By K. Tomita (auth.), Professor Dr. Yoshiki Kuramoto (eds.)
Read Online or Download Chaos and Statistical Methods: Proceedings of the Sixth Kyoto Summer Institute, Kyoto, Japan September 12–15, 1983 PDF
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Additional info for Chaos and Statistical Methods: Proceedings of the Sixth Kyoto Summer Institute, Kyoto, Japan September 12–15, 1983
Recently, KOHMOTO et al. (2) and OSTLUND et al. (3) showed that the transfer matrices of a special almost periodic Schrodinger equation obey the following recursion relation: (1) where M~ is a 2 x 2 real matrix with unit determinant. The purpose of this article is to review the derivation of this dynamical system and also to give a more general class of the equations than reported previously. 2. ,V(t+l) 52 z V(t), and w is an irrational number. It is traditional to introduce a transfer matrix in one-dimensional problems: ~n+l where ~n = M(nw}~n' = [ Wn ] and (3) M is the transfer matrix given by M(t} Wn-l The matrix satisfies M(t+l} matrices are also defined, = M(t} and detM(t} 1.
It is clear that as A-I, D - 2, hence ~ -1, and that chaos is approached without bifurcation. The facts are perhaps easier to visualize in terms of the parameter Po = 1/A and the variable u = 1/z. This change of variable does not change IF*. For I Pol <1, there is one limit point at u = O. As Po crosses 1, this limit point bifurcates into two limit points that coexist in a chaotic situation. 7. Third Path Beyond the Great Wave. The Siegel Scenario Figure 5 represents the IF* set for a value of A within the atom I A-I /2 I < 1/2, but very close to a point on boundary, namely AS = 1/2+(1/2) exp(2'ITiy), where y is the irrational number «1) whose continued fraction expansion is (4,1,1,1, ...
In both the Henon map and the rotor map (R) there is seen a special saddle periodic orbit whose collision with the chaotic attractor, as the parameter on the nonlinear term is increased, marks the destruction of the attractor. The stable manifold of this orbit delineates the boundary of the chaotic 27 Fig. 6 The attractor of map (R) very near destruction. A=1/2. 1986n. The two-piece chaotic attractor is shown relative to the unstable orbit of period' six (arrows). As the map is iterated the point cycles Pl+P2 ...