357 1040-7294/03/0700-0357/0 © 2004 Plenum Publishing Corporation Journal of Dynamics and Differential Equations, Vol. 15, Nos. 2/3, July 2003 (© 2004) Singular Perturbations, Transversality, and Sil’nikov Saddle-Focus Homoclinic Orbits* * Dedicated to Victor A. Pliss on the occasion of his 70th birthday. Flaviano Battelli 1 and Kenneth J. Palmer 2,3 1 Dipartimento di Scienze Matematiche, Facoltà di Ingegneria, Università Politecnica delle Marche, Via Brecce Bianche 1, 60100 Ancona, Italy. 2 Department of Mathematics, National Taiwan University, Taipei 106, Taiwan. 3 To whom correspondence should be addressed. E-mail: palmer@math.ntu.edu.tw Received April 2, 2003 We consider the singularly perturbed system x ˙=ef(x, y, e, l), y ˙ =g(x, y, e, l). We assume that for small (e, l), (0, 0) is a hyperbolic equilibrium on the nor- mally hyperbolic centre manifold y=0 and that y 0 (t) is a homoclinic solution of y ˙ =g(0, y, 0, 0). Under an additional condition, we show that there is a curve in the (e, l) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4 dimensional systems with Sil’nikov saddle-focus homoclinic orbits. KEY WORDS: Singular perturbations; transversality; homoclinic; centre mani- fold. 1. INTRODUCTION In this paper we consider a singularly perturbed system ˛ x ˙=ef(x, y, e, l) y ˙ =g(x, y, e, l) (1) where x ¥ R m , y ¥ R n , l ¥ R, and e ¥ R are small parameters and f(x, y, e, l), g(x, y, e, l) are C r -functions (r \ 2) in their arguments bounded with their derivatives. We suppose that the following conditions are satisfied: