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Keywords:
reaction-diffusion equation; attractor; invariant measure; entropy; Poincaré-Bendixson theorem
reaction-diffusion equation; attractor; invariant measure; entropy; Poincaré-Bendixson theorem
Summary:
We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms.
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