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
- Finite phase tori, local lifts and the flat metric
- Phasors, trigonometric identities and periodic derivatives
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
- Sum, product, quotient, and chain rules from difference quotients
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Linear independence: a finite list v : n → V is independent when ∑_i<n λᵢ vᵢ = 0_V forces every λᵢ = 0_F, and a subset S ⊆ V is independent when every injective finite list into S is independent
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Real and complex modulus product and triangle inequalities
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- The zero sets of sine and cosine and the least positive common period 2 pi
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Division with remainder in ℤ: for a ∈ ℤ and b > 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < b
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
- Real and imaginary parts, complex conjugation, and modulus
- The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products
- Point stability, attraction and stability of a set
- Normalized phase volume and almost-everywhere assertions
- Probability measures and probability spaces
- The Axiom of Countable Choice (AC_ω)
- The Borel sigma-algebra of a topological space
- Sigma-algebras are closed under countable intersections, differences, symmetric differences, and set limits
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- A product of two at most countable sets is at most countable
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- The principle of mathematical induction
- Empirical measure of observed units
- The empirical measure is a probability measure
- The Dirac set function at a point
- Weak convergence of borel probability measures
- The nonnegative Lebesgue integral
- The integral of a nonnegative simple function
- Integrable real and complex functions, and their integrals
- The indefinite integral of a nonnegative measurable function is a measure
- Sigma-algebras
- Measurable spaces and measurable sets
- Measures on sigma-algebras
- Measure spaces
- Extended-real-valued measurable functions
- A measurable function between measurable spaces
- Nonnegative simple measurable functions
- Closure properties of measurable functions used by the integral
- A continuous map has Borel preimages of Borel sets
- Integral over a measurable subset
- Phase chaos in coupled oscillators · Equation (1), pp. 065201-1–065201-2 and Figure 1
- Chaotic Attractor in the Kuramoto Model · Publisher abstract
- From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators · Section 2, p. 2; Section 3.1, p. 3
- Translate a local diffeomorphism and its spectral data to the origin
- Energy identity and displacement for Euclidean gradient trajectories
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- For n ≥ 1 every bounded sequence in ℝⁿ has a convergent subsequence
- 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
- Continuously differentiable maps, local inverses, and local diffeomorphisms
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- ℚ is countably infinite
- The sequential closure is contained in the closure, continuity implies sequential continuity, and sequential limits need not be unique
- 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)
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
Definition
Diagonal phase rotations and relative-phase coordinates
Proposition
Differentiation of symmetric periodic pair potentials
Proposition
Rigidity and rotation of a unit-phasor average
Definition
Point basins, equilibrium-set basins and aggregate volume
Definition
Empirical phase measures and Fourier order statistics
Remark
An independently attributed heterogeneous-model chaos report
Postulate
Scope of the phase-only sinusoidal oscillator idealization
Lemma
Translate every finite forward orbit without assuming global invertibility
Lemma