Double Pendulums Reveal Stable Periodic Orbits Across Chaotic Zones

2swap

Mapping double pendulums across angle and momentum coordinates reveals discrete islands of stability that resist divergence, demonstrating that chaotic behavior is not an inevitable consequence of starting position or elevated energy.

AI-assisted summary of Double Pendulums are Chaoticn't — original source. Check the original for full context.

Source published: 2025-07-11

Key ideas

While double pendulums are universally cited as textbook chaotic systems, specific configurations—such as the 'Pretzel Pendulum'—completely resist divergence and maintain stable, repeating trajectories despite minute differences in starting positions. (0:00)
Mapping double pendulum divergence rates across initial angle coordinates produces an infinite, periodic fractal landscape partitioned into a coherent black regime and a chaotic white sea. (2:13)
Isolated 'islands of stability' exist within the chaotic regime, yielding structured periodic angle-space trajectories dubbed the Pretzel, the Shoelace, and the Heart. (4:27)
Fixing starting angles at rest and varying initial angular momentum demonstrates that chaos is not strictly governed by high mechanical energy, as stable pockets and spirograph-like patterns reemerge at non-zero velocities. (5:35)

Detailed notes

## Background Published by 2swap on YouTube on July 11, 2025, this video explores the dynamics of double pendulums beyond their textbook reputation for chaos. The video description credits inspirations from Drew's Campfire, Cruz Godar's generative art color schemes, and scientific literature on double pendulum energy, while noting an erratum regarding analytical closed-form solutions for non-trivial starting positions.

## TL;DR While double pendulums are universally cited as quintessential examples of chaotic unpredictability, mapping their phase space reveals surprising pockets of absolute order. By tracking angle-space trajectories, sonifying motion data into stereo audio, and plotting divergence rates across high-density grids, the video visualizes intricate fractal boundaries between chaos and coherence. Within the sea of chaos lie isolated "islands of stability"—yielding closed, repeating loops nicknamed the Pretzel, Shoelace, and Heart. Finally, testing momentum-space demonstrates that this order is not merely an artifact of low mechanical energy, revealing a landscape where order and chaos interweave.

## Atmosphere & style The presentation is intellectually playful, meditative, and visually mesmerizing, blending rigorous computational physics with aesthetic appreciation. It moves smoothly from a disarming debunking of pop-science dogma into hypnotic visualizations and sonic experiments, creating an almost synesthetic experience when mapping angle data directly to stereo sound. The narration carries an understated, curious wonder, guiding the viewer through vast fractal landscapes to uncover delicate geometric gems hidden within apparent noise.

## Bulletin - [00:00](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=0s) While double pendulums are universally cited as textbook chaotic systems, specific configurations—such as the 'Pretzel Pendulum'—completely resist divergence and maintain stable, repeating trajectories despite minute differences in starting positions. - [02:13](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=133s) Mapping double pendulum divergence rates across initial angle coordinates produces an infinite, periodic fractal landscape partitioned into a coherent black regime and a chaotic white sea. - [04:27](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=267s) Isolated 'islands of stability' exist within the chaotic regime, yielding structured periodic angle-space trajectories dubbed the Pretzel, the Shoelace, and the Heart. - [05:35](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=335s) Fixing starting angles at rest and varying initial angular momentum demonstrates that chaos is not strictly governed by high mechanical energy, as stable pockets and spirograph-like patterns reemerge at non-zero velocities.

## Key takeaways - [00:00](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=0s) Popular science typically frames the double pendulum as chaotic due to extreme sensitivity to initial conditions, where micro-deviations amplify exponentially. - [00:00](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=0s) The 'Pretzel Pendulum' disproves the universal chaos claim by maintaining identical, repeating trajectories despite slightly varied initial conditions under standard deterministic physics. - [01:02](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=62s) Plotting state trajectories in angle-space (top arm angle vs. bottom arm angle) separates space-filling chaotic motion from closed, periodic Lissajous curves. - [01:02](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=62s) Sonifying pendulum angles into stereo audio channels translates chaotic motion into an erratic rumble and stable motion into structured periodic acoustic pitches. - [02:13](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=133s) Simulating one double pendulum per pixel across an angle grid generates an infinite periodic fractal that tiles every 2π radians. - [02:13](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=133s) Evaluating local divergence rates across the fractal grid reveals a dark coherent central regime surrounded by a high-divergence white chaotic regime. - [02:13](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=133s) Continuous coordinate movement inside the coherent region smoothly morphs Lissajous curves until crossing the white threshold, where chaotic dynamics trigger instantly. - [04:27](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=267s) Deep inside the chaotic white region, isolated 'islands of stability' harbor non-divergent trajectories, including the Pretzel, Shoelace, and Heart patterns. - [04:27](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=267s) The central coherent region overlaps closely with a low mechanical potential energy basin, prompting the hypothesis that low energy guarantees order. - [05:35](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=335s) Switching the analysis to momentum-space by varying initial angular velocities proves that stability is not solely a low-energy phenomenon. - [05:35](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=335s) Extreme starting momenta generate complex rotations and spirograph-like trajectories that weave between ordered modes and chaotic behavior. - [05:35](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=335s) Double pendulum differential equations lack closed-form analytic solutions for non-trivial positions, yet the underlying phase space supports profound pockets of order.

