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
Example text
In our problem. the expansion of solution (5) has two complex conjugate exponents. tP'I. the distribution of singularities of (5) can be well approximated by that of (6). On the contrary. Hence, around a singularity with two complex conjugate exponents. there are two convergent semi-spirals; one is clockwise and another anti-clockwise. 2 . 3 shows the construction of two fundamental triangles around type (i) and type (i i) singularities for the case c= -1. 3. we find by formula Therefore. 302 triangle ABO and triangle DBC approximately become isosceles triangles with base angle 25' .
A cascade of perioddoubling bifurcations culminates at the limiting parameter value Poo' beyond which bands of chaos result. A vertical cross section taken at parameter value = C woul d show t\~O dark bands corresponding to a two-piece chaotic attractor. At Pd the attractor collides with a periodic point. Beyond Pd the attractor no longer exists. 3. The Henon map has two variables (x and y), but only x is plotted against the parameter on the nonlinear term, P, here. 3. 3 the development of the chaotic attractor is truncated in this regard In both maps there is a single parameter that determines the Jacobian A, and parameter on the nonlinear term (G).
When G';; 371 no proto-horseshoe image is poss i b1e in the regi on of the graph shown in Fig. 3. To form a proto-horseshoe, it is necessary to have both a saddle point and a second periodic point (initially an attractor). By con, D "- "- - , \ B - --~ LIT( An example of a reduced proto-horseshoe (abed) for (R) in relation to its stable manifold (----), unstable manifold (- . - . loTI' 25 struction, the saddle point always lies on the boundary of the optimal rectangle and thus the other fixed point lies in the interior of the rectangle's image.