Arnold — *Ordinary Differential Equations* (Fast Track 0.4) — Audit + Gap Plan

audit_completeness: reduced. Full PDF could not be retrieved within the audit window: the loshijosdelagrange WordPress mirror exceeds WebFetch's 10 MB content limit; Springer's catalogue page sits behind an IDP authentication redirect; Internet Archive, Google Books, and Amazon previews returned only metadata. This plan is built from the canonical TOC (Springer GTM-tier "yellow book", 3rd Russian ed. translated by Roger Cooke, 1992 = ISBN 3-540-54813-0, 334 pp.) reconstructed from citation network + reviewer summaries + the parallel Arnold Mathematical Methods audit plan, and from the standard ODE-pedagogy cross-references (Hirsch-Smale-Devaney, Strogatz, Tenenbaum-Pollard, Coddington-Levinson). Per AGENTIC_EXECUTION_PLAN §6.6 this plan can trigger P2/P3/P4 against its punch-list, but P5 verification cannot mark the book equivalence-covered; only equivalence-partial. A full re-audit is queued in NEED_TO_SOURCE.md once a local PDF lands.

Book: V. I. Arnold, Ordinary Differential Equations, 3rd edition, translated by Roger Cooke from the Russian, Springer-Verlag 1992 (Springer-Textbook; ISBN 3-540-54813-0). 334 pages. (Distinct from the 1973 MIT-Press first English edition transl. R. A. Silverman, which is the Russian 2nd-edition translation.)

Fast Track entry: 0.4 — the canonical Section 0 prereq on ODEs, the last missing prerequisite audit per FASTTRACK_COVERAGE_ROADMAP.md Wave 5. Marked BUY in docs/catalogs/FASTTRACK_BOOKLIST.md.

Distinguish from sibling Arnold plan. A different Arnold book — Mathematical Methods of Classical Mechanics (Fast Track 1.11) — has its own audit plan at plans/fasttrack/arnold-mathematical-methods.md. That book is the symplectic / Hamiltonian / Lagrangian mechanics companion and is consulted here only where the two books' content overlaps (vector fields, flows, integral curves, geodesic flow as Hamiltonian). This plan is the canonical ODE-first stub.

Purpose: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite per the orchestration protocol). Output is a concrete priority-ordered punch-list of new units to write so that Arnold ODE is covered to the equivalence threshold (≥95% effective coverage; see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Not a full P1 audit (no line-number-level Problem inventory possible without the PDF).

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§1 What Arnold's ODE book is for

Arnold's Ordinary Differential Equations is the canonical geometric, qualitative introduction to ODEs. It inverts the standard power-series-and-tricks approach inherited from Tenenbaum-Pollard or Boyce-DiPrima: where those texts solve equations, Arnold studies vector fields on a phase space and their flows, integrates only when integration is illuminating, and uses the topology of phase portraits — rather than closed-form solutions — as the primary object of understanding. The book is foundational for the Fast Track because the geometric perspective is what is reused everywhere downstream: in Hamiltonian mechanics (FT 1.11), symplectic geometry (FT 1.11 / 1.04), dynamical systems (FT 1.12), and analysis on manifolds (FT 1.10).

Distinctive Arnold choices:

1. Phase space and flow are primitive. Chapter 1 introduces phase space, vector field, integral curve, phase flow, and one-parameter group of diffeomorphisms before writing down a single equation-in-coordinates. The ODE $\dot{x}=v(x)$ is the coordinate shadow of "the flow $g^t$ of the vector field $v$." This is the opposite framing from the standard US undergraduate ODE text (Boyce-DiPrima, Tenenbaum-Pollard) which writes equations first and reaches geometric interpretation only at the end if at all.

2. Existence-uniqueness via Picard iteration is in Chapter 4 — deliberately deferred. Chapters 1-3 build phase-portrait intuition, classification of equilibria, and linear-system technique; the existence-uniqueness proof via the contraction-mapping principle is pushed back so the reader sees what the theorem is for before seeing how it is proved. Pedagogical inversion vs Hirsch-Smale-Devaney Differential Equations, Dynamical Systems, and an Introduction to Chaos (3rd ed., 2013, Ch. 17) and Coddington-Levinson Theory of Ordinary Differential Equations (McGraw-Hill 1955) where the theorem is proved first.

3. Linear systems through the matrix exponential and the saddle / node / focus / centre classification of planar equilibria. Chapter 3 develops $\exp (At)$, classification of $2\times 2$ real linear systems by trace and determinant, and stability of linear equilibria. Strogatz Nonlinear Dynamics and Chaos (2nd ed., 2015, Ch. 5-6) covers the same material at a less rigorous level; Arnold gives the rigorous version with $e^{At}$ as a one-parameter group.

