Module 5 — Resonance overlap, cantori and phase-space diffusion

Lecture notes written against midi-sink as shipped (libsumi 1.12.0, Voxo 0.26.0, 2026-10-25) and checked claim by claim against the engine before this fold (2026-10-26; libsumi is unchanged since, and Voxo’s later versions add the sampler’s second generation and touch no synth claim here). Code citations and gate numbers refer to docs/PROJECT_SPEC.md (spec v6), docs/DECISIONS.md Part XIII (cited DECISIONS_13 #n), voxo/src/suzu.h, and site/src/content/docs/operators/chirikov.mdx. The lab beside the module is a standalone teaching re-implementation, never the engine; the engine’s own lab is the Suzu lab.

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Lecture Notes

5.1 Poincaré–Birkhoff Chains, Homoclinic Tangles & Chirikov’s Resonance Overlap Criterion

In Module 4, we proved two complementary theorems for a perturbed integrable Hamiltonian $H(I, \theta) = H_0(I) + \epsilon H_1(I, \theta)$ (or its stroboscopic twist map, the Chirikov Standard Map $p_{n+1} = p_n + K\sin\theta_n$, $\theta_{n+1} = \theta_n + p_{n+1}$):

1. KAM Theorem: Strongly irrational (Diophantine) tori $|k \cdot \omega| \ge \gamma \|k\|_1^{-\tau}$ survive for small $\epsilon$ (or $K < K_c(\omega)$), acting in 2D phase space as impenetrable topological barriers.
2. Poincaré–Birkhoff Theorem: Every rational torus $\omega = r / s \in \mathbb{Q}$ breaks into an alternating necklace of $s$ stable elliptic periodic points (surrounded by librating resonant islands) and $s$ unstable hyperbolic saddle points.

Now we ask: how does chaos actually born at those hyperbolic saddle points spread across phase space as $K$ increases, and what remains of a KAM torus after it breaks?

Splitting of Separatrices and the Homoclinic Tangle (Poincaré, 1890)

In an unperturbed 1-DoF pendulum $H_0(\theta, p) = \frac{1}{2}p^2 + K\cos\theta$, the unstable saddle point at $(\theta, p) = (0, 0)$ (equivalent to $(2\pi, 0)$ on the cylinder) is connected to itself by a smooth figure-eight separatrix $p = \pm 2\sqrt{K}\sin(\theta/2)$. Every trajectory leaving $(0, 0)$ in the distant past along the unstable manifold $W^u(0, 0)$ arrives smoothly back at $(0, 0)$ in the distant future along the stable manifold $W^s(0, 0)$; thus $W^u$ and $W^s$ coincide identically ($W^u = W^s$).

Under a discrete kick step or time-periodic perturbation, $W^u$ and $W^s$ are 1-dimensional invariant curves in 2D phase space that no longer coincide. Instead, they split apart (by an angle exponentially small in $1/\sqrt{K}$ for small $K$, governed by the Melnikov–Lazutkin integral) and cross each other transversely at a primary homoclinic point $z_\star \in W^u \cap W^s$!

Consider the geometric consequences of a single transverse intersection $z_\star \in W^u \cap W^s$ under an area-preserving symplectomorphism $M$ ($\det DM = 1$):

1. Infinite Cascade of Intersections: Because $W^u$ and $W^s$ are invariant sets ($M(W^u) = W^u$, $M(W^s) = W^s$), every forward and backward iterate $M^n(z_\star)$ ($n \in \mathbb{Z}$) is also a transverse homoclinic intersection point!
2. Symplectic Lobe Area Conservation Forces Exponential Stretching: Let $L_0$ be the closed region (homoclinic lobe) bounded by the segment of $W^u$ and the segment of $W^s$ between $z_\star$ and $M(z_\star)$. Because $M$ is a symplectomorphism, every iterate $L_n = M^n(L_0)$ must enclose the exact same symplectic area: $\operatorname{Area}(L_n) = \iint_{L_n} dq \wedge dp = \iint_{L_0} dq \wedge dp = \Delta A_{\text{lobe}} > 0 \qquad \forall n \in \mathbb{Z}$ Yet as $n \to +\infty$, the endpoints $M^n(z_\star)$ and $M^{n+1}(z_\star)$ converge exponentially along $W^s$ toward the saddle $(0, 0)$, so the base width of lobe $L_n$ shrinks as $\lambda_-^n \to 0$! To keep its area $\Delta A_{\text{lobe}}$ constant while its width shrinks to zero (and without ever self-intersecting, since $W^u$ is an embedded manifold), the lobe $L_n$ must stretch exponentially in length and fold back and forth across $W^s$ in a wild, self-similar oscillation — Poincaré’s homoclinic tangle!

