A behavior of generalized solutions of the Dirichlet problem by Borsuk M.V. PDF

By Borsuk M.V.

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Extra info for A behavior of generalized solutions of the Dirichlet problem for quasilienar elliptic divergence equations of second order near a conical point

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In Fig. 10(a), only one period-1 attractor of rule 200 is shown (labeled as point 1 ). The domain of the time-1 map ρ1 [200] in this trivial case consists of only the single point { 1 }, and all iterates 383 map trivially onto the fixed point 1 . One can interpret point 1 as the point where a planet intersects an imaginary Poincare cross-section once every revolution. Figure 10(b) shows a period-2 attractor (out of many others) of local rule 51 . The orbit of the circulating planet intersects the Poincare crosssection at two points.

162 and χ362 The graphs of the “time-1” characteristic function χ162 and “time-3 ” characteristic function χ362 of 62 are shown in Figs. 7(a) and 7(b), respectively. Observe that while there are no period-1 fixed points in χ162 , there are many vertical lines which landed on the main diagonal of χ362 . This implies that 62 has many period-3 attractors. Such local rules will be studied in Sec. 3. χ1240 The graph of the characteristic function χ1240 of 240 is shown in Fig. 4(b). 2 The “double-valued” appearance is only illusory because all red vertical lines terminate on the lower straight lines of slope = 1/2, and all 3.

A period-1 garden of Eden is therefore a truly unique specie worthy of its own name, henceforth dubbed an isle of Eden. Indeed, we can generalize this unique phenomenon, which does not exist in continuous dynamical systems (such as ODE), to define a “period-k ” isle of Eden from the kth iterated characteristic function χkN of N . A gallery of period-k isles of Eden of all one-dimensional cellular automata will be presented in Part V of this tutorial series. 1. Mapping CA attractors onto time-τ maps Since invariant orbits are not attractors, they are not robust in the sense that precisely specified initial states must be used to observe them.

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A behavior of generalized solutions of the Dirichlet problem for quasilienar elliptic divergence equations of second order near a conical point by Borsuk M.V.


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