4. Rectification of a vector field (straightening theorem). A non-vanishing smooth vector field is locally diffeomorphic to a constant vector field. Arnold's geometric statement of the existence-uniqueness-smooth-dependence package: in the neighbourhood of a non-equilibrium point the flow is trivial up to diffeomorphism. This is the local model, and it is the right way to state local ODE theory once one is doing geometry rather than calculation. Absent from standard ODE texts; appears explicitly in Arnold and in Spivak's Calculus on Manifolds (Ch. 5) for the straightening of a vector field.

5. Differential equations on manifolds (Ch. 5). Vector field on a smooth manifold, one-parameter group of diffeomorphisms $g^t$, Lie derivative, first integrals, the geometric statement that an ODE on ${\mathbb{R}}^n$ is a special case of an ODE on a manifold. The chapter foreshadows everything in 03-modern-geometry/ and 05-symplectic/. Lie derivative appears here in its primitive form as $\frac{d}{dt}{|}_{t=0}(g^t{)}^{\ast }f$ rather than the Cartan- formula form ${\iota }_{X}d+d{\iota }_X$ used in differential-forms texts.

6. Stability theory via Lyapunov functions. The Lyapunov direct method for stability of equilibria is presented geometrically as "a function decreasing along trajectories"; asymptotic stability via strict decrease. Originator-text status: Arnold cites Lyapunov 1892 Problème général de la stabilité du mouvement (Kharkov Math. Soc.; French translation Princeton 1947) as the foundational source. This is the right entry point to nonlinear stability before Hirsch-Smale-Devaney Ch. 9 or Khalil Nonlinear Systems Ch. 4.

7. Linear systems with periodic coefficients — Floquet theory. Chapter 3 closes with the Floquet-Lyapunov reduction theorem: $\dot{x}=A(t)x$ with $A(t+T)=A(t)$ admits a fundamental solution $X(t)=P(t)e^{Bt}$ with $P$ periodic, $B$ constant. Floquet 1883 (Ann. Sci. Éc. Norm. Sup.) is the originator. This is the gateway to parametric resonance, Hill's equation, and Mathieu equation. Stronger emphasis than in Hirsch-Smale-Devaney.

8. Examples-driven throughout. Arnold runs persistent worked examples: the population / radioactive-decay scalar ODE, the harmonic oscillator, the mathematical pendulum (and its phase portrait with separatrix), the predator-prey Volterra-Lotka system, the geodesic-flow scalar example as foreshadowing of geometric mechanics. Each is reused across chapters to illustrate theorem-after-theorem; the pendulum phase portrait in particular is the running visual.

Distinguished from the Mathematical Methods of Classical Mechanics (FT 1.11): Arnold ODE is the prereq, restricted to ODEs as a self-contained geometric theory; Hamiltonian / Lagrangian / symplectic content is touched only via the pendulum and geodesic-flow examples in the final chapter. Hamilton's equations, Poisson brackets, action-angle, and KAM live in the Mathematical Methods book and its audit plan.

The book is not a numerical-methods text (no Euler / Runge-Kutta analysis), is not a Sturm-Liouville / boundary-value / Bessel- function compendium (Tenenbaum-Pollard territory), and is not a PDE text. PDE coverage is a separate Fast Track entry and is explicitly deferred per §5 below.

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§2 Coverage table (Codex vs Arnold ODE)

Cross-referenced against the current Codex corpus (~313 units across nine chapters). Codex has substantial Hamiltonian / symplectic content in 05-symplectic/ (49 units shipped, including 05.02.01-hamiltonian-vector-field, 05.02.07-liouville-volume, 05.02.08-poincare-recurrence, 05.02.06-geodesic-flow-hamiltonian), which touches Arnold-ODE Chapter 5 material from the symplectic side but not from the ODE-as-flow side. The functional-analysis block in 02-analysis/11-functional-analysis/ contains 02.11.04-banach-spaces (which underwrites the contraction-mapping proof) but no contraction-mapping unit per se. The 03-modern-geometry/02-manifolds/ chapter contains only 03.02.01-smooth-manifold — no tangent bundle, no vector field, no flow unit.

✓ = covered at Arnold-equivalent depth, △ = topic present in some form but Codex unit shallower / framed differently, ✗ = not covered.

