arXiv research

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions fo…

New method constructs Morse-Novikov complex with infinite series coefficients.

We consider a compact manifold of dimension greater than 2 and a differential form of degree one which is closed but non-exact. This form, viewed as a multi-valued function has a gradient vector field with respect to any Riemannian metric. After S. Novikov's work and a complement by J.-C. Sikorav, under some genericity…

Study the evolution of the Lorenz strange set using Conley index theory.

GRUs exhibit diverse dynamical behaviors but cannot mimic continuous attractors.

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We develop the multicomponent functionalized Cahn-Hilliard (mFCH) free…

Homoclinic orbits found in geodesic flows on surfaces.

We compute the chains associated to the left-invariant CR structures on the three-sphere. These structures are characterized by a single real modulus $a$. For the standard structure $a=1$, the chains are well-known and are closed curves. We show that for almost all other values of the modulus $a$ either two or three ty…

We show that for a $C^1$ residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources.

The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested in investigating the problem of existence of brake orbits and homoclinic orbits,…

The study proves the uniqueness of entropy-maximizing measures for geodesic flows on specific manifolds.

We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian syste…

The study proves the existence of many geodesics on complex manifolds.

Symbolic dynamics for flows in high dimensions, extending previous work.

The study characterizes lamination limits and homeomorphisms in 3D handlebodies.

Study controls bifurcations in Eulerian flows with multiple Hopf singularities.

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

In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a generalization $cat(X,ξ)$ of the notion of Lusternik - Schnirelman category, depending…

The paper studies bifurcations in Lagrangian systems and geodesics.

Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.

MATLAB toolbox pde2path solves geometric PDEs and finds bifurcations in immersed surfaces.

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

The paper studies bifurcations in discrete dynamical systems on manifolds.

We use algebraic Backlund transformations (BTs) to construct explicit solutions of the modified 2+1 chiral model from $T^2\times R$ to SU(n), where $T^2$ is a 2-torus. Algebraic BTs are parameterized by $z\in C$ (poles) and holomorphic maps $π$ from $T^2$ to Gr$(k,C^n)$. We apply Bäcklund transformations with carefully…

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.

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

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…

Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must…

Study on bifurcations in Lagrangian systems and geodesics on manifolds.

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.

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

Study shows bifurcation in optimal retirement planning.

New theory shows how membranes can break symmetry.

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

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.

Extremely accurate prediction of dynamical system bifurcations using control inputs.

Study finds multiple periodic solutions to ODEs related to curvature problems.

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…

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.

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

We explain how to apply techniques from integrable systems to construct $2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times R^1$ to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the -1 flow equation associated to a compact …

Deep learning detects bifurcations in dynamical systems.

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.

Paper uses Simulated Bifurcation for quick asset allocation optimization.