Circle Like Strange Attractor in a Piecewise Smooth Map ⋆ Biswambhar Rakshit * Soumitro Banerjee ** Kazuyuki Aihara *** * Collaborative Research Center For Innovative Mathematical Modelling, Institute of Industrial Science, University of Tokyo (e-mail:hibiswa.iitkgp@gmail.com). ** Indian Institute of Science Education and Research, Mohanpur-741252, Nadia, West Bengal, India(e-mail:soumitro.banerjee@gmail.com ) *** Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan(e-mail:aihara@sat.t.u-tokyo.ac) Abstract: we explore the dynamics of a piecewise linear normal form map under the condition that the map is contractive in one compartment and expansive in the other. In particular, we analyze the transition from a mode-locked periodic orbit to a chaotic orbit. It occurs through the following sequence: first homoclinic contact followed by homoclinic intersection, which is again followed by a second homoclinic contact. We have shown that after the second homoclinic contact, a circular-shaped strange attractor with an infinite number of non-smooth folds is created. The mechanism of this chaotic behavior is explained in terms of tangencies with the stable foliation of the saddle fixed point. Keywords: Piecewise smooth maps, border-collision bifurcation, resonance tongues, quasiperiodicity, torus breakdown, chaos. 1. INTRODUCTION There are many routes for the transition from a periodic orbit to a chaotic orbit, the most well-known one being the period-doubling route and the quasiperiodicity route. In the second case, a two-frequency torus forms through a Neimark-Sacker bifurcation, which is subsequently de- stroyed to give rise to a chaotic orbit. The basic theorem for the two frequency torus destruction was given by Afraimovich and Shilnikov (1991), where three possible routes were described. The first possibility is the period doubling of the phase-locked periodic orbit. The second one is the saddle-node bifurcation in the presence of a homoclinic structure and the last one is the loss of smoothness of the torus owing to the formation of the wrinkles in the unstable manifold. Ostlund et al. (1983), Aronson et al. (1982) and Broer et al. (1998) give an overview of the possible topological transitions for the loss of smoothness of the torus. More recently these routes have been confirmed numerically as well as experimentally for both continuous and discrete-time systems (Kuznetsov (2004); Anishchenko et al. (1992); Maistrenko et al. (2003)). These results were obtained in the context of smooth maps. ⋆ This research is supported by the Aihara Innovative Mathematical Modelling Project, the Japan Society for the Promotion of Science (JSPS) through the ”Funding Program for World-Leading Innovative Research and Development on Science and Technology (FIRST Program),” initiated by the Council for Science and Technology Policy (CSTP). Recently, the creation and destruction of a torus has been described (Zhusubaliyev and Mosekilde (2006, 2008); Maity et al. (2007)) for piecewise smooth systems. In a recent development the occurrence of quasiperiodicity in a piecewise smooth map has been described by Zhusubaliyev et al. (2006) , Simpson and Meiss (2008) and Simpson and Meiss (2009). Torus destruction and the transition to chaos in a piecewise smooth map was investigated in (Zhusubaliyev et al. (2008)), and it was shown that the non-smoothness of the map causes some significant differences with the Afraimovich-Shilnikov mechanisms. In particular they have described two mechanisms of the torus destruction in non-smooth map. They have shown that the essential features like period-doubling of a mode- locked periodic orbit, or the creation of a homoclinic intersection followed by the disappearance of the orbit also occur in non-smooth systems. In this paper we have shown that the third mechanism proposed by Afraimovich and Shilnikov, i.e, wrinkling of the closed invariant curve, also occurs in non-smooth maps, of course with some major differences caused by the non-smoothness. In our present investigation we follow the bifurcations that take place within a 1 : 5 mode locked tongue. We show that the resonance torus is first destroyed through a homoclinic bifurcation and then quadratic tangencies with the stable foliation after the last homoclinc tangency is responsible for the occurrence of a large circle-like strange attractor. 2012 IFAC Conference on Analysis and Control of Chaotic Systems The International Federation of Automatic Control June 20-22, 2012. Cancún, México 978-3-902823-02-1/12/$20.00 © 2012 IFAC 81 10.3182/20120620-3-MX-3012.00023