Chapter 1 — Basic Concepts (phase space, vector field, flow)

Arnold topic	Codex unit(s)	Status	Note
Phase space, state of a process	—	✗	Gap. Foundational definition; absent.
Vector field on ${\mathbb{R}}^n$ / on a manifold	—	✗	Gap (high priority — load-bearing for whole curriculum).
Integral curve, phase curve, phase flow	—	✗	Gap.
One-parameter group of diffeomorphisms $g^t$	partial in 05.02.01 Hamiltonian vector field	△	Mentioned in Hamiltonian context; not its own unit.
Equilibrium / fixed point of a vector field	—	✗	Gap.
Worked examples: radioactive decay, population, pendulum, harmonic oscillator	—	✗	Gap.
Direction field on ${\mathbb{R}}^2$	—	✗	Gap (low priority — pedagogical).

Chapter 2 — Basic Theorems (existence, uniqueness, dependence)

Arnold topic	Codex unit(s)	Status	Note
Existence theorem (Peano / Picard)	—	✗	Gap (high priority).
Uniqueness theorem (Picard-Lindelöf, Lipschitz)	—	✗	Gap (high priority).
Extendability of solutions / maximal interval	—	✗	Gap.
Continuous dependence on initial data and parameters	—	✗	Gap.
Differentiable dependence on initial data (variational equation)	—	✗	Gap.
Rectification (straightening) theorem	—	✗	Gap (high priority — Arnold-distinctive geometric local model).

Chapter 3 — Linear Systems

Arnold topic	Codex unit(s)	Status	Note
Linear system $\dot{x}=Ax$ with constant $A$	—	✗	Gap.
Matrix exponential $e^{At}$ as one-parameter group	partial: spectral theorem in 02.11.*	△	Operator-exponential exists abstractly; ODE framing absent.
Classification of $2\times 2$ planar linear systems (saddle, node, focus, centre, improper, degenerate)	—	✗	Gap.
Stability of linear equilibria via eigenvalues	—	✗	Gap.
Inhomogeneous linear system / variation of constants	—	✗	Gap.
Linear system with periodic coefficients (Floquet theorem)	—	✗	Gap.
Parametric resonance (Mathieu / Hill)	—	✗	Gap (low priority).

Chapter 4 — Proofs of the Main Theorems

Arnold topic	Codex unit(s)	Status	Note
Contraction-mapping principle (Banach fixed-point)	—	✗	Gap (load-bearing across analysis). 02.11.04-banach-spaces provides the ambient setting only.
Picard iteration as fixed point of integral operator	—	✗	Gap; subsumed by existence unit above.
Gronwall inequality	—	✗	Gap.
Differentiable-dependence proof via variational equation	—	✗	Gap.

Chapter 5 — Differential Equations on Manifolds

Arnold topic	Codex unit(s)	Status	Note
Smooth manifold (definition)	03.02.01-smooth-manifold	✓	Already shipped.
Tangent vector / tangent space at a point	—	✗	Gap (already a known cross-strand gap).
Tangent bundle $TM$	—	✗	Gap (cross-strand).
Vector field on a manifold	—	✗	Gap.
Flow of a vector field on a manifold; existence on compact manifolds	partial via 05.02.01-hamiltonian-vector-field	△	Hamiltonian-only framing.
Lie derivative ${\mathcal{L}}_{X}f$	partial via 05.02.02-poisson-bracket, 03.04.04-exterior-derivative	△	Cartan-formula form present in symplectic chapter; primitive flow-pullback form absent.
First integral / conserved quantity	—	✗	Gap.
Geodesic flow on Riemannian manifold (as ODE example)	partial via 05.02.06-geodesic-flow-hamiltonian	△	Hamiltonian framing; pure-ODE / Lagrangian framing absent.
Lyapunov stability theorem (direct method)	—	✗	Gap.
Asymptotic stability via strict Lyapunov function	—	✗	Gap.
Poincaré-Bendixson theorem (planar limit sets)	—	✗	Gap (Arnold ODE-specific; foundational dynamical-systems result).
Limit cycle	—	✗	Gap.

Aggregate coverage estimate

• Theorem layer: ~5% topic-level (only smooth-manifold definition and partial Hamiltonian-side flow coverage). After priority-1 batch: ~55%; priority-1+2: ~85%; +priority-3: ~93%.
• Notation layer: ~50% aligned. Codex uses $X_H$ for Hamiltonian vector field; Arnold uses $v$ generically and $g^t$ for the flow. Both are standard; align via a §Notation paragraph in the basic vector-field unit. Lyapunov function $V$, first integral $f$, phase flow $g^t$ — adopt Arnold's notation directly.
• Sequencing layer: ~30% — Codex has no Section-0 ODE block at all. Arnold's Ch.1-2-3-4-5 sequence becomes a new Codex chapter family (proposed 02.12-ode/ or 02.07-ode/ — see §4).
• Application layer: ~25%. Pendulum, oscillator, predator-prey, geodesic flow partly visible in the symplectic chapter; ODE-side framing absent.

The gap is substantial but the intuition layer is partly prepared by the symplectic chapter: anyone who has read 05.02.* already has Hamiltonian flows; an ODE block formalises the general case those units secretly depend on.

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§3 Gap punch-list (priority-ordered)

Recommended chapter slot: a new 02.12-ode/ chapter (or 02.07-ode/ if the Section-0 ordering renumbers; see §4) inserted between the multivariable-differentiation chapter and the functional- analysis chapter, with a single forward edge into 05-symplectic/ from the manifold-flow unit.

Priority 1 — Arnold-equivalent backbone (the load-bearing units)

1. 02.12.01 Phase space, vector field, integral curve. Arnold Ch.1 §1-§3 anchor. Three-tier; ~1500 words. Beginner: scalar ODE $\dot{x}=v(x)$ as flow on $\mathbb{R}$; radioactive-decay / logistic worked examples. Intermediate: vector field on ${\mathbb{R}}^n$; integral curve; equilibrium. Master: smooth vector field on a smooth manifold (forward-references 03.02.01). Originator citation: Poincaré 1881-1886 Mémoire sur les courbes définies par une équation différentielle (J. de Math. Pures et Appl.). Load-bearing for whole curriculum — vector field appears downstream in 03.*, 05.*, 07.*.

2. 02.12.02 Phase flow / one-parameter group $g^t$. Arnold §4 anchor. Three-tier; ~1300 words. Beginner: scalar autonomous flow $g^t(x_0)=x(t;x_0)$. Intermediate: $g^t\circ g^s=g^{t+s}$, $g^0=\mathrm{id}$, smoothness. Master: complete vs incomplete flows; flow on compact manifold is complete. Lie derivative as $\frac{d}{dt}{|}_0(g^t{)}^{\ast }f$. Load-bearing for 05.02.*.

3. 02.12.03 Existence and uniqueness of solutions (Picard- Lindelöf). Arnold Ch.2 + Ch.4 anchor; statement in Ch.2, proof via contraction in Ch.4. Three-tier; ~2000 words. Beginner: statement and pendulum worked example. Intermediate: Lipschitz hypothesis; uniqueness vs Peano-only existence. Master: contraction-mapping proof; Picard iteration; Gronwall as consequence. Originator citations: Cauchy 1820s (lectures, not published), Lipschitz 1876 (J. f. Math. 82), Picard 1890 (Mémoire sur la théorie des équations aux dérivées partielles), Lindelöf 1894 (C. R. Acad. Sci.). Foundational across analysis.

4. 02.12.04 Continuous and differentiable dependence on initial data. Arnold Ch.2 §7-§8 + Ch.4 anchor. Three-tier; ~1500 words. Variational equation $\dot{Y}=(Dv)(x(t))Y$ for the derivative of the flow with respect to initial data. Master: Gronwall-style continuous-dependence bound; smoothness of $g^t$ in $(t,x)$ jointly. Worked example: sensitivity in the harmonic oscillator.

5. 02.12.05 Rectification (straightening) of a vector field. Arnold Ch.2 §7 (also called the flow-box theorem). Three-tier; ~1300 words. Statement: in a neighbourhood of a non-equilibrium point, a smooth vector field is diffeomorphic to $\partial / \partial x^1$. Master: proof via flow + transversal section. Arnold-distinctive; the geometric local model. Foundational for 03-modern-geometry/02-manifolds/ and the Frobenius / foliation theorems downstream.

6. 02.12.06 Linear system $\dot{x}=Ax$ and the matrix exponential $e^{At}$. Arnold Ch.3 anchor. Three-tier; ~1700 words. Beginner: scalar $\dot{x}=ax$ ⇒ $x=e^{at}{x}_0$. Intermediate: matrix exponential via series; $\frac{d}{dt} e^{At} = A e^{At}$; one-parameter group property. Master: spectral form $e^{At}={\sum }_{\lambda }{e}^{\lambda t}{P}_{\lambda }(t)$ on generalised eigenspaces; relation to Jordan form. Worked examples: harmonic oscillator, damped oscillator, coupled oscillators. Originator citation: Cauchy 1840s, Sylvester 1883 (matrix functions).

7. 02.12.07 Classification of planar linear systems (saddle / node / focus / centre). Arnold Ch.3 anchor. Three-tier; ~1800 words. The trace-determinant plane; phase portraits for each generic case (saddle, stable / unstable node, stable / unstable focus / spiral, centre); degenerate cases (improper node, parabolic / shear). Master: classification proof via Jordan form; topological vs linear equivalence. Worked example: pendulum linearised at equilibrium and at separatrix. Arnold's characteristic phase-portrait pedagogy in its purest form.

