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
- 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
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
- The zero sets of sine and cosine and the least positive common period 2 pi
- The derivatives of sine and cosine are cosine and minus sine
- Smooth atlases
- A product of finitely many compact spaces is compact in the product topology
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- ℚⁿ is a countable dense subset of ℝⁿ, and rational open boxes form a countable basis
- The product, Euclidean-metric and norm topologies on ℝⁿ agree, and for n=1 they agree with the real-line topology
- Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces
- Smooth manifolds and their smooth charts
- Each smooth atlas is contained in a unique maximal smooth atlas
- Riemannian metric and riemannian manifold
- Piecewise c one curve on a manifold
- Riemannian speed and length
- Riemannian length is independent of piecewise c one subdivision
- Riemannian distance on a connected manifold
- Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover
- A continuous piecewise-C¹ path is rectifiable and its length is the sum of the speed integrals over its pieces
- Every endpoint chord is no longer than the arc: ‖γ(b)-γ(a)‖₂≤ L(γ)
- Every path-connected space is connected, and every path component lies inside a component
- The complex exponential by its power series
- Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential
- 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
- Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives
- Sum, product, quotient, and chain rules from difference quotients
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Equivalence relation, equivalence class, and the quotient set A/∼
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- 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
- Local and strict local extrema for scalar fields on Euclidean open sets
- The absolute value makes ℝ a metric space: d(x,y) = |x-y| is a metric, its open balls are the intervals (x-r, x+r), and it is unbounded
- Continuity of a map of topological spaces at a point and globally
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Every nonempty set bounded below has an infimum
- Limits at +∞ and -∞, and infinite limits at a point
- The Axiom of Countable Choice (AC_ω)
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- For a nonzero real c, dilation by c multiplies Lebesgue outer measure by |c|ⁿ, and reflection in the origin preserves it
- The one-dimensional torus and its normalized Haar integral
- Probability measures and probability spaces
- Random elements and real random variables
- Law or distribution of a random element
- The Borel sigma-algebra of a topological space
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- Lebesgue measure on ℝⁿ is invariant under every orthogonal linear map
- Sigma-algebras
- Measures on sigma-algebras
- Measurable spaces and measurable sets
- Complete measure spaces
- Finite and countable subadditivity of measures
- The Riemannian gradient is the metric dual of the differential
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- The fundamental theorem of calculus and integration by parts
- Mean value theorem on a closed real interval
- Epsilon characterisation of the infimum
- Coordinate derivations at a point
- Coordinate derivations form a basis of the tangent space
- The differential of a smooth real-valued function
- The Hessian matrix and critical points of a scalar field
- Fermat's theorem: an interior differentiable local extremum has zero gradient
- Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+1/2h^TH_f(a)h+o(‖h‖²)
- Clairaut--Schwarz theorem for continuous second partial derivatives
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
Finite phase tori, local lifts and the flat metric
Definition
Phasors, trigonometric identities and periodic derivatives
Definition
Point stability, attraction and stability of a set
Definition
Normalized phase volume and almost-everywhere assertions
Proposition
Energy dissipation along a downward gradient curve
Proposition