Journal of Sound and < ibration (2000) 236(4), 725}729 doi:10.1006/jsvi.2000.2956, available online at http://www.idealibrary.com on LETTERS TO THE EDITOR LARGE-AMPLITUDE LIMIT CYCLES VIA A HOMOCLINIC BIFURCATION MECHANISM ANUPAM SHARMA AND N. ANANTHKRISHNAN Department of Aerospace Engineering, Indian Institute of ¹ echnology, Bombay, Powai, Mumbai 400076, India. E-mail: akn@aero.iitb.ernet.in (Received 14 July 1999, and in ,nal form 1 February 2000) 1. INTRODUCTION Unforced or self-excited periodic oscillations in non-linear dynamical systems are called limit cycles. Limit cycles usually arise at a Hopf bifurcation in non-linear systems with a varying parameter. In mechanical systems, the varying parameter is frequently a damping coe$cient. Examples of limit cycles in mechanical systems are #utter of aircraft wings, surge oscillations in axial #ow compressors, and wing rock in aircraft #ight dynamics. Regular or normal limit cycles were distinguished from large-amplitude limit cycles by Ananthkrishnan and Sudhakar [1]. Stable normal limit cycles are created at a supercritical Hopf bifurcation with the limit cycle amplitude building up gradually from nought as the parameter is varied from the Hopf bifurcation point. In contrast, stable large-amplitude limit cycles are either created with a "nite amplitude or show a sudden increase in amplitude after originating as a normal limit cycle at a Hopf bifurcation point. Stable large-amplitude limit cycles were characterized in terms of secondary bifurcations by Ananthkrishnan et al. [2]. The phenomena of "nite amplitude onset, and jump in amplitude were described in terms of secondary fold bifurcations. The presence of large- amplitude limit cycles was seen to be accompanied by hysteresis in the system response with varying parameter. Reference [2] also constructed low order models based on a non-linear damping mechanism that reproduced the essential dynamics associated with the primary Hopf-secondary fold bifurcation pairs characterizing large-amplitude limit cycles. These models were found to provide a suitable representation of the large-amplitude surge limit cycles in axial #ow compressors. However, the models in reference [2] based on a non- linear damping mechanism could not explain the large-amplitude wing rock limit cycles that had been characterized in terms of the same primary Hopf-secondary fold bifurcation pair [1]. Ananthkrishnan et al. [3], therefore, came up with another model based on an alternate mechanism involving a pair of resonantly coupled oscillators. The models in references [2, 3] represented large-amplitude limit cycles created at secondary fold bifurcations following a primary Hopf bifurcation. However, there could be mechanical systems where large-amplitude limit cycles are characterized by a di!erent combination of bifurcations. One such problem arises in the passage through resonance of  Currently at the Department of Aerospace Engineering, Pennsylvania State University, University Park, PA 16802, U.S.A. 0022-460X/00/390725#05 $35.00/0  2000 Academic Press