By S. Greco, R. Strano
Read or Download Complete Intersections. Proc. conf. Acireale (Catania), 1983 PDF
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Extra resources for Complete Intersections. Proc. conf. Acireale (Catania), 1983
0; 1Þ defined by 1 gðxÞ ¼ arcosð1 À 2xÞ p defines a topological equivalence between the logistic map and the dyadic tent map T: ½0; 1Þ ! ½0; 1Þ & 2x; for x 2 ½0; 12Þ TðxÞ ¼ 2ð1 À xÞ for x 2 ½12 ; 1Þ expressed through the formula S ¼ gÀ1 T g The transformation g transforms the Lebesgue measure, which is the invariant measure of the tent map to the invariant measure of the logistic map. The transformation G intertwines the Koopman operator V of the logistic map with the Koopman operator VT of the tent map: V ¼ GVT GÀ1 The intertwining transformations G and GÀ1 , when suitably extended, map the eigenvectors of VT onto the eigenvectors of V.
Time Evolution of Probability Density Now we consider the time evolution of the symmetric probability density, rn ðp; xÞ, in the space of the scaled variables, rn ðp; xÞ ¼ Umap Uop ðLnÀ1 Þ Á Á Á Umap Uop ðL0 Þr0 ðp; xÞ ð10Þ where Umap is the Frobenius–Perron operator of the generalized standard map, Umap rðp; xÞ ¼ rðp À f ðx À pÞ; x À pÞ ð11Þ thermodynamics of a simple hamiltonian chaotic system 27 and Uop ðLn Þ governs the time evolution of the probability density during the nth isothermal operation.
The following interesting conclusion thus follows from this discussion: Over a long range of intermediate times (1000 < t < 10000) the asymptotic Markovian density profile is non-Gaussian. The ‘‘pure’’ diffusive regime (36) only sets in after a time much longer than the relaxation time. VI. CONCLUSIONS In this work we presented a first step toward the construction of a kinetic theory of chaotic dynamical systems described by iterative maps. , Refs. 7 and 10). A closed equation for the density profile non-markovian effects in the standard map 19 due to Bandtlow and Coveney  is taken as a starting point.