## Detailed walkthrough ### [00:00](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=0s) The Pretzel Pendulum and Challenging Chaos

The standard narrative around the double pendulum is that it is the textbook example of chaos. Popular science videos frequently demonstrate this by setting up two seemingly identical double pendulums with microscopic initial differences; over time, those imperceptible deviations amplify exponentially until the two pendulums desynchronize entirely into an unpredictable mess. Because of this hyper-sensitivity to initial conditions, double pendulums are broadly declared to be chaotic.

```mermaid flowchart TD accTitle: Double Pendulum Sensitivity to Initial Conditions accDescr: Diagram showing how identical physics with tiny initial differences leads to either chaos or stable repeating patterns. subgraph Initial["Initial Conditions"] A["Top & Bottom Arm Angles"] B["Minute Starting Difference"] end subgraph Dynamics["Governing Physics"] C["Coupled Equations of Motion"] end subgraph Outcomes["Observed Behaviors"] D["Chaotic Desynchronization<br>(Complete Mess)"] E["Islands of Stability<br>(Pretzel Pendulum Pattern)"] end A --> C B --> C C --> D C --> E ```

However, this reputation ignores a striking counterexample. 2swap introduces a pair of double pendulums with slightly different starting positions that completely refuse to diverge: “These pendulums also have slightly different starting positions, but they will *not* diverge.” Instead of devolving into randomness, they trace out an identical, clean, repeating trajectory dubbed the "Pretzel Pendulum." Both the mess-making pendulums and the Pretzel Pendulums follow the exact same deterministic laws of physics—the only variable changed is the exact initial position where they were released. By systematically mapping how behavior changes as a function of starting position, one can construct intricate fractals where every coordinate indicates the chaoticity of that specific configuration.

[01:02](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=62s) Angle-Space Plots and Sonifying Pendulums

To unpack the mathematics, 2swap inspects individual specimens by tracking two core variables: the angle of the top bar relative to the vertical ($x$-axis) and the angle of the bottom bar relative to the vertical ($y$-axis). Plotting this state trajectory generates a phase-plane visualization in angle-space: - **The Chaotic Specimen**: The trajectory wanders continuously, filling space without ever settling into a regular loop or predictable geometric path. - **The Non-Chaotic Specimen**: The path cleanly traces a regular, closed geometric figure known as a Lissajous curve. - **The Pretzel Pendulum**: Traces a distinct, highly repetitive closed loop in angle-space.

```mermaid flowchart LR accTitle: Angle Data Sonification Pipeline accDescr: Flowchart illustrating how top and bottom angle signals are mapped to audio channels. subgraph Angles["Angle Signals"] X["Top Angle (X)"] Y["Bottom Angle (Y)"] end subgraph Audio["Stereo Audio Mapping"] L["Left Speaker"] R["Right Speaker"] end subgraph Soundscapes["Acoustic Output"] C["Clean / Periodic Pitch"] N["Harsh / Chaotic Noise"] end X --> L Y --> R L & R --> C L & R --> N ```

To make these patterns visceral, 2swap isolates the time-series signals of the top and bottom angles and routes them directly to audio, assigning the top angle to the left speaker and the bottom angle to the right speaker. When sonified, the regular Lissajous curve produces a harsh but clearly structured acoustic oscillation. The chaotic pendulum, by comparison, produces an erratic, noisy rumble. The Pretzel Pendulum, with its clean, tight periodic cycle, produces the purest and most consistent sound. While this audio mapping proves that one can listen to pendulums to diagnose chaos, doing so one by one is far too slow to explore the broader landscape.