Priority 2 — Lyapunov / Floquet / Poincaré-Bendixson (Arnold-distinctive depth)

8. 02.12.08 Lyapunov stability (direct method). Arnold Ch.5 anchor. Three-tier; ~1700 words. Statement: existence of a positive-definite $V$ with ${\mathcal{L}}_{v}V\le 0$ near equilibrium implies stability; strict decrease implies asymptotic stability. Master: proof sketch; energy functions for Hamiltonian systems are Lyapunov; converse Lyapunov theorem (Massera 1949) as pointer. Originator citation: Lyapunov 1892 Problème général de la stabilité du mouvement (Kharkov; reprinted Princeton 1947). Mandatory originator-prose per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10.

9. 02.12.09 Floquet theorem (linear systems with periodic coefficients). Arnold Ch.3 anchor. Three-tier; ~1500 words. Statement: $\dot{x}=A(t)x$ with $A(t+T)=A(t)$ has fundamental matrix $X(t)=P(t)e^{Bt}$ with $P$ $T$-periodic, $B$ constant. Monodromy matrix $X(T)$; Floquet multipliers and exponents; stability classification. Master: Hill's equation, Mathieu equation, parametric resonance. Originator citation: Floquet 1883 (Ann. Sci. Éc. Norm. Sup. 12).

10. 02.12.10 Poincaré-Bendixson theorem. Arnold Ch.5 anchor. Three-tier; ~1700 words. Statement: a bounded $\omega$-limit set of a planar flow with finitely many equilibria is either an equilibrium, a periodic orbit, or a chain of equilibria connected by heteroclinic / homoclinic orbits. Master: proof via Jordan curve theorem + transversal section; non-existence of chaos in 2D continuous flows. Worked example: Van der Pol oscillator limit cycle. Originator citations: Poincaré 1881-86 Mémoire sur les courbes définies par une équation différentielle, Bendixson 1901 (Acta Math. 24). Mandatory originator-prose for Poincaré.

11. 02.12.11 Vector field on a manifold; flow on a compact manifold is complete. Arnold Ch.5 anchor. Three-tier; ~1400 words. Vector field as section of $TM$; integral curves via local charts; gluing into global flow on $M$ compact. Bridge to 03-modern-geometry/. Forward-references the (currently-unwritten) tangent-bundle and Lie-derivative units.

12. 02.12.12 First integrals / conserved quantities. Arnold Ch.5 anchor. Three-tier; ~1200 words. $f$ is a first integral iff ${\mathcal{L}}_{v}f=0$; integration reduces dimension. Worked examples: energy for the pendulum, angular momentum for central-force motion, Volterra-Lotka conserved quantity. Bridge to Noether's theorem in 05.00.04.

Priority 3 — examples, applications, depth-completion

13. 02.12.13 Inhomogeneous linear ODE / variation of constants. Arnold Ch.3. Three-tier; ~1100 words. $\dot{x}=Ax+f(t)$; fundamental matrix; Duhamel formula. Standard but absent.

14. 02.12.14 Limit cycle and Liénard / Van der Pol systems. Three-tier; ~1300 words. Existence and uniqueness of limit cycles for $\ddot{x}+f(x)\dot{x}+x=0$ under Liénard conditions. Originator: Liénard 1928, Van der Pol 1926. Bridge from Poincaré-Bendixson (item 10).

15. 02.12.E1 Arnold ODE exercise pack. Three-tier exercise block; ~25-30 problems sampled from Arnold's Chapter exercises (pendulum phase portrait, classification problems, Lyapunov function construction, Floquet exponent computation, rectification near generic points). Defer to dedicated exercise-pack pass.

Priority 4 — optional Arnold-distinctive pointers (Master-tier only)

16. 02.12.15 Linearisation at a hyperbolic equilibrium (Hartman-Grobman theorem). Arnold mentions in passing; full statement and citation to Hartman 1960 / Grobman 1959. Master- only stub; ~900 words.

17. 02.12.16 Smooth dependence on parameters / structural stability survey. Arnold pointer to Andronov-Pontryagin 1937 structural stability of planar systems. Master-only; ~800 words.

18. 02.12.17 Bifurcation theory pointer. Saddle-node, Hopf, pitchfork bifurcations as the four canonical local bifurcations. Survey unit only — full bifurcation theory belongs to a nonlinear-dynamics text (Strogatz, Guckenheimer-Holmes), not to Arnold's ODE book. Master-only pointer; ~700 words.

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§4 Implementation sketch

Minimum Arnold-equivalence batch = priority 1 only (items 1-7): 7 new units in a new chapter family.

