Alphabeta MathPhase dynamics and oscillator models
Build phase geometry, analytic gradient dynamics and basin measure; place oscillator assumptions, historical observations and continuum context in their separate stated scopes.
1 · Prerequisites
- First-order systems, initial value problems, and solutions on intervals
- Picard-Lindelöf local existence and uniqueness for first-order systems
- Sum, product, quotient, and chain rules from difference quotients
- The Axiom of Countable Choice (AC_ω)
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Local Lipschitz continuity in the state variable, locally uniform in time and parameters
- Limits at +∞ and -∞, and infinite limits at a point
- The left and right limits of f at c, as limits of the restrictions of f to A ∩ (-∞, c) and A ∩ (c, ∞)
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- The tangent space to a regular level set
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
- Real and complex inner product spaces, with the inner product linear in the first argument
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Cᵏ Euclidean maps and diffeomorphisms
- Huygens’s clocks · Bennett, Schatz, Rockwood and Wiesenfeld, Proc. R. Soc. Lond. A 458 (2002), Sections 1 and 2(a), printed pp. 563–566
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- C k map between Banach spaces
- Fréchet derivative between Banach spaces
- Primary components ker q(T)ᵉ and generalised eigenspaces G_λ⁽ᵉ⁾(T)=ker(T-λ I)ᵉ
- Complexification as ℂ⊗_ℝV with its canonical real-linear embedding
- Complexification of a real-linear map
- Conjugations and real structures on a complex vector space
- The fixed real form of a conjugation
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- A Euclidean immersion is locally the canonical inclusion and is locally an embedding
- For n ≥ 1 all norms on ℝⁿ are equivalent
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Open ball, closed ball and sphere in a metric space
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- The reverse triangle inequality in a normed space
- The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞
- Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page
- For a differentiable scalar field, Dᵥf(a)=⟨∇ f(a),v⟩ and the unit direction of steepest ascent is the normalized gradient
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- For a ≤ b and f : [a,b] → ℝᵐ integrable when a<b, ‖∫ₐᵇ f‖₂ ≤ ∫ₐᵇ ‖ f‖₂; for a<b, ‖ f‖₂ is integrable
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- Continuously differentiable maps, local inverses, and local diffeomorphisms
- Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity
- Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous
- A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions
- Finite phase tori, local lifts and the flat metric
- Phasors, trigonometric identities and periodic derivatives
- Real analytic germs in several variables
- The intrinsic Hessian of a smooth function at a critical point
- The complex exponential by its power series
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- The binomial theorem in ℝ: (x+y)ⁿ = ∑_k<n+1 iotaC(n, k) xᵏ y^ n-k
- The Riemannian gradient is the metric dual of the differential
- Integral curves of a vector field
- Every smooth vector field on a compact manifold is complete
- C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade
- Cʳ and smooth maps between smooth manifolds
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Proposition
Explicit scalar quadratic descent and its forward domain
Definition
Sphere configurations and tangent variations
Experiment
Historical report of Huygens’s two clocks
Lemma
Translate a local diffeomorphism and its spectral data to the origin
Lemma
Translation preserves an embedded disc and its tangent space
Lemma
Adapted norm balls are ordinary Euclidean neighbourhoods
Lemma
Energy identity and displacement for Euclidean gradient trajectories
Definition
Analytic potentials on a finite phase torus
Definition