Study the geometry of bifurcation sets for specific types of functions.
problem Understanding the structure of bifurcation sets for specific types of functions.
method Using blow-ups and parametrization, investigate the Gaussian curvature, principal curvatures, and curve behavior.
result Bifurcation sets of D4±-functions can be parametrized as surfaces in R3. Study bifurcation diagrams of crosscap contours.
problem Understanding bifurcations in crosscap images.
method Determined precise bifurcation diagrams of crosscap contours.
result Three types of equivalences are crucial for bifurcations.
The paper determines the bifurcation set of a real polynomial function of two variables using Newton polygons.
problem Determining the bifurcation set of a real polynomial function of two variables.
method Using toric compactification and toric modifications to count singular phenomena at infinity.
result An upper bound of the number of elements in the bifurcation set is given in terms of its Newton polygon.
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.
Paper classifies bifurcations of Minkowski symmetry sets for plane curves.
problem Classifying bifurcations of Minkowski symmetry sets.
method Classification of bifurcations based on geometric criteria.
result Different bifurcation types for Minkowski symmetry sets compared to Euclidean symmetry sets.
Study bifurcation in Riemannian metrics and submersions.
problem Analyzing bifurcation points in Riemannian metrics.
method Examining a one-parameter family of metrics on total spaces of harmonic Riemannian submersions.
result Existence and characterization of bifurcation points, and sufficient conditions for bifurcation.
Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.
problem Minimal tori in ellipsoids.
method Analyzing 3D ellipsoids invariant under a 2-torus action.
result Infinitely many distinct minimal tori bifurcate from a 2-torus orbit.
Study the evolution of the Lorenz strange set using Conley index theory.
problem Understanding the evolution of the Lorenz strange set through parameter changes.
method Application of Conley index theory to analyze the global attractor and its Morse decompositions.
result Identification and analysis of bifurcations and the role of the strange set in these transformations.
Study on bifurcations in Lagrangian systems and geodesics on manifolds.
problem Investigating bifurcations in Lagrangian systems and geodesics on Finsler and Riemannian manifolds.
method Employing Morse index and nullity techniques, and refining the Gauss lemma.
result Derivation of precise conditions for bifurcations in geodesic flows.
Study detects P-type bifurcations in single system realizations using unreliable kernel density estimates.
problem Detecting P-type bifurcations in signals with unreliable kernel density estimates.
method Create persistence diagrams from single system realization, statistically analyze resulting set, compare point process modeling methods.
result Subsampling outperforms other point process modeling methods in predicting P-type bifurcations.
Let g:X -> Y be a smooth (i.e. C^\infty differentiable) map between two smooth manifolds. In analogy with the case of complex polynomial functions, we say that y_0 in Y is a typical value of g if there exists an open neighbourhood U of y_0 in Y, such that the restriction g:g^{-1}(U) -> U is a C^\infty trivial fibration…
A generalization of the Euler-Plateau problem to account for the energy contribution due to twisting of the bounding loop is proposed. Euler-Lagrange equations are derived in a parameterized setting and a bifurcation analysis is performed. A pair of dimensionless parameters govern bifurcations from a flat, circular gro…
Let F=(F1,F2,...,Fm):Cn→Cm be a polynomial dominant mapping with n>m. In this paper we give the relations between the bifurcation set of F and the set of values where F is not M-tame as well as the set of generalized critical values of F. We also construct explicitly a proper su…
Study controls bifurcations in Eulerian flows with multiple Hopf singularities.
problem Bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularities.
method Analysis of CW complex bifurcations of flow-invariant Clifford hypertori, using leaf-bifurcation varieties.
result Tertiary toral CW complex bifurcates from and persists outside a secondary toral CW complex.
We classify simple singularities of functions on space curves. We show that their bifurcation sets have properties very similar to those of functions on smooth manifolds and complete intersections [1,2]: the k(pi, 1)-theorem for the bifurcations diagram of functions is true, and both this diagram and the discriminant a…
Study how J+ invariants change under bifurcations of curves.
problem Understanding how J+ invariants of curves change under bifurcations. method Analyzing J+, J−, J1, and J2 invariants of curves under k-bifurcations. result Invariant J+ changes under bifurcations, preserving its essential properties. The paper studies bifurcations in Lagrangian systems and geodesics.
problem Investigating bifurcations in Lagrangian systems with various boundary conditions.
method Using Morse theory and nullity techniques, the paper establishes conditions for bifurcation in three configurations.
result Unified Morse-theoretic framework connecting geometric focal structure and analytic bifurcation behavior.
MATLAB toolbox pde2path solves geometric PDEs and finds bifurcations in immersed surfaces.
problem Finding bifurcations in geometric PDEs of immersed surfaces.
method Solving PDEs for surface displacement, updating surface, detecting and localizing bifurcations, and switching branches.
result Symmetry breaking bifurcations in various geometric surfaces.
Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.
problem Yamabe problem on Aloff-Wallach Spaces
method Constructing 1-parameter families of solutions and examining changes in the Morse index as the parameter varies.
result Identifies bifurcation and local rigidity points for homogeneous solutions to the Yamabe problem.
New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
problem Existence of contractible domains with specific boundary conditions.
method Local bifurcation argument around geodesic disks, anisotropic Hölder spaces, computer-assisted techniques.
result Existence of nontrivial contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
The paper studies bifurcations in discrete dynamical systems on manifolds.
problem Understanding bifurcations in discrete dynamical systems on manifolds.
method Topological techniques based on concentricity of manifolds.
result General result for attractors in n-dimensional manifolds.
