Abstract. This paper pursues the study carried out by the authors in Stability and Hopf bifurcation in the Watt governor system [14], focusing on the codimension one Hopf bifurcations in the centrifugal Watt governor differential system, as presented in Pontryagin's book Ordinary Differential Equations, [13]. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coeffi cients and bifurcation diagrams, illustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found a region in the space of parameters where an attracting periodic orbit coexists with an attracting equilibrium.Mathematical subject classification: 70K50, 70K20.

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In this paper, we study the existence of limit cycles for piecewise linear differential systems with two zones in the plane. More precisely, we prove the existence of piecewise linear differential systems with two zones in the plane with four, five, six and seven limit cycles. From our results we conjecture the existence of piecewise linear differential systems with two zones in the plane having exactly n limit cycles for all n ∈ ℕ.

The notion of principal configuration of immersions of surfaces into R 3 , due to Sotomayor and Gutierrez [16] for lines of curvature and umbilics, is extended to that of mean directional configuration for immersed surfaces in R 4. This configuration consists on the families of mean directionally curved lines, along which the second fundamental form points in the direction of the mean curvature vector, and their singularities, called here H-singularities. The concepts of H-singularities and periodic mean directionally curved lines are studied here in detail. Also the notion of principal structural stability of immersions of surfaces into R 3 is extended to that of mean directional structural stability, for the case of surfaces in R 4. Sufficient conditions for immersions to be mean directional structurally stable are provided in terms of H-singularities, periodic mean directionally curved lines and the asymptotic behavior of all the other mean directionally curved lines.