Hour estimates (mirroring the Arnold Mathematical Methods batch):

• ~3 hours per typical unit; ~4 hours for the existence-uniqueness unit (item 3, contraction-mapping proof) and ~4 hours for Poincaré-Bendixson (item 10, Jordan-curve subtlety).
• Priority 1 (items 1-7): 5 typical × 3 h + 2 heavier × 4 h = ~23 h.
• Priority 1+2 (items 1-12): ~23 + 4 typical × 3 h + 1 heavier × 4 h = ~39 h.
• Priority 1+2+3 (items 1-14, excl. exercise pack): ~39 + 2 × 3 h = ~45 h.

At 3-4 production agents in parallel, priority-1 fits in 2-3 days; priority-1+2 fits in 4-5 days with one integration agent. No single apex unit (KAM was the apex in Mathematical Methods; ODE has no comparable single-unit boss).

Chapter slot decision. Two candidates:

• 02.12-ode/ — inside the analysis chapter, after multivariable differentiation (02.05) and parallel to functional analysis (02.11). Natural Section-0 placement; ODEs are an analysis topic. Recommended.
• 02.07-ode/ — alternative interleaved numbering if 02.06-... 02.11 need stable-id stability.

Pick 02.12-ode/. Confirm by reading docs/catalogs/CONCEPT_CATALOG.md and docs/specs/UNIT_SPEC.md §3.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, originator-prose treatment is mandatory for:

• Poincaré 1881-1886 Mémoire sur les courbes définies par une équation différentielle (J. de Math. Pures et Appl., four memoirs) — items 1 and 10 (phase curves; Poincaré-Bendixson).
• Lyapunov 1892 Problème général de la stabilité du mouvement (Kharkov Math. Soc.; French Princeton 1947) — item 8.
• Picard 1890 + Lindelöf 1894 — item 3.
• Floquet 1883 — item 9.

Arnold himself is not the originator of this material (unlike for KAM in Mathematical Methods); originator citations are to the 19th and early-20th-century classics. Arnold's role is canonical synthesiser and the master-tier prose can cite him as the modern-geometric expositor.

Notation crosswalk. Arnold's notation:

• $v$ for vector field (Codex symplectic chapter uses $X$ for vector field, $X_H$ for Hamiltonian; standardise on $X$ generically and $v$ when Arnold-specific framing is invoked).
• $g^t$ for phase flow (adopt directly — no Codex conflict).
• $V$ for Lyapunov function (no conflict).
• $f$ for first integral (no conflict).

Pin in a §Notation paragraph of item 1 (02.12.01).

DAG edges. New prerequisites:

• 02.12.01 ← {02.05.04 implicit-inverse-function, 02.11.04 banach-spaces} (the latter prepares Banach-space ambient for §3 in item 3).
• 02.12.02 ← 02.12.01.
• 02.12.03 ← {02.12.01, 02.11.04}; introduces contraction mapping inline.
• 02.12.04 ← 02.12.03.
• 02.12.05 ← 02.12.04.
• 02.12.06 ← {02.12.01, 02.11.01 bounded-linear-operators}.
• 02.12.07 ← 02.12.06.
• 02.12.08 ← 02.12.05.
• 02.12.09 ← {02.12.06, 02.12.04}.
• 02.12.10 ← {02.12.07, 02.12.05}.
• 02.12.11 ← {02.12.05, 03.02.01 smooth-manifold}; forward edge into 05.02.01-hamiltonian-vector-field.
• 02.12.12 ← 02.12.11; forward edge into 05.00.04-noether- theorem.

Forward edges into already-shipped units. After priority-1, the following existing units gain a retroactive prerequisite:

• 05.02.01-hamiltonian-vector-field ← 02.12.11.
• 05.02.06-geodesic-flow-hamiltonian ← 02.12.11.
• 05.02.07-liouville-volume ← 02.12.02 (phase flow primitive).
• 05.02.08-poincare-recurrence ← 02.12.02.
• 05.00.04-noether-theorem ← 02.12.12 (Noether ↔ first integral).
• 05.09.01-kam-theorem ← 02.12.06, 02.12.07 (linear system classification is prerequisite for perturbation around equilibrium).

Composite Arnold-ODE + Arnold-MM batch. The two Arnold plans overlap at vector-field, flow, geodesic-flow, Lie-derivative, and first-integral / Noether. Production discipline:

• The general ODE / flow content is owned by this plan (items 1, 2, 5, 11, 12).
• The Hamiltonian / mechanics content is owned by the Mathematical Methods plan (Hamilton's principle, Legendre, Noether, KAM).
• No double-count: Lie-derivative-on-manifold lives in item 11 of this plan; the Hamiltonian-vector-field Lie derivative (Cartan formula form) is already in 05.02.02-poisson-bracket.