[02:13](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=133s) Fractal Landscapes and Divergence Rates

To scale up the analysis, 2swap constructs a 2D grid array of pendulums where the horizontal position corresponds to the top angle and the vertical position corresponds to the bottom angle, with colors assigned to specific angle positions. Pushing the simulation to a density of one pendulum per pixel reveals an expansive fractal pattern.

Because rotating either pendulum arm by a full circle ($2\pi$ radians) returns it to the exact same physical orientation, the resulting fractal perfectly tiles the plane periodically. Within this infinite landscape, two starkly separated regimes emerge: 1. **The Coherent Regime (Black)**: Centered around the origin $(0,0)$, where the pendulums are largely undisturbed and exhibit predictable, non-chaotic dynamics. 2. **The Chaotic Regime (White)**: Outer regions where initial configurations yield wild, unpredictable motion.

To quantify and visualize this boundary, each pixel is assigned two pendulums separated by an infinitesimal initial displacement. The color value of that pixel tracks how rapidly those two pendulums diverge over time. Selecting a coordinate inside the central black zone yields a stable, smooth Lissajous curve; sliding that coordinate continuously within the black region morphs the curve smoothly and gracefully. But the moment the coordinate crosses the threshold into the white zone: “All hell breaks loose.”

[04:27](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=267s) Islands of Stability and the Energy Hypothesis

Deep within the white sea of chaos lie isolated black enclaves: the "islands of stability." These pockets host special starting positions that resist divergence and settle into stable, repeatable orbits: - **The Pretzel**: Found inside the largest island of stability, maintaining its tight, woven trajectory. - **The Shoelace**: A more intricate closed pattern found in another stable enclave. - **The Heart**: A tiny island of stability whose angle-space trajectory traces out a literal heart shape.

Noticing that the well-behaved Lissajous pendulums all cluster near the coordinate $(0,0)$—where the arms hang down at rest—points toward a physical hypothesis: mechanical energy. Pendulums near $(0,0)$ possess very little potential energy. 2swap tests this by calculating the mechanical energy of a borderline-chaotic pendulum and highlighting in red every configuration on the map with equal or lesser mechanical energy. The highlighted energy basin overlaps almost perfectly with the central black Lissajous region, raising the question of whether chaos is simply a byproduct of high mechanical energy while low energy guarantees coherence.

[05:35](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=335s) Shifting to Momentum-Space and Order in the Rough

To test whether high energy strictly dictates chaos, 2swap shifts paradigms. Rather than releasing motionless pendulums from various angled configurations, the setup fixes the starting angles at $(0,0)$ (pointing straight down) and varies their initial starting momentum (angular velocities).

Mapping this momentum-space reveals that stability is not merely a low-energy phenomenon: - Islands of stability reappear across non-zero starting momenta. - While low-energy pendulums in momentum-space remain stable and slightly higher-energy ones become chaotic, ultra-high-energy configurations resist simple categorization. - At extreme energies, the arms whip through complex rotations, generating intricate spirograph-like trajectories that weave between order and chaos.

Reflecting on the nature of the system, 2swap notes that "chaos" itself is an elusive concept whose working definition shifts between aperiodicity, unpredictability, and trajectory divergence depending on the lens applied. Even though the governing differential equations lack an analytic closed-form solution for non-trivial starting points, the phase space remains profoundly rich: “...the differential equations describing double pendulums have no analytic solution for *any* nontrivial starting position, but even so, this system is sufficiently fertile to support little gems of order in the rough.”

## Notable quotes > [05:35](https://www.youtube.com/watch?v=dtjb2OhEQcU&t=335s) the differential equations describing double pendulums have no analytic solution for any nontrivial starting position, but even so, this system is sufficiently fertile to support little gems of order in the rough. — 2swap

## Mentioned 2swap, Brilliant, 6884, SwapTube, Drew's Campfire, Cruz Godar, Ricky Reuser

## Video description http://brilliant.org/2swap/ - Click here for a 30 day Brilliant free trial and 20% discount on an annual premium subscription!

