Multiple–Attractor Bifurcations and Quasiperiodicity in Nonsmooth Systems Zhanybai T. Zhusubaliyev, Erik Mosekilde and Soumitro Banerjee Piecewise-smooth maps typically arise as discrete-time models of dynamical systems when the continuous evolution in time is punctuated by impacts or discrete switching events that alter the form of the constitutive equations. Examples of such systems include power electronic converters and switch- ing circuits [1], [2], [3], mechanical systems with impacts and friction [4] as well as models of certain physiological [5] and economic systems [6]. As a parameter is varied, the fixed point for the Poincar´ e map of such a system may move in phase space and collide with the border between two smooth regions. When this happens, the eigenvalues of Jacobian matrix can change abruptly, leading to a special class of nonlinear dynamic phenomena known as border- collision bifurcations [7], [8], [9]. In a couple of recent publications, Kapitaniak and Maistrenko [10] and Duta et al. [11] showed that piecewise- smooth systems can exhibit a special type of border-collision bifurcation in which several attractors are created simul- taneously. A main feature of this type of bifurcation is that, close to the bifurcation point, the distance between the basins of attraction may be arbitrarily small. In the presence of noise, no matter how small, this leads to a fundamentally unpredictable behavior of the system when a system parameter is slowly varied through the bifurcation point. Many physical systems, including switching circuits and impact oscillators, are known to display quasiperiodicity and other forms of multimode dynamics [4], [3], [12], [13]. In our recent work [14], [15], [16] we have shown that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with a quasiperiodic or a periodic orbit. This phenomenon resembles the well-known Neimark-Sacker bifurcation in several respects. However, rather than through a continuous crossing of a pair of complex-conjugate multipliers of the periodic orbit through the unit circle, the border-collision bifurcation involves a jump of the multipliers from the inside to the outside of The work was supported by the Russian Foundation for Basic Research (grant 06-01-00811a) and by the Danish Natural Science Foundation through the Center for Modelling, Nonlinear Dynamics, and Irreversible Thermody- namics (MIDIT). Zh. T. Zhusubaliyev is with the Department of Computer Science, Kursk State Technical University, 50 Years of October Str., 94, Kursk 305040, Russia zhanybai@mail.kursk.ru. E.Mosekilde is with the Complex Systems Group, Department of Physics, The Technical University of Denmark, 2800 Lyngby, Denmark Erik.Mosekilde@fysik.dtu.dk. S. Banerjee is with the Department of Electrical Engineering and the Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur- 721302, India soumitro.banerjee@gmail.com. this circle. This leads to the questions: Can piecewise-smooth systems exhibit border-collision bifurcations in which a stable peri- odic orbit arises together with an attracting closed invariant curve? And, if this is the case what is the mechanism for the particular type of multiple-attractor bifurcation? In order to address these questions, we first consider the normal form map that represents the behavior of the piecewise-smooth systems in a close neighborhood of the border. f :  x y  →  f - (x, y), x ≤ 0; f + (x, y), x> 0, (1) where f - (x, y)=  τ - x + y + μ −δ - x  ; f + (x, y)=  τ + x + y + μ −δ + x  , (x, y) ∈ R 2 . In this representation the phase plane is divided into two regions, D - =  (x, y): x ≤ 0,y ∈ R  and D + =  (x, y): x> 0,y ∈ R  . τ - and δ - denote, respectively, the trace and the determinant of the Jacobian matrix in the half- plane D - , and τ + and δ + are the trace and determinant of the Jacobian matrix in D + . The theory of border-collision bifurcations developed so far assumes contractive dynamics on both sides of the discontinuity (i.e., |δ - | < 1 and |δ + | < 1)) [17], [18]. We consider a situation where an attracting fixed point changes into a spirally repelling fixed point as it moves across the border. This is ensured by assuming δ - < 1, δ + > 1, with −(1+δ - ) <τ - < 1+δ - and −2 √ δ + <τ + < 2 √ δ + (2) If μ< 0 then the map (1) has a single nontrivial stable fixed point with a negative x-coordinate. When μ changes sign, the x-coordinate of the fixed point also changes sign and the fixed point abruptly loses stability when a pair of complex-conjugate eigenvalues of the Jacobian matrix jump from the inside to the outside of the unit circle, i.e. the stable focus transforms into an unstable focus. This transition leads to the birth of a stable invariant curve, associated with quasiperiodic or phase-locked dynamics. However, more complicated bifurcation phenomena are also possible in such transitions. To study these phenomena, we have calculated the chart of dynamical modes in the pa- rameter plane (τ - ,τ + ) for positive values of μ> 0 (Fig. 1). As shown in Fig. 1, this chart is characterized by a dense set