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Continuation methods for principal foliations of embedded surfaces

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  • Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in $ \mathbb{R}^3 $ resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.

    Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in "Monge coordinates." Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  Splitting of an umbilic connection in $ S_\lambda $ when $ \lambda = 0.001 $. The separatrices of umbilics $ u_l $ and $ u_r $ are drawn green and blue. The line of curvature heading right from point $ w_0 $ is drawn black, and its first five crossings of cross-section $ \Sigma $ are labelled $ w_1,w_2,w_3,w_4,w_5 $. This line of curvature stays to the right of the green separatrix of $ u_l $ between $ w_0 $ and $ w_1 $, then passes left of $ u_l $ and right of the blue separatrix of $ u_r $ into the channel between the two separatrices, intersecting $ \Sigma $ from left to right at $ w_2 $. From $ w_2 $ it turns around $ u_r $ to the left of the separatrix of $ u_l $, next crossing $ \Sigma $ from right to left at $ w_3 $. From $ w_3 $ it turns around $ u_l $ with the separatrix of $ u_r $ on its right and the next turn of the separatrix of $ u_l $ on its left. It then crosses $ \Sigma $ from left to right at $ w_4 $ and finally crosses $ \Sigma $ again from right to left at $ w_5 $. Note that while the line of curvature originally headed right from $ \Sigma $ at $ w_0 $, it crosses $ \Sigma $ in the opposite direction at $ w_5 $

    Figure 2.  Graph of the return map $ \sigma $ and its fifth iterate for the surface $ S_{0.1857} $. The cross-section $ \Sigma $ is parameterized as $ [0,\sqrt{3}\cos(\theta),\sqrt{5}\sin(\theta)],\, \theta \in [-\pi,\pi] $. Values of the maps are displayed on a mesh of $ 1000 $ points

    Figure 3.  Graph of $ \sigma^5(\theta) - \theta $. Note that $ \sigma^5 $ has fixed points and breaks

    Figure 4.  The derivative of $ \sigma $, showing four jump discontinuities

    Figure 5.  Two closed lines of curvature on the surface $ S_{0.1857} $. The umbilics are drawn as red dots

    Figure 6.  Values of $ (\theta,\lambda) $ at which the return maps $ \sigma_{\lambda} $ of the surfaces $ S_{\lambda} $ have periodic orbits of period 5

    Figure 7.  The geometry of closed lines of curvature near an umbilic connection. The blue curve is an umbilic connection computed at $ \lambda = 0.185952861855609 $. The green curve $ \gamma $ is a closed line of curvature on $ S_{0.1857} $. Its intersections $ s_j $ with cross-section $ x = 0 $ (light black dotted curve) are marked by black dots. Umbilics are red dots. The point $ s_0 $ is the initial point of $ \gamma $, and its period $ 5 $ trajectory for the return map $ \sigma $ is $ s_{0},s_{2},s_{4},s_{6},s_{8},s_{10} = s_{0} $. Note that between $ s_2 $ and $ s_3 $, the line of curvature comes close to the umbilic $ u_1 $, following its separatrix in one direction before the turn and in the other direction after the turn. The $ \sigma $ trajectory of $ s_5 $ is $ s_{5},s_{7},s_{9},s_{1},s_{3} $, lying on the same line of curvature but following it with the opposite orientation

    Figure 8.  Values of $ (\theta,\lambda) $ at which the return maps $ \sigma_{\lambda} $ of the surfaces $ S_{\lambda} $ have periodic orbits of period 5 for $ \theta \in [-2.75,-2.65] $

    Figure 9.  The surface $ S_{0.24} $ defined by Equation (3) with $ b = 0.24 $. The surface is plotted in a tan color. The six umbilic points are red dots and segments of their separatrices up to intersections with the cross-section $ x = 2y $ are drawn as heavy blue curves. The cross-section is a thin black curve. A long segment of a single line of curvature is plotted as a brown curve. Note that the separatrix of the lowest umbilic in the figure passes close enough to another lemon umbilic that the gap between them cannot be seen at this scale

    Figure 10.  The $ z $ coordinate of umbilic points of the surfaces $ S_b $ plotted as a function of $ b $. Turning points on the curves occur at parameters where star and monstar umbilics merge and disappear, analogous to a saddle-node bifurcation of vector fields. Note that there are ranges of parameters with four, six and eight umbilic points

    Figure 11.  The $ z $ coordinate of umbilic points of the surfaces $ S_b $ plotted as a function of $ b $. Turning points on the curves, marked with blue crosses, occur at parameters where star and monstar umbilics merge and disappear. Large green dots mark monstar-lemon transitions

    Figure 12.  The return map $ \sigma $ of the surface $ S_{0.24} $. The cross-section $ \Sigma $ is a closed curve parameterized as $ (2r(\theta)\cos(\theta),r(\theta)\cos(\theta),r(\theta)\sin(\theta)) $ where $ r(\theta) $ is computed numerically so that $ \Sigma $ lies on $ S_{0.24} $. Lines of curvature starting at 1000 points on $ \Sigma $ are computed to their second return with $ \Sigma $. There are three discontinuities at intersections of the star umbilic separatrices with $ \Sigma $. Between each pair of discontinuities, $ \sigma $ is increasing. It has breaks (jumps in its first derivative) at intersections of the separatrices of lemon umbilics with $ \Sigma $. Apart from its discontinuities, $ \sigma $ is bijective

    Figure 13.  Successive returns of a long line of curvature on the surface $ S_{0.24} $. Over $ 42,000 $ returns are plotted as blue dots. If the line of curvature is dense in $ S_{0.24} $, its succesive returns would fill the graph of the return map $ \sigma $ densely. Here, quite large gaps remain despite the large number of returns. Nonetheless, the line of curvature does not seem to approach one that is closed. If that were the case, the successive returns would approach a discrete set of points in the intersection of the line of curvature and the cross-section. Here, a plot of only the last 1000 successive returns of this line of curvature produces a graph visually indistinguishable from this one

    Figure 14.  Principal foliation of $ S_{0.2233} $. Umbilic separatrices are heavy blue curves

    Figure 15.  Region near the star and monstar of $ S_{0.2233} $

    Figure 16.  The return map $ \sigma $ of $ S_{0.2233} $ to the cross-section $ x = 2y $. In comparison to the return map of $ S_{0.24} $ shown in figure 12, there are gaps in the domain and {image of $ \sigma $.} These come from lines of curvature that tend to the monstar umbilic and do not return to the cross-section

    Figure 17.  A (near) umbilic connection (blue) between a star umbilic (large red dot) and a lemon umbilic for $ S_{0.2447609874} $. The second and third separatrices tend to the same closed line of curvature, plotted green

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