In Les Méthodes Nouvelles de la Mécanique Céleste (Vol. 3, 1899), Poincaré wrote: “One will be struck by the complexity of this figure, which I shall not even attempt to draw.” Today, in Part 1 of our WebGL2 lab, your GPU draws 262,144 particles threading through that exact homoclinic tangle at 60 frames per second!

Chirikov’s Resonance Overlap Criterion (1959, 1979)

When $K$ is small, the homoclinic tangle around each resonant island chain is confined to an exponentially thin stochastic layer sandwiched between surviving KAM tori. How large must $K$ become before adjacent resonant islands collide and destroy the intervening KAM barriers?

Write the Chirikov Standard Map $p_{n+1} = p_n + K\sin\theta_n$, $\theta_{n+1} = \theta_n + p_{n+1}$ as the exact stroboscopic map of the periodically kicked rotor Hamiltonian:

$H(\theta, p, t) = \frac{1}{2}p^2 + K\cos\theta \sum_{m=-\infty}^{\infty} \delta(t - m) = \frac{1}{2}p^2 + K \sum_{m=-\infty}^{\infty} \cos(\theta - 2\pi m t)$

where we used the Poisson summation identity $\sum_{m \in \mathbb{Z}} \delta(t - m) = \sum_{m \in \mathbb{Z}} e^{-2\pi i m t}$. Look at the Fourier harmonics $\cos(\theta - 2\pi m t)$:

• Each integer $m \in \mathbb{Z}$ defines a primary $1:1$ resonance where the phase $\psi_m = \theta - 2\pi m t$ is stationary: $\dot{\theta} = p = 2\pi m =: p_m^*$.
• Neglecting all harmonics except $m$ reduces the Hamiltonian near $p \approx 2\pi m$ to the single pendulum $H_m(\psi_m, \delta p) = \frac{1}{2}(\delta p)^2 + K\cos\psi_m$, whose separatrix has momentum half-width: $\Delta p_{\text{sep}} = 2\sqrt{K}$
• The spacing in momentum between two neighbouring integer resonances ($p_0^* = 0$ and $p_1^* = 2\pi$) is $\delta p_{01} = 2\pi$.

Boris Chirikov defined the Resonance Overlap Parameter $s_{\text{overlap}}$ as the ratio of the sum of the two unperturbed separatrix half-widths to the distance between the resonance centres:

$\boxed{\; s_{\text{overlap}} := \frac{\Delta p_{\text{sep}}^{(0)} + \Delta p_{\text{sep}}^{(1)}}{|p_1^* - p_0^*|} = \frac{2\sqrt{K} + 2\sqrt{K}}{2\pi} = \frac{2\sqrt{K}}{\pi} \;}$

• Naive First-Order Overlap ($s_{\text{overlap}} = 1$): Setting $2\sqrt{K}/\pi = 1$ predicts global chaos at $K_1 = (\pi/2)^2 \approx 2.467$.
• Secondary Resonance Hierarchy ($K_c \approx 0.971635$): Why does the last golden KAM curve actually break earlier, at Greene’s threshold $K_c \approx 0.971635$ (Greene 1979 — the engine quotes no $K_c$: it caps a canvas throw at $K = 1.25$ a step and the rotor’s $K$ at 2.5 on the wheel)? Because at second order in perturbation theory, the interaction between the $m=0$ and $m=1$ harmonics generates a period-2 ($1:2$) resonance halfway between them at $p = \pi$ with half-width $\Delta p_{1/2} = K/2$, plus an infinite hierarchy of Fibonacci resonances $F_{k-1}/F_k$ whose separatrices overlap at $K_c \approx 0.9716354$!

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5.2 Aubry–Mather Theory, Cantori & MacKay–Meiss–Percival Turnstiles

What happens to an irrational KAM torus $\mathcal{T}_\omega$ the moment $K$ exceeds its critical breakup threshold $K_c(\omega)$? Does the quasi-periodic orbit with rotation number $\omega$ vanish into thin air?

In 1982–1983, Serge Aubry (studying the Frenkel–Kontorova crystal dislocation model) and John Mather (studying monotone area-preserving twist maps) independently proved one of the deepest theorems in modern Hamiltonian dynamics: the quasi-periodic motion never disappears; when a KAM curve breaks, it turns into an invariant Cantor set — a Cantorus!