ERRATUM: At the end of the video, I said: "there is no closed-form solution for non-trivial starting positions." The trivial cases I was referring to are those in which the two arms are collinear, and those in which both arms are vertical but pointing in opposite directions. However, my general claim may not be precisely correct. It is indeed the case that there exists no closed-form function which maps a pendulum's starting angles and a time duration onto the pendulum propagated out for that duration of time. However it is not yet proven that no _specific_ starting condition has a function that maps duration runtime to future position. That said, no such configuration has been found either, and very well may not exist.

If you liked this, please support me on Patreon! https://www.patreon.com/2swap You can support the musician, 6884, on Kofi: https://ko-fi.com/iam6884 Check out his music here! https://6884.bandcamp.com/

SOURCES AND CREDITS Video animated with SwapTube: https://github.com/2swap/swaptube This was largely inspired by this video from Drew's Campfire: https://www.youtube.com/watch?v=OIHyN7TzY6A Pretzel drawing by https://www.instagram.com/litaapollo/ Discussion of energy was inspired by this paper: https://www.famaf.unc.edu.ar/~vmarconi/fiscomp/Double.pdf. Notice that they find slightly different results, since they plot whether or not a pendulum arm peforms a flip as opposed to divergent behavior, which I focused on. The color scheme I used in this video was generously provided by Cruz Godar, who has a website packed full of cool generative art here, double pendulums included! https://cruzgodar.com/gallery

After rendering this video, I investigated the internal structure of the islands of stability, such as their oscillatory modes. Maybe I'll make another video showing my findings...

Here's an interactive online version by Ricky Reuser: https://observablehq.com/@rreusser/the-double-pendulum-fractal

Join our discord server about math, computer science, and other miscellaneous nerd stuff! https://discord.gg/XDb7x2BZAE

[Source](https://www.youtube.com/watch?v=dtjb2OhEQcU)

中文简报

双摆系统在混沌区域中展现稳定周期轨道通过在角度和动量坐标系中绘制双摆图谱，揭示了能够抵御发散的离散“稳定岛”，表明混沌行为并非初始位置或高能量必然导致的结果。尽管双摆被普遍视为混沌不可预测性的典型范例，但绘制其相空间却揭示了令人惊讶的绝对有序区域。通过追踪角度空间轨迹、将运动数据转化为立体声音频，并在高密度网格上绘制发散率图，该视频直观呈现了混沌与有序之间错综复杂的分形边界。在混沌之海中存在着孤立的“稳定岛”——形成了被称为“椒盐卷饼”（Pretzel）、“鞋带”（Shoelace）和“爱心”（Heart）的封闭重复循环。最后，对动量空间的测试表明，这种有序性不仅仅是低机械能的副产物，从而揭示了一个秩序与混沌相互交织的图景。尽管双摆常被奉为教科书级的混沌系统，但某些特定构型（例如“椒盐卷饼摆”）却能完全抵御发散，在初始位置存在微小差异的情况下仍能保持稳定的重复轨迹。 (0:00) 在初始角度坐标系中绘制双摆的发散率，会生成一个无限的周期性分形图景，该图景被划分为有序的黑色区域和混沌的白色海洋。 (2:13) 混沌区域内存在孤立的“稳定岛”，从而产生被称为“椒盐卷饼”、“鞋带”和“爱心”的结构化周期性角度空间轨迹。 (4:27) 在固定初始静止角度并改变初始角动量的情况下，测试表明混沌并非严格由高机械能决定，因为在非零速度下，稳定口袋和类似万花尺的图案会再次出现。 (5:35)

