A search for the simplest chaotic partial differential equation |
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Authors: | Charles D. Brummitt |
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Affiliation: | Department of Physics, University of Wisconsin, Madison, WI 53706, USA |
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Abstract: | A numerical search for the simplest chaotic partial differential equation (PDE) suggests that the Kuramoto-Sivashinsky equation is the simplest chaotic PDE with a quadratic or cubic nonlinearity and periodic boundary conditions. We define the simplicity of an equation, enumerate all autonomous equations with a single quadratic or cubic nonlinearity that are simpler than the Kuramoto-Sivashinsky equation, and then test those equations for chaos, but none appear to be chaotic. However, the search finds several chaotic, ill-posed PDEs; the simplest of these, in the discrete approximation of finitely many, coupled ordinary differential equations (ODEs), is a strikingly simple, chaotic, circulant ODE system. |
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Keywords: | 05.45.Ra 02.30.Jr 02.60.Lj |
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