Ein gesuchter, dennoch bislang unbekannter elementarer Satz - Mathematics > Dynamical SystemsReport as inadecuate




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Abstract: If and only if each point of a set of the phase-space is in the topologicalhull of a trajectory running through any other point of this set, we call thisset a quasiergodic set. But which are these so defined quasiergodic sets in thecase of a given continuous dynamical system, which has piecewise differentiabletrajectories in a finit-dimensional real phase-space, which is compact? Let itstrajectories define a field of normed tangents, which is continuous in almosteach point of the phase-space: Then the topological hulls of all trajectoriesof the phase-space form a partition of it. Thus the elements of this partitionare the quasiergodic sets of the given continuous dynamical system. This is theimportant but rather trivial statement of the elementary theorem 1.1, which wepresent in this tractatus. We call this theorem elementary, because it islimited to finit-dimensional real phase-spaces. We also find, that there is alinear homogeneous partial differential equation of first order, describing theinvariants of the system, which allow us the construction of the quasiergodicsets. Furthermore we show, that any quasiergodic set is a sensitive attractor,if it is neither a closed trajectory or a fixed point.



Author: Andreas Johann Raab

Source: https://arxiv.org/







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