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Dynamical Systems

A lot of time series data are generated by dynamical systems. One of the most cited examples is the coordinates \(x(t)\), \(y(t)\), \(z(t)\) as functions of time \(t\) in a Lorenz system.

Lorenz System

A Lorenz system is defined by the Lorenz equations1

\[ \begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma (y - x), \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\ \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z, \end{align} \]

where \(x\), \(y\), and \(z\) are the coordinates of a particle.

It is a chaotic system that is very sensitive to the initial conditions.

Dynamical Systems

Many real-world systems are dynamical systems. Differential equation is a handy tool to model a dynamical system. For example, the action potentials of a squid giant axon can be modeled by the famous Hodgkin-Huxley model.

A naive philosophy to model time series is to come up with a set of differential equations to model the time series. However, finding clean and interpretable differential equations is not easy. It has been the top game in physics for centuries.

In the following sections, we will discuss a few solutions to model data as dynamical systems.

  1. Wikipedia contributors. Lorenz system — Wikipedia, the free encyclopedia. 2023.\system\oldid=1186188179

Contributors: LM