Jorge Sotomayor

Bifurcation analysis of the Watt governor system

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 Pontryagins book Ordinary Differential Equations, [13]. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients 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.

Bifurcations of cuspidal loops

A cuspidal loop for a planar vector field X consists of a homoclinic orbit through a singular point p, at which X has a nilpotent cusp. This is the simplest non-elementary singular cycle (or graphic) in the sense that its singularities are not elementary (i.e. hyperbolic or semihyperbolic). Cuspidal loops appear persistently in three-parameter families of planar vector fields. The bifurcation diagrams of unfoldings of cuspidal loops are studied here under mild genericity hypotheses: the singular point p is of Bogdanov - Takens type and the derivative of the first return map along the orbit is different from 1. An analytic and geometric method based on the blowing up for unfoldings is proposed here to justify the two essentially different models for generic bifurcation diagrams presented in this work. This method can be applied for the study of a large class of complex multiparametric bifurcation problems involving non-elementary singularities, of which the cuspidal loop is the simplest representative. The proofs are complete in a large part of parameter space and can be extended to the complete parameter space modulo a conjecture on the time function of certain quadratic planar vector fields. In one of the cases we can prove that the generic cuspidal loop bifurcates into four limit cycles that are close to it in the Hausdorff sense.

Structural Stability of Asymptotic Lines on Surfaces Immersed in R3

In this paper are studied immersions of surfaces into to R3 whose nets of asymptotic lines are topologically undisturbed under small perturbations of the immersion. These immersions are called structurally asymptotic stable. Sufficient conditions to belong to this class are established here. These conditions focus on the stable patterns around parabolic points, parabolic separatrix connections, periodic asymptotic lines (including those that intercept the parabolic lines) as well the exclusion of recurrent asymptotic lines. The class of immersions that are structurally stable in this sense is open in the C5-topology.

Global asymptotic stability of differential equations in the plane

The problem of determining the basin of attraction of equilibrium points is of paramount importance for applications of stability theory. Local conditions which guarantee the existence of small basins of attraction, such as tr L 0, where L is the linear part of the planar system at an equilibrium point, are well known. This paper is concerned with sufficient conditions which guarantee that the basin of attraction of an equilibrium point of a Q?’ planar system of differential equations x’ =f(x) is the whole x-space R*. In this context, the fundamental problem, yet unsolved, is the following: Consider an autonomous system of differential equations x’=f(x) (’ = d/dt),

Structurally Stable Discontinuous Vector Fields in the Plane

LetM be a plane domain, partitioned into sub domainsN andS, with common borderD. InN andS are defined vector fieldsX andY, respectively, leading to a discontinuous vector fieldZ=(X, Y). This work pursues the stability and transition analysis of solutions betweenN andS, started by Filippov and Kozlova and reformulated by Sotomayor and Teixeira in terms of the regularization method. This method consists in defining a one parameter family of continuous vector fieldsZε, by averagingX andY. This family approachesZ when the parameter goes to zero. The results of Sotomayor and Teixeira providing conditions for the regularized vector fields to be structurally stable are extended and shown to be generic.

Lines of axial curvature on surfaces immersed in R4

Abstract The main notions concerning umbilics and lines of principal curvature, traditionally studied on surfaces in R 3 , are extended in this paper to surfaces immersed in R 4 . Departing from the classical second fundamental form and the ellipse of curvature of the immersion of a surface into R 4 here are studied in detail the concepts of a) axiumbilic points, analogous to classical umbilics since at them the ellipse of curvature has equal axes, and b) periodic lines of axial curvature, called here axial cycles, corresponding both to principal and mean curvature cycles in the classical R 3 case. To any immersed surface in R 4 its axial configurations: the principal configuration and the mean curvature configuration are associated. For surfaces in R 3 , the first one reduces to the configuration by umbilics and principal lines, while the second one gives the configuration by umbilics and integral foliations of the mean curvature line fields. Also the notion of principal structural stability of immersions of surfaces into R 3 is extended to that of axial structural stability, for the case of surfaces in R 4 . Sufficient conditions for the axial structural stability are provided in terms of axiumbilics, axial cycles and the asymptotic behavior of all the other lines of axial curvature

Structurally stable configurations of lines of mean curvature and umbilic points on surfaces immersed in R3

In this paper we study the pairs of orthogonal foliations on oriented surfaces immersed in R3 whose singularities and leaves are, respectively, the umbilic points and the lines of normal mean curvature of the immersion. Along these lines the immersions bend in R3 according to their normal mean curvature. By analogy with the closely related Principal Curvature Configurations studied in [S-G], [GS2], whose lines produce the extremal normal curvature for the immersion, the pair of foliations by lines of normal mean curvature and umbilics, assembled together, are called Mean Curvature Configurations. This paper studies the stable and generic cases of umbilic points and mean curvature cycles, with their Poincare map. This provides two of the essential local ingredients to establish sufficient conditions for mean curvature structural stability, the analog of principal curvature structural stability, [S-G], [GS2].

Limit cycles of vector fields of the form X(v) = Av + f(v) Bv

where A and B are 2 x 2 matrices, det A # 0 and f: R* + [w is a smooth real function such that its expression in polar coordinates is f(r cos 8, r sin 0) = r”‘(e) with D > 1 (note that if f is a homogeneous function then f(0) =f(cos 8, sin 0)). In this case we shall say that / is a homogeneous function of degree D. If f is such that f(Lx, Ay) = LDf(x, y) we shall say that f is homogeneous in the usual sense. This class of vector fields have been studied by C. Chicone [l] as an important extension of a less general class of quadratic vector fields considered by D. E. Koditschek and K. S. Narendra [3,4]. There are two hypotheses Hi (i = 1,2), one for the matrices A and B, the other for the function j For a 2 x 2 matrix C let C’ denote the transpose of C. Then, the symmetric part of C is given by (C), = &C + C’). If J is the sympletic 2 x 2 matrix (O -l , o ), then the hypothesis H, states that (JB), and (B’JA), are definite and have the same sign. Note that if these two matrices associated to X are definite with opposite sign, then the system -X satisfies hypothesis Hi. 90 0022-0396187

BIFURCATIONS OF UMBILIC POINTS AND RELATED PRINCIPAL CYCLES

3.00

Lines of mean curvature on surfaces immersed inR3

The simplest patterns of qualitative changes on the configurations of lines of principal curvature around umbilic points on surfaces whose immersions into ℝ3 depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper.Global effects, due to umbilic bifurcations, on these configurations such as the appearance and annihilation of periodic principal lines, called also principal cycles, are also studied here.