Reading summary

While the double pendulum is conventionally treated as the textbook archetype of chaos, 2swap demonstrates that the same deterministic laws produce both extreme divergence and highly stable, periodic trajectories depending strictly on starting conditions. Identifying Order in Angle-Space and Sound (0:00) Popular science demonstrations frequently highlight double pendulums to illustrate sensitive dependence on initial conditions, showing how imperceptible initial variations exponentially diverge into unpredictable motion. 2swap counters this narrative using the "Pretzel Pendulum," a pair of pendulums launched from slightly different starting angles that trace an identical, closed trajectory without diverging. Tracking the top and bottom arm angles on a phase-plane visualization separates these behaviors: chaotic setups wander aimlessly and fill space, whereas non-chaotic configurations trace regular Lissajous figures or tight periodic loops. When 2swap routes the top angle signal to the left audio speaker and the bottom angle to the right, chaotic trajectories produce an erratic rumble, while the Pretzel Pendulum emits a consistent, pure periodic tone. Fractal Landscapes of Trajectory Divergence (2:13) To evaluate behavior at scale, 2swap maps initial configurations onto a 2D grid where the horizontal and vertical axes represent the top and bottom arm angles, setting simulation density at one pendulum per pixel. Because a 2π rotation returns the arms to their initial physical state, the resulting map tiles periodically across the plane. By measuring how rapidly two infinitesimally separated pendulums diverge at each coordinate, the simulation produces a fractal landscape. A coherent central black region around the origin (0,0) exhibits smooth, predictable Lissajous movement, while the surrounding white region marks rapid divergence where small shifts cause chaotic motion. Islands of Stability and Energy Constraints (4:27) Isolated pockets of order, termed "islands of stability," appear embedded within the chaotic white expanse. These enclaves produce repeatable, non-diverging orbits, including the wide loops of the Pretzel Pendulum, the complex interlocking paths of the "Shoelace," and a tiny enclave tracing a "Heart" shape. Because the large, stable Lissajous region centers on (0,0)—the resting point of lowest potential energy—2swap tests whether chaos is simply dictated by total mechanical energy. Mapping all coordinates possessing mechanical energy equal to or below a borderline-chaotic threshold roughly matches the central stable basin, suggesting at first that low energy enforces stability while high energy induces chaos. Momentum-Space Dynamics and Nuanced Chaos (5:35) To test whether high energy inevitably triggers chaos, 2swap holds initial angles fixed at the downward rest position (0,0) and varies the initial angular velocities across momentum-space. This setup reveals that stability is not purely a low-energy trait, as distinct islands of stability recur at non-zero starting momenta. At extreme energies, pendulum arms rotate rapidly and produce intricate, spirograph-like trajectories that weave between order and chaos rather than devolving purely into disorganized noise. This demonstrates that energy level alone does not define chaotic behavior. 2swap concludes that double pendulum differential equations lack analytic solutions for nontrivial positions, yet support persistent pockets of order within chaotic regimes. Furthermore, the working definition of chaos itself shifts depending on whether a researcher evaluates trajectory divergence, unpredictability, or aperiodicity.

