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Mathematics LibreTexts

9.4: Chaos in Continuous-Time Model

As we reviewed above, chaos is really easy to show in discrete-time models. But the discovery of chaos was originally made with continuous-time dynamical systems, i.e., differential equations. Edward Lorenz, an American mathematician and meteorologist, and one of the founders of chaos theory, accidentally found chaotic behavior in the following model (called the Lorenz equations) that he developed to study the dynamics of atmospheric convection in the early 1960s [5]:

\[ \begin{align*} \dfrac{dx}{dt} &= s(y−x) \label{9.9} \\[5pt] \dfrac{dy}{dt} &= rx−y−xz \label{9.10}\\[5pt] \dfrac{dz}{dt} &= xy−bz \label{9.11} \end{align*} \]

Here \(s\), \(r\), and \(b\) are positive parameters.This model is known to be one of the first that can show chaos in continuous time. Let’s simulate this model with s = 10, r = 30, and b = 3, for example: