TAILIEUCHUNG - Báo cáo sinh học: " Research Article Determining Consecutive Periods of the Lorenz Maps"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Determining Consecutive Periods of the Lorenz Maps | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 985982 15 pages doi 2010 985982 Research Article Determining Consecutive Periods of the Lorenz Maps Fulai Wang School of Mathematics and Statistics Zhejiang University of Finance and Economics Hangzhou 310012 China Correspondence should be addressed to Fulai Wang flyerwon@ Received 18 October 2009 Revised 27 February 2010 Accepted 19 May 2010 Academic Editor Roderick Melnik Copyright 2010 Fulai Wang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Based on symbolic dynamics the paper provides a satisfactory and necessary condition of existence for consecutive periodic orbits of the Lorenz maps. In addition a new algorithm with computer assistance based on symbolic dynamics is proposed to find all periodic orbits up to a certain number with little computer time. Examples for consecutive periods of orbits are raised for the Lorenz maps. With a little variation the theorems and algorithm can be applied to some other dynamic systems. 1. Introduction The Lorenz system of introduced by Lorenz in 1 is one of the chaotic dynamic systems discussed early. It is a deterministic chaos x ơ y - x y rx - y - xz z xy - bz. On the Poincaré section some geometrical structure of the Lorenz flow may be reduced to a one-dimensional Lorenz map 2 3 f x yL yR fL x 1 - yL x . x 0 fR x -1 R x x ỉ . x 0 2 Advances in Difference Equations Figure 1 a Lorenz map b Lorenz map c Lorenz map . where ị is a constant greater than 1. Generally a Lorenz map with a discontinuity point is as follows f x b fi x Jr x x b x b where f is piecewise increasing but undefined at x b the point b limx b f xỴ is a discontinuity point and denoted by C x e I c b U b d and f is a map from c d into c d . .

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