精炼阅读摘要

创作者 2swap 指出，双摆常被视作对初始条件极度敏感的经典混沌教科书范例，但在完全相同的确定性物理定律下，特定初始配置会产生拒绝发散并持续重复的稳定轨迹；系统相空间中不仅存在大范围混沌，更蕴含着精致的“稳定岛”。挑战双摆的纯粹混沌叙事 (0:00) 在通俗科学的普遍叙事中，双摆被广泛用作展示混沌现象的基准工具。常见的演示是将两个肉眼看似完全一致的双摆置于存在微观差异的初始位置，随着时间推移，这种难以察觉的微小偏差会被指数级放大，最终导致系统彻底失去同步并陷入无法预测的无序状态，双摆也因此被普遍贴上不可预测的混沌标签。然而，2swap 提出了一个打破该常识的反例：一对初始位置存在微小差异的双摆完全没有发生发散，而是稳定绘制出重复且干净的轨迹，该现象被称为“椒盐卷饼摆”（Pretzel Pendulum）。混沌双摆与椒盐卷饼摆完全遵循相同的确定性运动方程，唯一的变量在于释放时的确切初始坐标。这表明双摆系统并非处处混沌，而是依赖于初始状态在相空间中的具体分布。角度空间轨迹与双声道声音化表征 (1:02) 为了在数学层面深入剖析动力学状态，2swap 追踪了两个核心变量：顶端摆臂相对于垂线的摆角（X轴）与底端摆臂相对于垂线的摆角（Y轴），并在角度空间（angle-space）中绘制相平面轨迹。测试显示，混沌样本的轨迹在角度空间中无序游荡并逐渐填满空间，永不形成规则回路；非混沌样本呈现为规则闭合的利萨茹曲线（Lissajous curve）；而椒盐卷饼摆则展现出高度重复且对称的紧致闭环轨迹。为进一步感官化呈现这些模式，2swap 将两根摆臂的角度时间序列信号直接映射到音频通道（顶角对应左声道，底角对应右声道）。声音化测试表明，利萨茹曲线产生虽然粗糙但结构清晰的周期性声学振荡；混沌双摆表现为完全失序的轰鸣噪声；而周期极其稳定的椒盐卷饼摆则输出了最纯净、最一致的音调。这种音频映射证明通过听觉可以直接诊断动力学系统的混沌程度。分形发散全景与混沌海中的稳定岛 (2:13) 逐一测试摆锤无法窥见全局，2swap 进而建立了单像素对应单双摆的高分辨率二维网格，水平和垂直坐标分别代表顶角与底角。由于摆臂旋转 2π 弧度（一整圈）会回到完全相同的物理朝向，整张动力学图谱在平面上呈现出无限周期性平铺的分形图案。网格中每个像素被赋予一对存在微小初始位移的双摆，其颜色深浅直接对应两个摆随时间发散的速率。该全景图呈现出清晰的区域分离：以原点 (0,0) 为中心的连贯区域表现为黑色，表明摆锤运动发散极慢且遵循可预测的利萨茹轨迹；向外跨过阈值后则进入快速发散的白色混沌海。引人注目的是，在白色混沌海深处散布着独立的黑色飞地，即“稳定岛”。这些孤岛中的初始条件能够完全抵御发散，包括位于最大稳定岛内的“椒盐卷饼”模式、结构更交织的“鞋带”（Shoelace）模式，以及角度轨迹精准呈现为心形轮廓的微型稳定岛“爱心”（Heart）。能量假说检验与动量空间秩序 (4:27) 观察到表现良好的利萨茹双摆集中在原点 (0,0) 附近——即双臂静止垂下的最低点，2swap 提出了物理机械能假说，探讨混沌是否仅仅由高机械能诱发，而低能量则天然保证连贯性。通过计算临界混沌摆的机械能并标注等能量或更低能量的区域，该能量盆地与中心黑色的利萨茹区域几乎完全重合，初步支持了能量门槛的猜想。为了彻底检验高能量是否必然导致混沌，2swap 将实验范式转向动量空间：将初始角度固定在垂直向下的原点 (0,0)，转而改变摆臂释放时的初始动量（角速度）。映射结果否定了简单的能量决定论：离散的稳定岛在非零初始动量区域依然存在；在极端高能量下，剧烈旋转的摆臂形成了类似繁花曲线（spirograph）的复杂轨迹，在秩序与混沌之间反复穿梭交织，证明相空间中的稳定结构并不受制于单一能量界限。 2swap 总结指出，“混沌”本身的定义随着非周期性、不可预测性或轨迹发散等不同视角而有所浮动。尽管描述双摆的微分方程对任何非平凡初始状态都不存在解析闭式解，但该确定性动力系统在看似杂乱无章的混沌表象之下，依然具备支撑精致局部秩序的丰饶数学空间。

Source quotes

[5:35] 2swap: the differential equations describing double pendulums have no analytic solution for any nontrivial starting position, but even so, this system is sufficiently fertile to support little gems of order in the rough.

