A new proof for existence of horseshoe in the R€ ossler system q Xiao-Song Yang a,b,c, * , Yongguang Yu b , Suochun Zhang b a Department of Automation, Xiamen University, Xiamen, Fujian 361005, China b Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Science, Beijing 100080, China c Institute for Nonlinear Systems, Chongqing University of Posts and Telecomm, Chongqing 400065, China Accepted 13 December 2002 Abstract A new simple computer-assisted proof based on a newly established topological horseshoe theorem is given for the existence of horseshoe in the well known R€ ossler system. Ó 2003 Elsevier Science Ltd. All rights reserved. 1. Introduction The R€ ossler system _ x ¼y  z; _ y ¼ x þ ay ; _ z ¼ xz  bz þ c; 8 < : ð1Þ where a ¼ 0:2, b ¼ 5:7, c ¼ 0:2, is one of the simplest continuous time systems that exhibit chaos. Since it was proposed [1], a lot of numerical and analytic investigations [2–4] have been carried out to beat out the complex behavior of this system. Recently Zgliczy nski [5] gave a computer-assisted proof of chaos in the R€ ossler system with the same pa- rameters given above. To study the R€ ossler system, Zgliczy nski made use of his new result on topological shifts, which is complicated and hard for non-specialist to use, because the Poincare map he obtained is of rather complex feature to be dealt with by means of elementary horseshoe theory. In this paper, we give a new simple proof based on the second return Poincare map and a topological horseshoe theorem by Kennedy and York [6]. This proof is easy to understand even to non-specialists. The attractor of the R€ ossler system is shown in Fig. 1. To study the complex behavior of (1), we consider the plane C ¼ fðx; y ; zÞ2 R 3 : y ¼ 0g. We choose a Poincare section on this plane as follows: jABCDj¼fx; y ; zÞ2 C : 8 6 x 6  3; 0:014 6 z 6 0:024g; and will study the corresponding Poincare map P : jABCDj! C; ð2Þ q This paper was written when the first author was visiting Academy of Mathematics and Systems Sciences, Chinese Academy of Science. * Corresponding author. Address: Institute for Nonlinear Systems, Chongqing University of Posts and Telecomm, Chongqing 400065, China. 0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00641-0 Chaos, Solitons and Fractals 18 (2003) 223–227 www.elsevier.com/locate/chaos