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§5 What this plan does NOT cover

• PDE techniques. Separation of variables, Sturm-Liouville, Fourier methods, characteristic-method for first-order PDE. Arnold has a separate Lectures on Partial Differential Equations (Springer 2004) which is the canonical sequel; PDE coverage is a separate Fast Track entry and is deferred. Arnold ODE touches Hamilton-Jacobi only as a remark; the full HJ-PDE unit is in the Mathematical Methods punch-list (item 8 there).
• Numerical ODE methods. Euler, Runge-Kutta, implicit / stiff solvers, symplectic integrators, error analysis. Numerics is a separate curriculum thread; Hairer-Wanner Solving ODEs would be the canonical book if it were ever added.
• Boundary-value problems / Sturm-Liouville / special functions. Bessel, Legendre, hypergeometric, Mathieu functions in full detail. Tenenbaum-Pollard / Coddington-Levinson territory; Arnold dispatches as remarks. Mathieu equation gets a single mention in item 9 only.
• Hamiltonian / Lagrangian mechanics in full. Owned by plans/fasttrack/arnold-mathematical-methods.md. The geodesic- flow worked example in item 11 of this plan is the ODE-side pointer; the symplectic-side framing is the Mathematical Methods plan's territory.
• KAM theory / Nekhoroshev / Aubry-Mather. All in Mathematical Methods (FT 1.11) plan.
• Bifurcation theory in full. Pointer-only in item 18. Guckenheimer-Holmes Nonlinear Oscillations or Kuznetsov Elements of Applied Bifurcation Theory would be the canonical text; not on the current FT booklist.
• Chaos / strange attractors / ergodic theory beyond Poincaré-Bendixson. Strogatz / Wiggins / Katok-Hasselblatt territory. Item 10 closes the planar-flow gap; higher-dim chaos is deferred.
• Line-number Problem inventory across Arnold's ~150 problems. Cannot do without the PDF. Defer; exercise-pack production is P3 priority-3 (item 15) and can be sourced from Hirsch-Smale-Devaney
  • Strogatz as substitute exercise pool until the Arnold PDF lands.
• Stochastic ODEs / SDEs. Øksendal territory; not in Arnold.
• Delay-differential equations / functional-differential equations. Not in Arnold.
• A notation/arnold-ode.md standalone file. Crosswalk in §4 is sufficient.

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§6 Acceptance criteria for FT equivalence (Arnold ODE)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, the book is at equivalence-coverage when:

• ≥95% of Arnold's named theorems map to Codex units at Arnold- equivalent proof depth (currently ~5%; after priority-1 ~55%; after priority-1+2 ~85%; after priority-3 ~93%; priority-1+2+3+4 ~96%). The phase-flow → linear-classification → Poincaré- Bendixson → Lyapunov backbone closes most of the gap.
• ≥80% of Arnold's ~150 Problems have a Codex equivalent (currently 0%; closing requires the 02.12.E1 exercise pack — item 15). Until the PDF lands, exercise-pack can be sourced from Hirsch-Smale-Devaney + Strogatz as substitute pool.
• ≥90% of Arnold's worked examples reproduced (currently ~10%; priority-1+2 batch brings to ~70%; remainder needs worked-example densification across items 6, 7, 10, 14).
• Notation alignment recorded inline in 02.12.01 per §4.
• DAG edges land per §4: in particular the new chapter's forward edges into 05.02.* make the symplectic-chapter prerequisites honest (currently 05.02.01-hamiltonian-vector-field is shipped without a vector-field prerequisite — priority-1 item 1 fixes this retroactively).
• Pass-W weaving connects 02.12-ode/ to: 02.05-* (multivariable differentiation, for the variational equation in item 4), 02.11-* (functional analysis, for the contraction-mapping ambient in item 3), 03.02-manifolds/ (item 11 bridge), 05.02-hamiltonian/ (the symplectic chapter's silent flow dependency).
• audit_completeness: reduced cannot be marked equivalence-covered. Per AGENTIC_EXECUTION_PLAN §6.6, P5 verification on this book can mark it only equivalence-partial until a full PDF audit re-runs. The re-audit is queued in NEED_TO_SOURCE.md. Hard rule.

Honest scope. Substantial equivalence gap: Codex has no Section-0 ODE block at all; every priority-1 unit is net-new. Once priority-1 ships (7 new units, ~23 hours), the symplectic chapter's silent ODE-prereq dependencies become explicit, and the curriculum gains a load-bearing vector-field / flow / Picard-Lindelöf / matrix- exponential foundation that is currently absent. This is the last missing prerequisite audit per FASTTRACK_COVERAGE_ROADMAP Wave 5 — its closure unblocks the Section-0 → Section-1 transition in the Fast Track DAG.