Original transcript

[0:00] Double pendulums are NOT chaotic. [0:02] Or, at least, not all of them. [0:04] You've probably seen videos like these, [0:07] where a tiny deviation in similar double pendulums amplifies over time, [0:11] until they completely desynchronize. [0:13] This system is so sensitive to initial conditions that it's practically unpredictable, so we call it chaotic. [0:20] But you probably haven't seen this: [0:22] These pendulums also have slightly different starting positions, but they will _not_ diverge. [0:27] They even trace a repeating pattern, [0:30] for which I call this the Pretzel Pendulum. [0:33] A stark contrast with the first ones, which are... making a complete mess. [0:38] So... what's the deal? [0:40] These pendulums follow the same laws of physics. [0:43] The only difference is the position from which they started. [0:47] And behavior as a function of starting position can be graphed, [0:51] revealing fractals like these, [0:53] where each point shows how chaotic a certain pendulum is. [0:56] But before diving into the fractals, let's get to know a few individual specimen. [1:02] This pendulum is one of the chaotic ones. [1:05] We are particularly interested in the angles that separate each bar from the vertical. [1:10] Plotting the top angle as X and the bottom angle as Y, we can make a graph like this. [1:15] This pendulum is evidently chaotic from the graph. [1:18] It follows no consistent pattern. [1:24] Here's a non-chaotic pendulum. [1:32] It's drawing a shape known as a Lissajous curve. [1:35] Here's the signal from the top and bottom angles, separated out. [1:40] We can run this pendulum for a long time, reinterpret these signals as [1:44] sound waves on the left and right speaker, and 'listen' to the pendulum! [1:48] [Computer Generated Sound] [1:52] It doesn't precisely sound beautiful, but compare that with the chaotic pendulum! [1:56] [Computer Generated Sound] [1:59] And here's the pretzel pendulum. [2:01] [Computer Generated Sound] [2:04] This one traces a repetitive curve in angle-space, so it sounds the cleanest. [2:08] Listening to a pendulum tells us if it's chaotic, but we have to listen one-by-one. [2:13] Instead, we can make a large array of pendulums like this. [2:18] The pendulum's x position in the grid corresponds to the top angle, [2:22] and its y position corresponds to the bottom angle. [2:26] By associating angle positions with colors, it's easier to tell what's going on. [2:30] As a bonus, we can add points in angle space for these pendulums and see how they move. [2:53] Let's increase the resolution to one pendulum per pixel. [3:00] A nice feature of this fractal is that it tiles the plane. [3:04] Rotating either pendulum arm by 2pi yields the exact same position, so the fractal is periodic. [3:11] Pay attention to how there are two distinct modes of behavior here. [3:15] There's a region of chaotic pendulums sensitive to their initial conditions, [3:19] as well as a region of coherent ones which are not. [3:23] Now, for each pixel, we track two pendulums, separated by a [3:26] slight starting difference, and plot their difference over time. [3:31] So, this plot shows how quickly the pendulums in our grid diverge. [3:35] Let's reset the pendulums and watch that again from the start. [3:38] Here we go! [3:48] Now, I'm gonna pick a certain point, corresponding to a pendulum [3:52] in the black region, meaning its behavior is non-chaotic. [3:55] We can plot its path in angle-space just like we did before! [3:59] Moving the point around in the black region, this curve moves smoothly and cleanly. [4:13] This main black region is home to all the lissajous style pendulums. [4:17] But as soon as you leave and step into the chaotic region... [4:23] All hell breaks loose. [4:27] Let's look closer at the chaotic region in white. [4:29] There are a few spots of black in here. [4:32] These 'islands of stability' contain special pendulums which follow stable paths. [4:37] This big island of stability contains the Pretzel we saw earlier. [4:41] [Computer Generated Sound] [4:47] This one, which I call the shoelace, traces a more complex pattern. [4:51] [Computer Generated Sound] [4:57] This one draws a picture of a heart. This island is particularly small. [5:02] [Computer Generated Sound] [5:08] We've seen how the pendulums which start near the angle zero-zero are very well-behaved. [5:12] Those pendulums have extremely low mechanical energy. [5:16] So maybe energy is somehow involved? [5:19] Taking this borderline-chaotic pendulum, [5:21] If I color red all the pendulums with less mechanical energy than this one [5:26] It overlaps nicely with the lissajous-esque pendulums. [5:29] So, is that it? Are high-energy pendulums chaotic, while low-energy pendulums are coherent? [5:35] It's time for a change in perspective. [5:37] So far, I've been dropping the pendulums from a motionless state. [5:41] What if, instead, we start the pendulum off with some momentum? [5:53] Sure enough, there are islands of stability for other starting momenta too. [5:57] [Computer Generated Sound] [6:03] I'm now reorienting the axes of our fractal to be in momentum-space instead of angle-space. [6:09] In other words, we're looking at a grid of pendulums all with starting angles at 0, but with different starting speeds. [6:16] It's still the case that the pendulums with low energy are stable, [6:19] [Computer Generated Sound] [6:25] and the pendulums with slightly higher energy are chaotic, [6:28] [Computer Generated Sound] [6:34] But these ultra-high-energy pendulums don't nicely fit into one box or another. [6:39] [Computer Generated Sound] [6:45] [Computer Generated Sound] [6:51] [Computer Generated Sound] [6:57] [Computer Generated Sound] [7:04] [Computer Generated Sound] [7:09] Here's some spirographs from all over momentum-space. [7:27] Chaos is a hard thing to define. [7:30] You may have even noticed that I subtly changed definitions [7:33] throughout the video: aperiodicity, unpredictibility, divergence... [7:38] These definitions usually match up... but, the differential equations describing [7:42] double pendulums have no analytic solution for _any_ nontrivial starting position, [7:47] but even so, [7:49] this system is sufficiently fertile to support little gems of order in the rough. [7:54] This has been 2swap. [7:56] with music by 6884