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
- Energy identity and displacement for Euclidean gradient trajectories
- Continuity and derivatives of positive-base real powers
- Picard-Lindelöf local existence and uniqueness for first-order systems
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Real powers for positive bases, with the zero-base positive-exponent convention
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- Integrable functions on [a,b] form a set closed under sums and scalar multiples, and ∫ₐᵇ(λ f+μ g) = λ∫ₐᵇ f + μ∫ₐᵇ g
- The integral function F(x) := ∫ₐˣ f of an integrable f
- The integral function of a bounded integrable f is Lipschitz, hence uniformly continuous
- Cᵏ maps and multi-index derivative notation in Euclidean space
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- Empirical phase measures and Fourier order statistics
- Phasors, trigonometric identities and periodic derivatives
- From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators · §§3.1–3.2 and 4, equations (3.1)–(3.3) and (4.1)–(4.6), printed pp. 3–8
- Translate a local diffeomorphism and its spectral data to the origin
- Translate every finite forward orbit without assuming global invertibility
- Translation preserves an embedded disc and its tangent space
- Adapted norm balls are ordinary Euclidean neighbourhoods
- Point stability, attraction and stability of a set
- Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- First-order systems, initial value problems, and solutions on intervals
- Open ball, closed ball and sphere in a metric space
- Fermat's theorem: an interior differentiable local extremum has zero gradient
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Every Picard–Lindelöf initial value problem has one maximal solution on an open interval
- Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system
- Local Lipschitz continuity in the state variable, locally uniform in time and parameters
- A compactly trapped gradient trajectory has a critical accumulation point
- Continuously differentiable maps, local inverses, and local diffeomorphisms
- Extensions, maximal solutions, maximal intervals, and global solutions
- On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂
- Real analytic germs in several variables
- Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The exponential function is strictly increasing
- P.-A. Absil and K. Kurdyka, On the stable equilibrium points of gradient systems (accepted 2006 preprint) · Section 3, Lemma 4, printed/PDF p. 4 (inequality); Theorem 3, p. 2, proof pp. 4–5 (stability application)
- Primary components ker q(T)ᵉ and generalised eigenspaces G_λ⁽ᵉ⁾(T)=ker(T-λ I)ᵉ
- Michael Shub, Global Stability of Dynamical Systems (1987) · Paragraph immediately preceding Theorem III.7 on printed p. 65; Theorem III.7, printed pp. 65–66, PDF pp. 78–79; center-stable case (2)
- Ioannis Panageas and Georgios Piliouras, Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions (ITCS 2017) · Theorem 8, printed p. 2:7 (PDF p. 7), equation (2)
- The reals form a totally ordered field
- A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions
- Sum, product, quotient, and chain rules from difference quotients
- The Axiom of Countable Choice (AC_ω)
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- The class L¹(μ) of integrable functions
- Integrable real and complex functions, and their integrals
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
Lemma
Conditional gradient inequality gives a length bound
Remark
A scoped continuum self-consistency input
Lemma
Transport local invariance and forward trapping to a fixed point
Lemma
A local minimum with a gradient inequality is stable
Lemma
Nearby critical values force nonminimum escape
Theorem
Łojasiewicz inequality and analytic point stability
Theorem
Local center-stable trapping disc for a time map
Proposition