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báo cáo hóa học:" Research Article Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 542073 13 pages doi 10.1155 2010 542073 Research Article Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control Chengmin Hou 1 Lili Han 1 and Sui Sun Cheng2 1 Department of Mathematics Yanbian University Yanji 133002 China 2 Department of Mathematics Tsing Hua University Hsinchu 30043 Taiwan China Correspondence should be addressed to Chengmin Hou houchengmin@yahoo.com.cn Received 2 June 2010 Revised 8 September 2010 Accepted 14 November 2010 Academic Editor Ondrej Dosly Copyright 2010 Chengmin Hou et al. 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. We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold 1. Treating the threshold as a bifurcation parameter that varies between 0 and TO we work out a complete asymptotic and bifurcation analysis. Among other things we show that all solutions either tend to a limit 1-cycle or to a limit 2-cycle and we find the exact regions of attraction for these cycles depending on the size of the threshold. In particular we show that when the threshold is either small or large there is only one corresponding limit 1-cycle which is globally attractive. It is hoped that the results obtained here will be useful in understanding interacting network models involving piecewise constant control functions. 1. Introduction Let N 0 1 2 . . In 1 Ge et al. obtained a complete asymptotic and bifurcation analysis of the following difference equation xn axn-2 bfx xn-i n e N 1.1 where a e 0 1 b e 0 to and fx R R is a nonlinear signal filtering control function of the form fxx 1 0 x e 0 1 x e -TO 0 u 1 to 1.2 in which the positive number 1 can be regarded as a threshold