arXiv research

Deep learning detects bifurcations in dynamical systems.

The paper studies bifurcations in discrete dynamical systems on manifolds.

Study controls bifurcations in Eulerian flows with multiple Hopf singularities.

Topological method detects Hopf bifurcations from time series.

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…

We consider the static and dynamic models of Cournot duopoly with tax evasion. In the dynamic model we introduce the time delay and we analyze the local stability of the stationary state. There is a critical value of the delay when the Hopf bifurcation occurs.

We characterize stationary solutions to McKean-Vlasov equations on the circle.

In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We also analyze some natural Morse decompositions of the global attractor of the syst…

We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-Δ_g u + λu = λu^q$, with $λ>0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1 \geq d_2$ we show that for any $q<\frac{ n-d_2+2 }{n - d_2…

Agent-based market shows herding cycles with square-root price impact.

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

Solve supercritical Yamabe problem on manifolds with non-umbilic boundary.

New energy identity found for biharmonic maps into spheres.

Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.

Paper proves estimates for Lagrangian flow singularities.

Let $A \subset \mathbb{R} ^2$ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus $|u|=1$ on $\partial A$ and with prescribed degrees on the two connected components o…

Solves supercritical dHYM on projective manifolds with specific conditions.

Paper confirms conjecture for projective manifolds in supercritical phase.

Stable blowup solutions found for supercritical Yang-Mills equations.

Conservation law for weakly harmonic mappings in high dimensions.

Paper proves gradient estimates for Lagrangian mean curvature equation.

We propose a reduced form set of two coupled continuous time equations linking the price of a representative asset and the price of a bond, the later quantifying the cost of borrowing. The feedbacks between asset prices and bonds are mediated by the dependence of their "fundamental values" on past asset prices and bond…

Solutions grow for a special type of math problem on curved spaces.

Wave maps with noise can lead to self-similar blowup from arbitrary initial data.

Derives Hessian estimates for Lagrangian mean curvature equation.

We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.

Study on self-similar solutions of supercritical Fujita equation, proving entropy and energy gap.

We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can…

We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak …

We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, whi…

Paper proves solvability condition for complex equation on special submanifolds.

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

The paper studies bifurcations in Lagrangian systems and geodesics.

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

Smooth Yang-Mills fields proved in supercritical dimensions.

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

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 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.

Recently, large-scale cascading failures in complex systems have garnered substantial attention. Such extreme events have been treated as an integral part of the self-organized criticality (SOC). Recent empirical work has suggested that some extreme events systematically deviate from the SOC paradigm, requiring a diffe…

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 disproves conjecture about Hermitian-Yang-Mills solutions.

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.