LOCAL STABILITY OF CONTINUOUS DYNAMICAL SYSTEMS IN PRESENCE OF NON-HYPERBOLIC EQUILIBRIA F. BALIBREA Department of Mathematics, University of Murcia, 30100 Murcia, Spain E-mail: balibrea@um.es A. MARTINEZ, J. C. VALVERDE Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain E-mail: Jose.Valverde@uclm.es April 14, 2008 In this work we analyze what happens when the generalized conditions given in [Balibrea et al., 2008], which produce the appearance of local bifurcations of continuous dy- namical systems, fail. As a result, we are able to find out some situations of local stability in presence of non-hyperbolic equilibria. 1. Introduction Consider a uniparametric family of continuous-time dynamical systems ˙ x = f (x,μ) (1) where μ ∈ R is the parameter and x ∈ R n is the variable in the state space under consideration. An experimental phenomenon which depends on a parameter could be represented by this family (see [Cugno & Montrucchio, 1984]). Thus, we are interested in knowing if systems which are related in terms of the parameter, present similar dynamics. Roughly speaking, if there is a qualitative change when crossing a particular value of the pa- rameter μ 0 ∈ R, then μ 0 is said to be a bifurcation value or it is said that a bifurcation occurs at μ 0 . More rigorously, we can state the following defini- tion. Definition 1.1. A dynamical system {T, R n ,ϕ} is called locally topologically equivalent near an equi- librium x 0 ∈ R n to another dynamical system {T, R n ,ψ} near an equilibrium y 0 ∈ R n if there ex- ists a homeomorphism h : R n → R n such that i) h is defined in a small neighborhood U ⊂ R n ; ii) h(x 0 )= y 0 ; iii) h maps orbits of the first system in U onto orbits of the second system in h(U ) ⊂ R n , preserving the direction of time. This definition let us assure that, in the neigh- borhoods U and h(U ) the number and stability of the equilibria is the same. In other case, we have (locally) topologically non-equivalent systems Definition 1.2. The appearance of a topologically non-equivalent system under variation of the pa- rameter is called a bifurcation. 1