Variational Principle & The Aubry–Mather Theorem

Recall from Module 3 (§3.3) that the Chirikov Standard Map (and any monotone twist map where $\frac{\partial \theta_{n+1}}{\partial p_n} > 0$) is generated by a Type-1 generating function $S(\theta_n, \theta_{n+1})$:

$S(\theta_n, \theta_{n+1}) = \frac{1}{2}(\theta_{n+1} - \theta_n)^2 + V(\theta_n), \qquad V(\theta) = K\cos\theta$

so that $p_n = -\frac{\partial S(\theta_n, \theta_{n+1})}{\partial \theta_n}$ and $p_{n+1} = \frac{\partial S(\theta_n, \theta_{n+1})}{\partial \theta_{n+1}}$. Consequently, a bi-infinite sequence of angles $\{\theta_n\}_{n \in \mathbb{Z}}$ is a true dynamical trajectory if and only if every finite segment is a stationary point of the discrete Lagrangian action:

$\mathcal{A}\big[\{\theta_n\}\big] = \sum_{n} S(\theta_n, \theta_{n+1}) \quad \Longleftrightarrow \quad \frac{\partial \mathcal{A}}{\partial \theta_n} = \frac{\partial S(\theta_{n-1}, \theta_n)}{\partial \theta_n} + \frac{\partial S(\theta_n, \theta_{n+1})}{\partial \theta_n} = p_n - (\theta_{n+1} - \theta_n) - V'(\theta_n) = 0$

Even when $K$ is arbitrarily large so that no smooth KAM curves exist, Aubry and Mather proved that the global minimum of $\mathcal{A}$ over the space of monotone configurations with mean spacing $\lim_{N\to\infty}(\theta_N - \theta_0)/N = 2\pi\omega$ always exists:

Theorem 5.1 (Aubry–Mather Theorem, 1982): Let $M$ be an exact area-preserving monotone twist map of the cylinder $\mathbb{T}^1 \times \mathbb{R}$. For every rotation number $\omega \in \mathbb{R}$ and every perturbation strength $K \ge 0$, there exists a closed, $M$-invariant action-minimizing set $\mathcal{M}_\omega \subset \mathbb{T}^1 \times \mathbb{R}$ projecting injectively onto $\mathbb{T}^1$ (a Lipschitz graph $p = P_\omega(\theta)$ over $\pi_1(\mathcal{M}_\omega) \subset \mathbb{T}^1$) such that every orbit in $\mathcal{M}_\omega$ preserves cyclic angular order and has exact rotation number $\omega$:

1. Subcritical ($K < K_c(\omega)$, $\omega$ irrational): $\mathcal{M}_\omega$ is a smooth, connected KAM circle ($\pi_1(\mathcal{M}_\omega) = \mathbb{T}^1$) with zero gaps.
2. Supercritical ($K > K_c(\omega)$, $\omega$ irrational): $\mathcal{M}_\omega$ is a nowhere-dense, perfect Cantor set ($\pi_1(\mathcal{M}_\omega) \subsetneq \mathbb{T}^1$) — called an Aubry–Mather Cantorus (Percival, 1979)!

MacKay–Meiss–Percival Turnstiles & Symplectic Lobe Flux $\Delta W$ (1984)

Think about what a cantorus $\mathcal{M}_\omega$ looks like in phase space for $K$ slightly above $K_c(\omega_{\text{gold}}) \approx 0.971635$:

• Because $\mathcal{M}_\omega$ is a Cantor set on a curve wrapping around the cylinder, its complement along that curve is a countable union of open gaps $\{G_n\}_{n \in \mathbb{Z}}$, where $G_{n+1} = M(G_n)$ is the image of gap $G_n$ under one step of the map!
• Because the total circumference of the cylinder is finite ($2\pi$), the sum of all gap lengths $\sum_{n=-\infty}^\infty |G_n| \le 2\pi$ converges, meaning the gap widths $|G_n|$ shrink exponentially to zero as $n \to \pm\infty$!

In 1984, Robert MacKay, James Meiss, and Ian Percival asked: if we close all the gaps of a cantorus to form a continuous closed loop $\mathcal{C}_\omega$ around the cylinder, how much phase-space area leaks across $\mathcal{C}_\omega$ per iteration?

