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Tweak README
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IlyaOrson committed Jan 3, 2025
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# The Kicked Rotor: Chaos on a Swing

Welcome to the curious dance of the [kicked rotor](https://en.wikipedia.org/wiki/Kicked_rotator), a fun model that thrives with chaos.
Welcome to the curious world of the [kicked rotor](https://en.wikipedia.org/wiki/Kicked_rotator), a physical system that thrives with chaos!

Imagine a pendulum (rotor) that gets periodically kicked on a fixed direction, like a child on a swing receiving pushes with a specific rhythm and strength.
A rotor is similar to a rigid pendulum, which can rotate without friction around a fixed point.
A kicked rotor is a gravity-free rotor that gets periodically hit with a specific rhythm, direction and strength, like a child on a swing receiving pushes.

This idealised system is governed by 2 equations known as the [Chirikov Standard Map](http://www.scholarpedia.org/article/Chirikov_standard_map):

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Here, θ represents the angle, p the angular momentum, and K the kick strength - the chaos tuner.

Each point on the interactive map represents a particular position and velocity of the pendulum.
While the pendulum receives kicks at a constant rhythm, it's the strength of each kick what determines whether the next states of the pendulum will remain predictable or evolve chaotically.
Each point on the interactive map represents a particular position and velocity of the rotor.
While the rotor receives kicks at a constant rhythm, it's the strength of each kick what determines whether the next states of the rotor will remain predictable or evolve chaotically.
The kicked rotor demonstrates how captivating complexity can emerge from simplicity, a theme that resonates throughout nature.

## Interactive exploration 🧑‍🔬

- **Click new paths**: Each click sets a pendulum on a particular angle (θ) and velocity of rotation (p). Points of the same colour show where the pendulum ends up after each kick. This trajectory of points is calculated from the equations above and animated to be able to follow where the pendulum lands after many consistent kicks of the same strength. Watch closely how unexpected symmetries unfold!
- **Click new paths**: Each click sets a rotor on a particular angle (θ) and velocity of rotation (p). Points of the same colour show where the rotor ends up after each kick. This trajectory of points is calculated from the equations above and animated to be able to follow where the rotor lands after many consistent kicks of the same strength. Watch closely how unexpected symmetries unfold!
- **Play with K**: Start with K = 0.5 and watch ordered, predictable motion. Crank it up past K ≈ 0.971635... (the critical parameter) and witness the onset of global chaos.
- **Hunt for islands**: Even in the chaotic sea (high K), you can find stable regions of tangible symmetry (called *islands* or *KAM tori*). These [KAM tori](http://www.scholarpedia.org/article/Kolmogorov-Arnold-Moser_theory) are the last survivors of order in a chaotic world. Can you find them?

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- Scientists at the NIST created a real quantum kicked rotor using cesium atoms in a pulsed optical lattice. They discovered that quantum mechanics can actually suppress chaos - a phenomenon called *dynamical localization* ([Moore et al., 1995](https://doi.org/10.1103/PhysRevLett.73.2974)). Turns out this counterintuitive phenomenon has practical implications for quantum computing ([Pizzamiglio et al., 2021](https://doi.org/10.3390/e23060654)).

- In fusion reactors, plasma (super-hot ionized gas) is confined in a donut-shaped magnetic field. The plasma particles receive kicks from magnetic field perturbations as they orbit, just like our pendulum system, which lead to regions of stability known as *magnetic islands* and to chaos regions on their border ([Willensdorfer et al., 2024](https://doi.org/10.1038/s41567-024-02666-y)), affecting our ability to contain the plasma. Luckily the [Chirikov criterion](http://www.scholarpedia.org/article/Chirikov_criterion) helps to predict under which conditions will chaos emerge.
- In fusion reactors, plasma (super-hot ionized gas) is confined in a donut-shaped magnetic field. The plasma particles receive kicks from magnetic field perturbations as they orbit, just like the kicked rotor system, which lead to regions of stability known as *magnetic islands* and to chaos regions on their border ([Willensdorfer et al., 2024](https://doi.org/10.1038/s41567-024-02666-y)), affecting our ability to contain the plasma. Luckily the [Chirikov criterion](http://www.scholarpedia.org/article/Chirikov_criterion) helps to predict under which conditions will chaos emerge.

## Final thoughts 💭

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