arXiv research

Example shows not all conjugate points are bifurcation points in semi-Riemannian geodesics.

Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.

Study bifurcation diagrams of crosscap contours.

Study bifurcation in Riemannian metrics and submersions.

We give two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to methods for constructing constant mean curvature surfaces, we conclude that the bifurcation point in the analytic method of Mazzeo-Pacard is the same as a limiting point encountered in the integrable systems met…

Study describes bifurcations of gradient flows on 2-sphere with holes.

Study bifurcations of curves on surfaces in Minkowski 3-space.

We obtain an estimate for the covering dimension of the set of bifurcation points for solutions of nonlinear elliptic boundary value problems from the principal symbol of the linearization of the problem along the trivial branch of solutions.

Study detects P-type bifurcations in single system realizations using unreliable kernel density estimates.

The paper explains how the conjugate locus of a point on a surface can gain or lose cusps as the point moves.

Study describes Morse flows on a torus with up to six singular points.

We give a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functional…

Study how $J^+$ invariants change under bifurcations of curves.

The simplest patterns of qualitative changes on the configurations of lines of principal curvature} around umbilic points on surfaces whose immersions into $\mathbb R^3$ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper. Global effects, due to umbilic bifurcations, o…

Study bifurcations and local rigidity on flag manifolds for Yamabe solutions.

The paper studies bifurcations in discrete dynamical systems on manifolds.

Study the geometry of bifurcation sets for specific types of functions.

We give a classification of generic bifurcations of intersections of wavefronts generated by different points of a hypersurface with or without boundaries.

New solutions found for Ginzburg-Landau equations on complex manifolds.

The paper studies bifurcations in Lagrangian systems and geodesics.

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…

Let (ρ_λ)_{λ\in Λ} be a holomorphic family of representations of a finitely generated group G into PSL(2,C), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λdescribing the bifurcations of this family of representations in a quantit…

We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $β(f)$ which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index $β(f)$ is derived from the index bundle of the linearization of the …

Here are described the axiumbilic points that appear in generic one parameter families of surfaces immersed in R4. At these points the ellipse of curvature of the immersion, Little, Garcia - Sotomayor has equal axes. A review is made on the basic preliminaries on axial curvature lines and the associated axiumbilic poin…

Study on positive solutions of Yamabe-type equation on spheres.

Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic value of the parameter are also studied. A determinant for singularity types and con…

Let Riemannian metrics $g$ and $\bar g$ on a connected manifold $M^n$ have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the famous Levi-Civita's Theorem, the metrics have a certain standard form near the…

Local Neural Operators enable efficient system-level analysis of complex PDEs.

New symmetric Willmore tori emerge from Clifford torus in Berger spheres.

Study on bifurcations in Lagrangian systems and geodesics on manifolds.

Study stability and bifurcation of liquid interfaces in cylindrical supports.

Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing first integral cohomology group. We improve this result in two directions: topologically and analytically. From the analy…

In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spa…

We consider the Reidemeister torsion associated with SL(2, C)-representations of a knot group. A bifurcation point in the SL(2, C)-character variety of a knot group is a character which is given by both an abelian SL(2, C)-representation and a non-abelian one. We show that there exist limits of the non-acyclic Reidemei…

Study topological properties of integrable case on Lie algebra so(4).

Study local topological types of binary differential equations for surface families.

Machine learning detects tipping points in complex systems.

Topological method detects Hopf bifurcations from time series.

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local…

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non homothetic solutions of the Yamabe problem. Using local rigi…

Cohen et al. (2021) show GD trajectories align on a bifurcation diagram.

We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic fixed point in a flow on $\real^3$ originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or ``pleat,'' we may apply the theory of templates, or branched …

Study reveals different types of critical points in shallow neural networks.

The paper extends curve deformation methods in Minkowski plane.

Study controls bifurcations in Eulerian flows with multiple Hopf singularities.

We prove a criterion for stability of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map. This generalizes a theorem of G.W. Patrick. The method of the proof is also useful in studying the bifurcation of relative equilibria.

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describi…

Extremal metrics studied with new bifurcation technique.