Here is their famous Turnstile Construction:

1. Let $z_n^-, z_n^+ \in \mathcal{M}_\omega$ be the left and right endpoints of gap $G_n$. As $n \to +\infty$, $|z_n^+ - z_n^-| \to 0$ along the stable manifold $W^s$ of the gap; as $n \to -\infty$, $|z_n^+ - z_n^-| \to 0$ along the unstable manifold $W^u$ of the gap.
2. Join $z_n^-$ to $z_n^+$ along the unstable manifold segment $W^u_n$ for all $n \le 0$, and along the stable manifold segment $W^s_n$ for all $n > 0$. This forms a continuous closed barrier curve $\mathcal{C}_\omega$ around the cylinder!
3. Under one iteration of the map $M$, every gap segment $W^u_{n-1}$ ($n \le 0$) maps exactly onto $W^u_n$, and every segment $W^s_n$ ($n \ge 1$) maps exactly onto $W^s_{n+1}$ — except at the widest gap $G_0$! Inside gap $G_0$, the unstable segment $W^u_0$ and the stable segment $W^s_0$ cross at a homoclinic point (a minimax orbit $\{\theta_n^{\text{max}}\}$ sitting in the middle of the gap!), forming a figure-eight Turnstile with two lobes:
   • Upward Lobe $L_{\text{up}}$: Points that sit below $\mathcal{C}_\omega$ at step $n$ and land above $\mathcal{C}_\omega$ at step $n+1$.
   • Downward Lobe $L_{\text{down}}$: Points that sit above $\mathcal{C}_\omega$ at step $n$ and land below $\mathcal{C}_\omega$ at step $n+1$.

By Stokes’ Theorem and exact symplecticity ($\oint_{\mathcal{C}_\omega} p\,dq = \oint_{M(\mathcal{C}_\omega)} p\,dq$), the upward and downward turnstile lobes have identical symplectic area, given by Mather’s $\Delta W$ action difference between the gap-centre minimax orbit $\{\theta_n^{\text{max}}\}$ and the cantorus endpoint minimizing orbit $\{\theta_n^{\text{min}}\}$:

$\boxed{\; \Delta W(\omega) := \operatorname{Area}(L_{\text{up}}) = \operatorname{Area}(L_{\text{down}}) = \oint_{\text{turnstile lobe}} p\,dq = \sum_{n=-\infty}^{\infty} \Big( S\big(\theta_n^{\text{max}}, \theta_{n+1}^{\text{max}}\big) - S\big(\theta_n^{\text{min}}, \theta_{n+1}^{\text{min}}\big) \Big) \;}$

Look at how $\Delta W(\omega_{\text{gold}})$ behaves across Greene’s critical threshold $K_c \approx 0.971635$:

• For $K \le K_c$, the golden KAM torus has no gaps: $\Delta W = 0$ (flux is strictly zero!).
• For $K > K_c$, renormalization group theory at the golden fixed point proves that $\Delta W$ turns on with a universal critical exponent $\eta \approx 3.011722$: $\Delta W(\omega_{\text{gold}}) \approx C\,(K - K_c)^{3.011722}$ Because the exponent is $\eta \approx 3$, just above the breakup threshold (say at $K = 1.05$, where $K - K_c \approx 0.078$), the turnstile area $\Delta W \propto (0.078)^3 \approx 4.7 \times 10^{-4}$ is less than one twentieth of one percent of the phase-space area! Therefore, even though the golden KAM curve is technically broken at $K = 1.05$, trajectories initialized below the golden cantorus are trapped behind the cantorus for thousands of iterations, queuing up until they happen to land inside the tiny turnstile lobe $L_{\text{up}}$ and pivot across!

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5.3 Normal vs. Anomalous Action Diffusion ($\langle \Delta I^2(t) \rangle \sim D t^\alpha$)

Suppose we initialize a large ensemble of $M$ particles along a single action line $I(0) = I_0$ (with uniform angles $\theta_m(0) \in [0, 2\pi)$) and iterate the symplectic map for $t = N$ steps. How does the mean-squared action spread

$\sigma_I^2(t) := \big\langle \big(I(t) - \langle I(t) \rangle\big)^2 \big\rangle \sim D\,t^\alpha$

evolve over time? Depending on $K$ and the local cantorus/island hierarchy, Hamiltonian phase space exhibits four distinct transport regimes:

