Chapter 21
Differential Equations as Dynamic Coherence Systems

Flow Algebra

flow "Differential Equations as Dynamic Coherence Systems"

# ---------- CONCEPT ----------
# A differential equation encodes how a state changes with respect to its variables.
# It specifies dynamic coherence: rules that link present configuration to its infinitesimal evolution.
# Solutions are flows (trajectories/fields) that satisfy both the local change law and global constraints.

# ---------- STATES ----------
state DE_System(Σ):
  State:
    description: "x(t) ∈ ℝⁿ or u(x,t) — the evolving configuration (ODE or PDE)."
  LawOfChange:
    description: "F(x, x', x'', …, t) = 0 or 𝓛[u] = s — relation between state and derivatives."
  Constraints:
    description: "Initial/Boundary data: x(t₀)=x₀, u|_{∂Ω}=g, u(x,0)=u₀ — anchors of coherence."
  Operator:
    description: "Linear/nonlinear operator 𝓛 acting on u; may be time- or space-dependent."
  SolutionManifold:
    description: "Set of trajectories/fields satisfying law + constraints."

# ---------- PARAMETERS ----------
params:
  type: ["ODE","PDE"]
  order: "Highest derivative present"
  linearity: ["linear","affine","nonlinear"]
  domain: "t ∈ I ⊆ ℝ, x ∈ Ω ⊆ ℝᵈ"
  data: "Initial/Boundary conditions (IC/BC)"
  regularity: "Required smoothness of data/coefficients"
  symmetry: "Invariants/conservation laws"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Existence–Uniqueness (Picard–Lindelöf): Lipschitz in state ⇒ unique local ODE flow."
  - "Superposition for linear problems: if u₁, u₂ solve 𝓛[u]=0, so does αu₁+βu₂."
  - "Conservation form: d/dt Q(x(t)) = 0 ⇒ Q is invariant along trajectories."
  - "Energy dissipation/production laws for gradient flows or forced systems."
  - "Well-posedness (Hadamard): solution exists, is unique, and depends continuously on data."
  - "Dimensional/scale invariance constrains admissible solutions."

# ---------- TRANSFORMS ----------
transforms:
  ReduceOrder:
    rule: "Introduce new variables to lower order: x' = v, etc."
  Linearise(around=x*):
    rule: "Approximate nonlinear F by Jacobian dynamics: ẏ = A y + … near equilibrium."
  SeparateVariables:
    condition: "Structure allows u(x,t)=X(x)T(t)."
    rule: "Solve eigenproblem for space; evolve coefficients in time."
  IntegratingFactor:
    rule: "For linear first-order ODE: (e^{∫p} y)' = e^{∫p} q."
  GreenFunction:
    rule: "For linear 𝓛: u = G * s + homogeneous solution (with BCs)."
  SpectralDecompose:
    rule: "Expand in eigenbasis/orthogonal modes; evolve amplitudes."
  VariationOfParameters:
    rule: "Construct particular solution using fundamental matrix/base."
  Nondimensionalise:
    rule: "Rescale to reveal controlling groups (Pe, Re, ε, …)."
  NumericalScheme:
    options: ["Euler","RK4","Implicit Euler","Crank–Nicolson","Finite Difference/Element/Volume"]
    rule: "Discretise domain; enforce stability/consistency/convergence."

# ---------- MONITORS ----------
monitors:
  StabilityIndex:
    measure: "Eigenvalues of linearised A or spectrum of 𝓛."
    meaning: "Re(λ)<0 ⇒ decay; >0 ⇒ growth; =0 ⇒ neutral."
  LyapunovFunctional:
    meaning: "V̇ ≤ 0 certifies stability/attractor behaviour."
  ErrorTriplet:
    track: "[consistency, stability, convergence]"
    meaning: "Lax equivalence: stability + consistency ⇒ convergence (linear problems)."
  CFL_Criterion:
    meaning: "Time-step restriction for explicit PDE schemes."
  Residual:
    measure: "R = 𝓛[u]−s"
    meaning: "Zero (or below tol) indicates solution coherence."
  ConservationCheck:
    meaning: "Numerically track invariants (mass, energy, momentum)."

# ---------- EQUATIONS ----------
equations:
  - "ODE: \\; \\dot{x} = f(x,t), \\quad x(t_0)=x_0"
  - "Higher-order ODE: \\; x^{(k)} = F(x^{(k-1)},…,x,t)"
  - "PDE exemplar (diffusion): \\; u_t = D \\nabla^2 u"
  - "Wave: \\; u_{tt} = c^2 \\nabla^2 u"
  - "Advection–diffusion: \\; u_t + \\mathbf{v}\\cdot\\nabla u = D\\nabla^2 u + s"
  - "Linear operator form: \\; 𝓛[u] = s, \\; 𝓛 = a_0 + \\sum a_i \\partial_i + \\sum a_{ij}\\partial_{ij} + …"
  - "Variation-of-constants: \\; x(t)=\\Phi(t,t_0)x_0 + \\int_{t_0}^t \\Phi(t,τ)b(τ)\\,dτ"
  - "Green's solution: \\; u(x)=\\int_Ω G(x,ξ)s(ξ)\\,dξ + u_h"
  - "Lyapunov: \\; V(x)>0, \\; \\dot V = \\nabla V \\cdot f(x,t) \\le 0"

# ---------- NOTES ----------
notes:
  - "Differential equations define evolution rules; IC/BC anchor them to reality."
  - "Linear problems admit superposition and spectral methods; nonlinear systems need phase portraits, invariants, or numerics."
  - "Classification (elliptic/parabolic/ hyperbolic) predicts qualitative behaviour: smoothing, propagation, or constraint."
  - "Scaling and nondimensionalisation expose controlling parameters and regimes."
  - "Well-posed discretisations require balance of accuracy and stability; monitor invariants to prevent drift."
  - "In Flow Algebra, a DE is a coherence propagator: local derivative laws + boundary anchors ⇒ global evolving form."