When a real homotopy is used for solving a polynomial system with real coefficients, bifurcation of some of the homotopy paths at singular points is inevitable. The main result of this paper shows that, generically, the solution set of a real homotopy contains no singular point other than a finite number of quadratic turning points. At a quadratic turning point, the bifurcation phenomenon is qu...

The regularity of the conjugacy between two onedimensional maps with singular points is considered. We prove that the conjugacy between two nice and mixing quasi-hyperbolic one-dimensional maps is a diffeomorphism if it is an absolutely continuous homeomorphism and the exponents and the asymmetries of these two maps at all corresponding singular points are the same. We also discuss the applicat...

This paper proposes a robust method for directional field estimation from fingerprint images that combines estimates at multiple scales. The method is able to provide accurate estimates in scratchy regions, while at the same time maintaining correct estimates around singular points. Compared to other methods, the penalty for detecting false singular points is much smaller, because this does not...

The nonlinear fiber coupler is considered as a Hamiltonian dynamical system with an infinite number of degrees of freedom, with the soliton states of the coupler being the singular points of this dynamical system. Numerical simulations show that arbitrary initial conditions give rise, asymptotically, to oscillations around some of the stable singular points and some amount of radiation. Example...

We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to the one of computing planar arrangements of algebraic curves. We succeed in locating all event p...

We study dynamics of the Duffing–Van der Pol driven oscillator. Periodic steady-state solutions of the corresponding equation are determined within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude on forcing frequency as an implicit function, referred to as resonance curve or amplitude profile. Equations for singular points of resonance curves are solved exactly. We i...

Let f be a polynomial in n+1 complex variables. In this paper, we study the topology of the set V = f−1(0). While V is a differentiable manifold in the neighborhood of simple points, the topology is more complicated at singular points. We use the following construction to study the topology at singular points. Let S be a small sphere around a singular point, and let K = S ∩V . We will prove tha...

In this paper, we use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram–Levine signatures of links, and the associated Murasugi-type inequalities, we reprove (in a slightly weaker form) a result obtained by Steenbrink ...

The focal points of a curve traced by a planar linkage capture essential information about the curve. In a previous paper, we showed how to determine the singular foci of planar linkages from an expression for the tracing curve derived by use of the Dixon determinant. This paper gives an alternative approach to finding the singular foci, one that lends itself to simple geometric interpretations...