Transport Regime	Kick Range $K$	Scaling Exponent $\alpha$	Physical & Geometric Mechanism
1. KAM Confinement (Bounded)	$K < K_c \approx 0.9716$	$\alpha = 0$ as $t \to \infty$ ($\sigma_I^2(t) \le \sigma_{\max}^2$)	Unbroken golden-ratio KAM curves ($\Delta W = 0$) partition the cylinder into isolated bands. $\sigma_I^2(t)$ saturates at a finite ceiling!
2. Cantorus Turnstile Trapping (Subdiffusion)	$K_c < K \lesssim 1.4$	$0 < \alpha < 1$ (transient / multiscale)	Trajectories are bottlenecked behind golden and noble cantori with tiny turnstile flux $\Delta W \propto (K - K_c)^{3.01}$, pausing on quasi-stationary plateaus before leaking through.
3. Chaotic Sea (Normal Quasilinear Diffusion)	$K \gtrsim 1.5$ (away from accelerator modes)	$\alpha = 1$ ($\sigma_I^2(t) = 2 D(K)\,t$)	Phases $\theta_n$ decorrelate rapidly; $\Delta p = \sum_{n=0}^{t-1} K\sin\theta_n$ behaves like a diffusive random walk with $D_{\text{QL}} = K^2 / 4$ (modulated by Bessel correlations $J_2(K)$).
4. Sticky Islands & Accelerator Modes (Superdiffusion)	Near $K \approx 2\pi m$ (or island boundary stickiness)	$1 < \alpha \le 2$ (Lévy flights)	Orbits sticking to cantorus hierarchies around accelerator-mode islands step ballistically by $\Delta p = \pm 2\pi$ per kick, producing heavy-tailed Lévy flights!

Derivation of the Quasilinear Diffusion Coefficient $D_{\text{QL}}$ & Rechester–White Oscillations

After $N$ steps of the Chirikov Standard Map, the total momentum change is $\Delta p_N = p_N - p_0 = K \sum_{n=0}^{N-1} \sin\theta_n$. Squaring and averaging over the uniform initial angle ensemble $\theta_0 \in [0, 2\pi)$ gives the exact Green–Kubo / Taylor diffusivity sum:

$\big\langle \Delta p_N^2 \big\rangle = K^2 \sum_{n=0}^{N-1} \big\langle \sin^2\theta_n \big\rangle + 2 K^2 \sum_{0 \le n < m \le N-1} \big\langle \sin\theta_n \sin\theta_m \big\rangle$

1. Random-Phase (Quasilinear) Approximation ($D_{\text{QL}} = K^2 / 4$): If we neglect multi-step angle correlations ($\langle \sin\theta_n \sin\theta_m \rangle \approx 0$ for $n \neq m$) and use $\langle \sin^2\theta_n \rangle = \frac{1}{2}$, we obtain linear Fickian growth $\langle \Delta p_N^2 \rangle = \frac{1}{2}K^2 N = 2 D_{\text{QL}} N$, where: $D_{\text{QL}}(K) := \frac{\langle \Delta p_N^2 \rangle}{2 N} = \frac{K^2}{4}$
2. Rechester–White Fourier-Path / Bessel Correlation Correction (1980): Even at large $K$, $\theta_{n+2} = \theta_n + 2p_{n+1} + K\sin(\theta_n + p_{n+1})$ retains a non-zero 2-step correlation $\langle \sin\theta_n \sin\theta_{n+2} \rangle = -\frac{1}{2}J_2(K)$ (where $J_m(K)$ is the Bessel function of order $m$)! Summing the leading Fourier paths gives the famous Rechester–White oscillatory diffusion coefficient: $\boxed{\; D(K) = \frac{K^2}{2}\left[ \frac{1}{2} - J_2(K) - J_1^2(K) + J_2^2(K) + J_3^2(K) + \cdots \right] \;}$ Notice that near the peaks of $-J_2(K)$ (and at $K \approx 2\pi m \approx 6.283$ where accelerator modes appear), $D(K)$ spikes far above the quasilinear parabola $K^2/4$!

How libsumi & Suzu Gate Chaotic Diffusion (chirikov.mdx, suzu::Rotor)

Look at the Chirikov page and suzu::Rotor (voxo/src/suzu.h):

