International Journal of Bifurcation and Chaos, Vol. 12, No. 12 (2002) 2799–2820 c  World Scientific Publishing Company A NOTE ON THE TRIPLE-ZERO LINEAR DEGENERACY: NORMAL FORMS, DYNAMICAL AND BIFURCATION BEHAVIORS OF AN UNFOLDING E. FREIRE, E. GAMERO and A. J. RODR ´ IGUEZ-LUIS Department of Applied Mathematics II, E. S. I., University Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain A. ALGABA Department of Mathematics, E. P. S., University Huelva, Crta. Palos-Huelva s/n, 21819 La R´ abida, Huelva, Spain Received August 17, 2001; Revised November 1, 2001 This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension- two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,...) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the R¨ ossler equation. Keywords : Local bifurcations; triple-zero degeneracy; normal forms; global connections. 1. Introduction The analysis of bifurcations in codimension-one and -two linear degeneracies of equilibria of au- tonomous systems can be found in several text- books on nonlinear dynamics (see [Guckenheimer & Holmes, 1983; Wiggins, 1990; Kuznetsov, 1995]). These cases correspond to a simple-zero eigenvalue (saddle-node bifurcation), a pair of pure imagi- nary eigenvalues (Hopf bifurcation), a double-zero eigenvalue with geometric multiplicity one (Takens– Bogdanov bifurcation), a zero and a pair of pure imaginary eigenvalues (Hopf-zero bifurcation), and two pairs of pure imaginary eigenvalues rationally independent (Hopf–Hopf interaction). Our goal here is to analyze the codimension-three case corre- sponding to a triple-zero eigenvalue with geometric multiplicity one. This case will be referred in this paper as triple-zero bifurcation. On this subject, only partial results have been obtained. Namely, [Medved, 1984] showed the ap- pearance of saddle-node, Hopf, Takens–Bogdanov and Hopf-zero linear degeneracies in the triple-zero bifurcation, but without characterizing them. Yu and Huseyin [1988] analyzed the Hopf bifurcation (their study is also useful to determine the character of the quasiperiodic motions born in the Hopf-zero bifurcation) and applied the results to the analysis of an electronic circuit. More recently, [Dumortier & Ib´ a˜ nez, 1996, 1998] have classified the triple-zero singularity up to codimension four. Ib´ a˜ nez and Rodr´ ıguez [1995] have analyzed a four-parameter unfolding corre- sponding to a codimension-four degenerate case, that arises when a second-order normal form co- efficient vanishes. In this situation, the presence of a saddle-focus homoclinic connection satisfying the Shil’nikov condition is proved. 2799