We explore the high-dimensional chaotic dynamics of the Lorenz-96 model bycomputing the variation of the fractal dimension with system parameters. TheLorenz-96 model is a continuous in time and discrete in space model firstproposed by Edward Lorenz to study fundamental issues regarding the forecastingof spatially extended chaotic systems such as the atmosphere. First, we explorethe spatiotemporal chaos limit by increasing the system size while holding themagnitude of the external forcing constant. Second, we explore the strongdriving limit by increasing the external forcing while holding the system sizefixed. As the system size is increased for small values of the forcing we finddynamical states that alternate between periodic and chaotic dynamics. Thewindows of chaos are extensive, on average, with relative deviations fromextensivity on the order of 20%. For intermediate values of the forcing we findchaotic dynamics for all system sizes past a critical value. The fractaldimension exhibits a maximum deviation from extensivity on the order of 5% forsmall changes in system size and decreases non-monotonically with increasingsystem size. The length scale describing the deviations from extensivity andthe natural chaotic length scale are approximately equal in support of thesuggestion that deviations from extensivity are due to the addition of chaoticdegrees of freedom as the system size is increased. As the forcing is increasedat constant system size the fractal dimension exhibits a power-law dependence.The power-law behavior is independent of the system size and quantifies thedecreasing size of chaotic degrees of freedom with increased forcing which wecompare with spatial features of the patterns.

Extensive Chaos in the Lorenz-96 Model

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