Numerical Analysis of the Flip Bifurcation of Maps* zyxwvutsrqponmlkjihgfed Yu. A. Kuznetsov zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB Research Computing Centre USSR Academy of Sciences Pushchino, Moscow Region, 142292, USSR and S. Rinaldi Dipartimento di Elettronica Politecnico di M ilan0 20133 Milano, ltaly zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A BSTRA CT A numerical procedure for the analysis of the flip bifurcation of maps in R” is described. The procedure is based on normal forms and the center-manifold approach and can be applied to study period doubling of limit cycles in autonomous systems as well as period doubling of periodic solutions of time-periodic systems. The procedure has been programmed in FORTRAN-77, and the code is available on request from the second author. 1. INTRODUCTION Consider a discrete-time dynamical system dt +1> =fa(dt)) defined by a smooth map depending upon a parameter, fi =f,(x> = A,x + g,(x), (1) *This work has been supported by the Italian Ministry of Scientific Research and Technology, contract MURST 40% Teoria dei sistemi e de1 controllo. APPLIED MATHEMATlCS AND COMPUTATION 43:231-236 (1991) 0 Elsevier Science Publishing Co., Inc., 1991 231 655 Avenue of the Americas, New York, NY 10010 0096-3003/91/$03.50