fixed-point — not in Mathlib at the current pin; use @lean-research to confirm before authoring.

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See also

• ../../references/lean4-math-dynamical.md — Nonlinear Dynamics Encyclopaedia (full encyclopaedia, extracted from this skill)
• ../../templates/Template_Dynamics.md — Template: Lyapunov / Markov / contraction mappings
• ../../references/lean4-proof-strategy.md — Proof strategy: real-valued contraction patterns