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…
Study bifurcations and local rigidity on flag manifolds for Yamabe solutions.
problem Yamabe problem on maximal flag manifolds
method Determine bifurcation and local rigidity points for 1-parameter families of solutions
result Identify specific points of bifurcation and local rigidity
We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifu…
Example shows not all conjugate points are bifurcation points in semi-Riemannian geodesics.
problem Determining which conjugate points in semi-Riemannian geodesics are bifurcation points.
method Revisiting and correcting an example by Musso, Pejsachowicz, and Portaluri.
result Every conjugate point on the improved example is a bifurcation point.
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 …
New bifurcation found in perturbations of non-generic closed self-shrinkers.
problem Understanding the behavior of perturbations in non-generic closed self-shrinkers.
method Analyzing the mean curvature flow singularity transitions.
result Different types of singularity transitions based on perturbation direction.
Study describes bifurcations of gradient flows on 2-sphere with holes.
problem Analyzing gradient flows on a 2-sphere with up to six singular points.
method Using separatrix diagrams to specify saddle-node and saddle connections.
result Identified all possible topological structures of bifurcations.
Study shows bifurcation in optimal retirement planning.
problem Optimal consumption and retirement planning model.
method Cobb-Douglas utility, simple model with wealth bifurcation.
result Critical wealth level leads to a continuum of retirement trajectories.
New theory shows how membranes can break symmetry.
problem Understanding symmetry breaking in membranes with boundaries.
method Applied bifurcation theory and reduced membrane equation.
result Existence of symmetry breaking bifurcation in membrane solutions.
New solutions found for Ginzburg-Landau equations on complex manifolds.
problem Finding solutions to Ginzburg-Landau equations on closed manifolds.
method Bifurcation theory applied to Laplace-type operator eigenvalues.
result First nonminimal and irreducible solutions on nontrivial line bundles.
Extremely accurate prediction of dynamical system bifurcations using control inputs.
problem Predicting complex bifurcation structures in dynamical systems.
method Extending extreme learning machines with control inputs to model system dynamics.
result The model can nearly reproduce the entire structure of bifurcations using only a few parameter values.
Study finds multiple periodic solutions to ODEs related to curvature problems.
problem Finding multiple positive periodic solutions to quasilinear ODEs.
method Global bifurcation techniques applied to second order quasilinear ODEs.
result Bifurcation-theoretic proof of nonuniqueness for conformal metrics with constant scalar curvature.
In this paper mechanisms of reversion - momentum transition are considered. Two basic nonlinear mechanisms are highlighted: a slow and fast bifurcation. A slow bifurcation leads to the equilibrium evolution, preceded by stability loss delay of a control parameter. A single order parameter is introduced by Markovian cha…
Study bifurcations of curves on surfaces in Minkowski 3-space.
problem Understanding the behavior of curves on surfaces in Minkowski 3-space.
method Analyzing the degeneracy of induced pseudo metric, discriminant of principal curvatures, parabolic curve, and mean curvature vanishing points.
result Bifurcations of robust features on surfaces in Minkowski 3-space.
Study finds multiple constant curvature metrics in product spaces.
problem Finding multiple constant scalar curvature metrics in product spaces.
method Equivariant bifurcation theory.
result Multiplicity of constant scalar curvature metrics discovered.
We extend the notion of reticular Legendrian unfoldings in order to investigate multi-time bifurcations of wavefronts generated by an r-corner. We give a classification list of generic and stable bifurcations with two time parameter and give all generic figures in the plane and the space.
Deep learning detects bifurcations in dynamical systems.
problem Predicting catastrophic changes in dynamical systems across sciences.
method Data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries.
result Extracts topologically invariant features to detect bifurcation boundaries in unseen systems.
Paper uses Simulated Bifurcation for quick asset allocation optimization.
problem Optimal asset allocation selection.
method Simulated Bifurcation algorithms applied to 441 S&P500 assets.
result Unrivaled timescale for optimal sub-allocation selection.
We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.
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…
The paper explains how the conjugate locus of a point on a surface can gain or lose cusps as the point moves.
problem Understanding the behavior of the conjugate locus as a point moves on a surface.
method Explains bifurcations in terms of vanishing higher derivatives of the exponential map and classifies them using scalar invariants.
result Simple equations for higher derivatives of the exponential map are derived, and the bifurcations of cusps are classified based on the local structure of the conjugate locus.
Study on positive solutions of Yamabe-type equation on spheres.
problem Existence and multiplicity of conformal metrics with constant scalar curvature.
method Reduction to an ODE and application of bifurcation theory.
result Existence of degenerate solutions and multiplicity results.
Study describes Morse flows on a torus with up to six singular points.
problem Understanding the structure of Morse flows on a torus with a hole.
method Used separatrix diagrams to describe topological structures and saddle-node bifurcations.
result Identified all possible topological structures of Morse flows with at most six singular points.
Researchers find multiple ways to deform manifolds with specific curvature properties.
problem Finding distinct conformal deformations of manifolds with boundary conditions.
method Using bifurcation results from Case, Moreira, and Wang, the researchers construct geometrically distinct solutions.
result There are multiple solutions to the conformal deformation problem in a finite set of dimensions.
The two phase behavior in financial markets actually means the bifurcation phenomenon, which represents the change of the conditional probability from an unimodal to a bimodal distribution. In this paper, the bifurcation phenomenon in Hang-Seng index is carefully investigated. It is observed that the bifurcation phenom…
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…
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…