• On the libsumi canvas, chirikov_fs applies the standard map driven by mod-wheel deltas $\delta$ (so a slow wheel sweep applies many tiny steps with $\delta^2 K_{\max} \ll K_c$, sliding ink smoothly along integrable sheets, while a fast one-frame throw jumps across $K_c \approx 0.9716$ into cantorus turnstiles and homoclinic filaments!). Because chaotic stretching above $K_c$ narrows filaments down to sub-texel width, libsumi caps per-step throws in the core at K = 1.25 — right inside the cantorus turnstile regime!
• In Suzu, suzu::Rotor (voxo/src/suzu.h) runs a single trajectory $(\theta_n, p_n)$ at one kick per fundamental period $T_0 = 1/f_0$ and re-bases its magic-circle cell onto $f_0(1 + p_n / 2\pi)$. In Part 2 of this module’s lab, we generalize suzu::Rotor to a 512-Trajectory Granular Action-Diffusion Synthesizer: each particle $m \in \{0, \dots, 511\}$ carries its own action $I_m(t)$ and angle $\theta_m(t)$, mapping $I_m(t)$ to oscillator frequency $f_m(t) = f_0 \exp(\gamma I_m(t))$ and $\theta_m(t)$ to stereo pan $\cos\theta_m(t)$!

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Lab Instructions (module_05_lab.html)

Open the module’s lab (a standalone page — a teaching re-implementation, never the engine) in a WebGL2 + WebAudio browser. Both workspaces run live ensemble simulations of the Chirikov Standard Map across the KAM / Cantorus / Diffusion transition.

Part 1: Shader Lab — 262,144-Particle GPU Turnstile Flux & Density Diffusion

The top workspace uses two alternating $512 \times 512$ RG32F WebGL2 floating-point textures (512 × 512 = 262,144 particles!) to integrate a quarter-million trajectories in parallel on the GPU:

• Each texel $(u, v)$ stores a particle’s exact floating-point phase-space coordinates $(q_m, p_m)$ (RG32F).
• At $t = 0$, all 262,144 particles are initialized along a single horizontal action line $p = I_0$ (default $I_0 = \pi$, right between the two Golden-Ratio KAM / Cantorus barriers at $p_{\text{gold}^-} = 2\pi(1 - \varphi^{-1}) \approx 2.400$ and $p_{\text{gold}^+} = 2\pi\varphi^{-1} \approx 3.883$!).
• A GPU Point-Splatting Additive Histogram Shader (gl.POINTS with additive blending gl.ONE, gl.ONE) renders all 262,144 particles onto the phase portrait alongside the golden cantorus barriers and turnstile lobe markers!

1. Watch Cantorus Trapping & Turnstile Leakage ($K = 1.05$ vs. $K = 0.85$ vs. $K = 1.60$):
   • Click $K = 0.85$ (KAM Trapped): All 262,144 particles spread inside the central stochastic zone but hit a brick wall at the two gold horizontal curves ($p \approx 2.40$ and $p \approx 3.88$). Notice in the HUD that Cantorus Leakage % stays at 0.00% forever, and $\langle \Delta I^2(t) \rangle$ saturates flat!
   • Now click $K = 1.06$ (Cantorus Turnstile Leak) and click Reset 262,144 Line: Here $K > K_c \approx 0.9716$, so the golden KAM curves have broken into Aubry–Mather Cantori with tiny MacKay–Meiss–Percival turnstile lobes of area $\Delta W \propto (K - K_c)^{3.01} \approx 5 \times 10^{-4}$! Watch the density histogram: the 262,144 particles pile up behind the golden cantorus barrier for hundreds of iterations, and then thin jets of particles begin leaking through the turnstile gap (marked by the glowing turnstile portals) into the upper and lower cylinders!
   • Click $K = 1.65$ (Normal Diffusion) or $K = 6.35$ (Accelerator Superdiffusion): Watch the cantorus dam wash away as the cloud diffuses across the cylinder with $\langle \Delta I^2(t) \rangle \sim D t^\alpha$, and read the live power-law exponent $\alpha$ and empirical diffusivity $D_{\text{emp}}$ vs. $D_{\text{QL}} = K^2/4$ on the Log–Log Variance $\langle \Delta I^2(t) \rangle$ Scope!

Part 2: Sound Lab — 512-Voice Granular Action-Diffusion Synthesizer at 48 kHz

The bottom workspace evolves an ensemble of 512 diffusing trajectories $\{(I_m(t), \theta_m(t))\}_{m=0}^{511}$ all initialized along $I_m(0) = 0$ (with tiny thermal spread $10^{-4}$ and uniform angles $\theta_m(0) = 2\pi m / 512$) and synthesizes all 512 sinusoidal oscillators in real time at 48 kHz:

$f_m(t) = f_0 \exp\!\big(\gamma\,I_m(t)\big), \qquad \operatorname{pan}_m(t) = \cos\!\big(\theta_m(t)\big) \in [-1, +1]$