No apex unit. Unlike Mathematical Methods (where KAM is the apex), Arnold ODE has no single oversize unit. The book is uniformly load-bearing; production risk is spread across the seven priority-1 units rather than concentrated.

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§7 Sourcing

Status: BUY. Per docs/catalogs/FASTTRACK_BOOKLIST.md, this book is not available via the project's free-PDF channels. Audit performed under the §6.6 reduced fallback.

Attempted sources (this audit, 2026-05-17):

• loshijosdelagrange WordPress mirror (https://loshijosdelagrange.files.wordpress.com/2013/04/vladimir-i-arnold-vladimir-i-arnold-roger-cooke-ordinary-differential-equations-1992.pdf). PDF exists at the redirected URL https://loshijosdelagrange.wordpress.com/wp-content/uploads/2013/04/vladimir-i-arnold-vladimir-i-arnold-roger-cooke-ordinary-differential-equations-1992.pdf but exceeds WebFetch's 10 MB content-length limit. Recommended manual-download target for re-audit. Likely legality grey; treat as study-only mirror.
• Springer catalogue (link.springer.com/book/9783540548133 ≈ DOI 10.1007/978-3-642-58694-1 for the 1992 Springer-Textbook edition). Sits behind IDP authentication redirect. WebFetch-inaccessible without credentials.
• Internet Archive (archive.org/details/ordinarydifferen0000arno). Metadata only; no preview content available to WebFetch.
• Google Books / Amazon previews. Catalogue pages reached; front-matter preview not exposed to WebFetch.

Recommendation for full re-audit:

1. Purchase Springer-Textbook 3rd-edition reprint (~$50-70 new, $20-30 used). Add to reference/fasttrack-texts/00-prereq/ as Arnold-OrdinaryDifferentialEquations.pdf once digitised.
2. Or: download from the loshijosdelagrange mirror via browser (which has no content-length limit) and add locally under the same path.
3. Or: borrow via ILL — Arnold's ODE is widely held by university libraries; ISBN 3-540-54813-0 (1992 reprint) or 0-262-51018-9 (MIT Press 1973 first English ed., transl. Silverman, which is the Russian 2nd-edition translation and is content-equivalent for the priority-1 punch-list).

Once a PDF lands, re-run P1 audit; promote audit_completeness from reduced to full; re-verify §2 line-number coverage and §3 priority-1 punch-list against the actual chapter / section / problem layout (this stub used reconstructed TOC, and Cooke's 1992 edition has minor section renumbering vs Silverman's 1973; priority-1 punch-list robust to both).

Peer-source anchors used to build §1 (cited ≥3):

• M. W. Hirsch, S. Smale, R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd ed., Academic Press / Elsevier 2013. The standard modern peer text; Lipschitz existence / linear-system classification / Lyapunov stability / Poincaré-Bendixson are all in chapters 7, 4-6, 9, 10 respectively. Used to cross-check item-sequence in §3.
• S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview / CRC Press 2015. The standard applied-pedagogy text; trace-determinant plane / phase portraits / Van der Pol limit cycles in Ch. 5-7. Used as worked-example anchor.
• M. Tenenbaum, H. Pollard, Ordinary Differential Equations, Dover 1985 reprint (original Harper & Row 1963). The contrasting classical equation-solving text; used as foil to articulate Arnold's distinctive geometric framing in §1.
• E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill 1955. The classical rigorous text; existence-uniqueness and Floquet-theory references match Arnold's content directly. Cited for item 3 originator cross-check.

Citation network for originator-prose:

• H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle," J. de Math. Pures et Appl., 4 memoirs, 1881-1886.
• A. M. Lyapunov, "Problème général de la stabilité du mouvement," doctoral dissertation, Kharkov Math. Soc. 1892; French translation Princeton Univ. Press, Annals of Math. Studies 17, 1947.
• É. Picard, "Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives," J. Math. Pures Appl., 1890.
• E. Lindelöf, "Sur l'application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre," C. R. Acad. Sci. Paris, 1894.
• G. Floquet, "Sur les équations différentielles linéaires à coefficients périodiques," Ann. Sci. Éc. Norm. Sup. 12, 1883, 47-88.
• I. Bendixson, "Sur les courbes définies par des équations différentielles," Acta Math. 24, 1901, 1-88.

License note. The loshijosdelagrange mirror is unauthorised; re-audit must use a properly licensed copy. Springer reprints are in print as of 2026.