Lyapunov exponents measure: rate of divergence of nearby trajectories in a dynamical system

How do tiny differences grow over time in chaotic systems?

Imagine you're planting seeds in a garden. If two seeds start just a millimeter apart, under certain conditions, they might end up miles apart after a while.

In our garden, the seeds' final distance apart shows how much they spread out. This spreading rate tells us how chaotic the garden's growth is.

Example

If seeds start 1mm apart and after a month they're 10km apart, the garden's growth is extremely chaotic.

Remember this

The Lyapunov exponent measures how quickly tiny differences grow, indicating chaos.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

Related concepts

Chaos theory

Butterfly effect demonstrates sensitive dependence on initial conditions

Langevin dynamics does: adds noise to gradient descent to sample from a distribution

How can we simulate randomness in physics?

Entropy H = -Σ p(x) log₂ p(x) measures average surprise in bits

How do we measure uncertainty in everyday decisions?

the reverse process learns: p_θ(x_{t-1}|x_t)

Can we trace back the roots of life?

the condition number κ(A) measures: sensitivity of Ax=b to perturbations

How small changes affect big outcomes

AdaGrad's learning rate decays to zero

Why does a car's speed drop when it goes uphill?

Swipe through 100 ML concepts daily

Open Pocket Polymath