1. Hear the Unison Tone Fracture Across $K_c \approx 0.9716$:
   • Click Start 512-Voice Granular Synth and click Reset 512 Unison ($I_m = 0$) at $K = 0.00$. All 512 voices sit on the exact same action line $I_m = 0$, playing a pure, laser-focused unison sine tone at $f_0 = 220\text{ Hz}$ while rotating smoothly around the stereo field!
   • Raise $K$ to 0.75 (Subcritical KAM): The unison fractures into a bounded, shimmering chorus trapped strictly between the inner KAM barriers — the pitch spread never widens past the KAM dam!
   • Raise $K$ to 1.05 (Cantorus Turnstile Pausing) and click Reset 512 Unison: Listen closely! First, the 512 voices spread into a tight cluster trapped behind the first cantorus barrier — holding a stationary microtonal cluster chord for several seconds — and then, as trajectories squeeze one by one through the MacKay–Meiss–Percival turnstile lobes, you hear discrete microtonal tendril voices peel away upward and downward into the next cantorus chamber!
   • Raise $K$ to 1.55 (Global Chaotic Cloud): The cantori dissolve and the 512 voices diffuse outward as $\exp(\gamma I_m(t))$ into a vast, stereo-panned stochastic Ligeti-style cloud!

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Problem Set

Problem 5.1: Chirikov’s Resonance Overlap Criterion & Secondary $1:2$ Islands

Consider the continuously driven rotor Hamiltonian $H(\theta, p, t) = \frac{1}{2}p^2 + K\cos\theta \sum_{m=-\infty}^\infty \cos(2\pi m t)$.

1. Primary Integer Resonances & Overlap Threshold: Near the $m$-th primary resonance $p \approx 2\pi m$, define the resonant phase $\psi_m = \theta - 2\pi m t$ and momentum deviation $\delta p = p - 2\pi m$.

   • Average $H$ over the fast non-resonant terms $m' \neq m$ to obtain the pendulum Hamiltonian $H_m(\psi_m, \delta p) = \frac{1}{2}(\delta p)^2 + K\cos\psi_m$.
   • Show that the maximum momentum excursion on the separatrix $H_m = +K$ is $\Delta p_{\text{half}} = 2\sqrt{K}$, and verify that two adjacent integer resonances ($m$ and $m+1$, separated by $\delta p = 2\pi$) touch when: $s_{\text{overlap}} = \frac{2\Delta p_{\text{half}}}{2\pi} = \frac{2\sqrt{K}}{\pi} = 1 \quad \Longleftrightarrow \quad K_{\text{overlap}}^{(1)} = \frac{\pi^2}{4} \approx 2.4674$

2. Secondary $1:2$ Resonance at $p = \pi$: Using canonical perturbation theory to second order in $K$ midway between $m = 0$ and $m = 1$ (at $p = \pi$), the interaction of $\cos\theta$ and $\cos(\theta - 2\pi t)$ produces an effective half-integer resonance in $2\theta - 2\pi t$ whose separatrix half-width is $\Delta p_{1/2} = K/2$ (§5.1; derive the resonant potential’s amplitude on the way). Show that requiring the primary separatrices ($\Delta p_0 = 2\sqrt{K}$) and the secondary $1:2$ separatrix ($\Delta p_{1/2} = K/2$) to fill the half-spacing $\pi$ gives the improved overlap equation: $2\sqrt{K} + \frac{K}{2} = \pi \quad \Longrightarrow \quad K_{\text{overlap}}^{(2)} \approx 1.46$ Explain how including the infinite Fibonacci chain of higher-order resonances $F_{k-1}/F_k$ around $\omega_{\text{gold}} = \varphi^{-1}$ brings the overlap threshold down to Greene’s exact $K_c \approx 0.971635$.

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Problem 5.2: Discrete Lagrangian Action, Cantori & Turnstile Flux $\Delta W$

The Chirikov Standard Map $p_{n+1} = p_n + K\sin\theta_n$, $\theta_{n+1} = \theta_n + p_{n+1}$ is generated by the Type-1 generating function:

$S(\theta_n, \theta_{n+1}) = \frac{1}{2}(\theta_{n+1} - \theta_n)^2 + K\cos\theta_n$

1. Period-1 & Period-2 Action-Minimizing vs. Minimax Orbits: For rotation number $\omega = 0/1$ ($\theta_{n+1} = \theta_n$), the 1-step action is $W(\theta_0) = S(\theta_0, \theta_0) = K\cos\theta_0$.

   • Find the action-minimizing fixed point $\theta_0^{\text{min}}$ and the minimax (saddle/maximum) fixed point $\theta_0^{\text{max}}$, and compute the linearized Jacobian trace $\operatorname{tr}(DM)$ at each to verify that the action minimum at $\theta_0^{\text{min}} = \pi$ is hyperbolic ($\operatorname{tr} DM = 2 + K > 2$ for $K > 0$), while $\theta_0^{\text{max}} = 0$ is elliptic ($\operatorname{tr} DM = 2 - K \in (-2, 2)$ for $0 < K < 4$).
   • Explain why Poincaré–Birkhoff island chains always pair an action-minimizing hyperbolic saddle with an action-minimax elliptic centre!

2. Exact Turnstile Lobe Area from Generating Functions (MacKay–Meiss–Percival Theorem): Suppose $z^{\text{min}}_n = (\theta_n^{\text{min}}, p_n^{\text{min}})$ and $z^{\text{max}}_n = (\theta_n^{\text{max}}, p_n^{\text{max}})$ are two homoclinic orbits (such as the gap endpoints and gap-centre minimax point of a cantorus) that converge to the same asymptotic limits as $n \to \pm\infty$, connected by an unstable curve $\gamma_u$ at step 0 and a stable curve $\gamma_s$ at step 0 that bound a turnstile lobe $L$. Using $p_n\,d\theta_n - p_{n+1}\,d\theta_{n+1} = -dS(\theta_n, \theta_{n+1})$ and telescoping the integral from $n = -M$ to $n = +M$, prove that the symplectic area of the turnstile lobe is given by Mather’s $\Delta W$ sum: $\operatorname{Area}(L) = \oint_{\partial L} p\,d\theta = \sum_{n=-\infty}^{\infty} \Big( S\big(\theta_n^{\text{max}}, \theta_{n+1}^{\text{max}}\big) - S\big(\theta_n^{\text{min}}, \theta_{n+1}^{\text{min}}\big) \Big)$

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Problem 5.3: Quasilinear Diffusion, Bessel Correlations & Accelerator Modes

1. Two-Step Angle Correlation & The $-J_2(K)$ Correction (Rechester & White, 1980): Starting from a uniform distribution of $(\theta_0, p_0) \in [0, 2\pi)^2$, write $\theta_1 = \theta_0 + p_0 + K\sin\theta_0$ and $\theta_2 = \theta_1 + p_0 + K\sin\theta_0 + K\sin\theta_1$.

   • Using the Jacobi–Anger Bessel identity $e^{i z \sin\phi} = \sum_{m=-\infty}^\infty J_m(z)\,e^{i m \phi}$, integrate over $(\theta_0, p_0) \in [0, 2\pi)^2$ to prove that: $\big\langle \sin\theta_0\,\sin\theta_1 \big\rangle = 0, \qquad \big\langle \sin\theta_0\,\sin\theta_2 \big\rangle = -\frac{1}{2}J_2(K)$
   • Substitute this into the Green–Kubo diffusivity formula $D(K) = \frac{K^2}{2}\Big(\frac{1}{2} + \sum_{m=1}^\infty \langle \sin\theta_0 \sin\theta_m \rangle\Big)$ to obtain the leading oscillatory diffusion law: $D(K) \approx \frac{K^2}{4}\big(1 - 2 J_2(K)\big)$

2. Ballistic Accelerator Modes & Superdiffusion ($\alpha = 2$): Look for a period-1 accelerator mode of the Chirikov Standard Map where the angle returns to the same value $\theta_{n+1} = \theta_n = \theta_* \pmod{2\pi}$ while the momentum increases by $+2\pi m$ every single step ($p_{n+1} = p_n + 2\pi m$, $m \in \mathbb{Z} \setminus \{0\}$).

   • Show that such a fixed point exists at $p_* = 0 \pmod{2\pi}$ and $\sin\theta_* = \frac{2\pi m}{K}$ whenever $K \ge 2\pi |m|$.
   • Compute the Jacobian trace $\operatorname{tr}(DM) = 2 + K\cos\theta_*$ at this accelerator point and prove that it is linearly stable (elliptic, $|\operatorname{tr} DM| < 2$) inside the window: $2\pi |m| < K < \sqrt{(2\pi m)^2 + 16}$
   • Explain why trajectories initialized inside (or sticking to the cantori around) this stable island have $\Delta p_N \propto 2\pi m N$, driving anomalous superdiffusion $\langle \Delta p_N^2 \rangle \sim N^\alpha$ with $\alpha \to 2$.