arXiv subjects

Konstantinos Kourliouros

Publications and source records attributed to Konstantinos Kourliouros.

12 recordsLinked to original sources

arXiv2026

Adiabatic Invariant Actions for Partially Integrable Systems

We introduce action integrals for partially integrable Hamiltonian systems that interpolate between the non-integrable and completely integrable theory. We show that under the assumption of ergodicity of the Hamiltonian system on the common level sets of its integrals, these actions become constants of motion for the averaged dynamics and, consequently, adiabatic invariants when the Hamiltonian function is allowed to slowly change with time.

math.DS↗

arXiv2026

Invariant Sets and Boundary Systems of Nonautonomous Differential Inclusions

In this paper we propose a finite-dimensional and deterministic approach to the study of invariant sets of certain nonautonomous differential inclusions naturally arising in the context of random and control dynamical systems, as well as in systems modeling the dynamical propagation of uncertainty. In particular, to any such differential inclusion, we associate a finite-dimensional and deterministic system of nonautonomous ordinary differential equations, which we call the boundary system, due to its following characteristic property: invariant sets of the differential inclusion lift in a unique way to backward invariant unit normal cones of the associated boundary system, and these are even invariant if the boundary is smooth. We further illustrate the strength of this approach in the study of minimal attractors of nonautonomous linear differential inclusions. Under the natural assumption of exponential stability for the unperturbed problem, we establish existence and uniqueness of a minimal attractor for the differential inclusion with fiberwise strictly convex, closed, and continuously differentiable boundaries. We also show that the unit normal bundle is in fact the pullback attractor for the skew-product flow associated to the boundary system which extends to the global attractor when the underlying admits a compact base.

math.DS↗

arXiv2024

Modules of derivations, logarithmic ideals and singularities of maps on analytic varieties

We introduce the module of derivations $Θ_{h,M}$ attached to a given analytic map $h:(\mathbb C^n,0)\to (\mathbb C^p,0)$ and a submodule $M\subseteq \mathcal O_n^p$ and analyse several exact sequences related to $Θ_{h,M}$. Moreover, we obtain formulas for several numerical invariants associated to the pair $(h,M)$ and a given analytic map germ $f:(\mathbb C^n,0)\to (\mathbb C^q,0)$. In particular, if $X$ is an analytic subvariety of $\mathbb C^n$, we derive expressions for analytic invariants defined in terms of the module $Θ_X$ of logarithmic vector fields of $X$.

math.AG↗

arXiv2022

Bruce-Roberts Numbers and Quasihomogeneous Functions on Analytic Varieties

Given a germ of an analytic variety $X$ and a germ of a holomorphic function $f$ with a stratified isolated singularity with respect to the logarithmic stratification of $X$, we show that under certain conditions on the singularity type of the pair $(f,X)$, the following relative analog of the well known K. Saito's theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce-Roberts numbers) is equivalent to the relative quasihomogeneity of the pair $(f,X)$, i.e. to the existence of a coordinate system such that both $f$ and $X$ are quasihomogeneous with respect to the same positive rational weights.

math.CV↗

arXiv2020

Sections of Hamiltonian Systems

A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e.g. a boundary, an obstacle or a set of admissible states). In this paper we give local classification results for all typical singularities of sections of regular (non-singular) Hamiltonian systems, a problem equivalent to the classification of typical singularities of Hamiltonian systems with one-sided constraints. In particular we give a complete list of exact normal forms with functional invariants, and we show how these are related/obtained by the symplectic classification of mappings with prescribed (Whitney-type) singularities, naturally defined on the reduced phase space of the Hamiltonian system.

math.SG↗

arXiv2020

Relative Logarithmic Cohomology and Nambu Structures of Maximal Degree

We present local classification results for isolated singularities of functions with respect to a Nambu structure (multi-vector field) of maximal degree, in a neighbourhood of a smooth point of its degeneracy hypersurface. The results depend on a logarithmic version of the Brieskorn-Sebastiani theorem, which guarantees the finiteness and freeness of the corresponding deformation module. This relates the functional moduli of the classification problem with the integrals of logarithmic forms along the vanishing cycles of the complement of the Milnor fibers of the restriction of the function on the degeneracy hypersurface of the Nambu structure, inside the Milnor fibers of the function itself.

math.AG↗

arXiv2018

The Milnor-Palamodov Theorem for Functions on Isolated Hypersurface Singularities

In this note we give a simple proof of the following relative analog of the well known Milnor-Palamodov theorem: the Bruce-Roberts number of a function relative to an isolated hypersurface singularity is equal to its topological Milnor number (the rank of a certain relative (co)homology group) if and only if the hypersurface singularity is quasihomogeneous. The proof relies on an interpretation of the Bruce-Roberts number in terms of differential forms and the Lê-Greuel formula.

math.AG↗

arXiv2018

Local Diffeomorphisms in Symplectic Space and Hamiltonian Systems with Constraints

In this paper we obtain exact normal forms with functional invariants for local diffeomorphisms, under the action of the symplectomorphism group in the source space. Using these normal forms we obtain exact classification results for the first occurring singularities of Hamiltonian systems with one-sided constraints, a problem posed by R. B. Melrose in his studies of glancing hypersurfaces.

math.SG↗

arXiv2018

First Occurring Singularities of Functions in Symplectic Semi-Space

We explain how a classical theorem by Arnol'd and Melrose on non-singular functions on a symplectic manifold with boundary can be proved in few lines, and we use the same method to obtain a new result, which is a normal form with functional invariants for the first occurring singularities

math.SG↗

arXiv2015

Gauss-Manin Connections for Boundary Singularities and Isochore Deformations

We study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular we prove the relative analog of J. Vey's isochore Morse lemma, J. -P. Françoise's generalisation on the local normal forms of volume forms with respect to the boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's theorem on the isochore version of Mather's versal unfolding theorem.

math.AG↗

arXiv2014

A Converse to a Theorem on Normal Forms of Volume Forms with Respect to a Hypersurface

In this note we give a positive answer to a question asked by Y. Colin de Verdière concerning the converse of the following theorem, due to A. N. Varchenko: two germs of volume forms are equivalent with respect to diffeomorphisms preserving a germ of an isolated hypersurface singularity, if their difference is the differential of a form whose restriction on the smooth part of the hypersurface is exact.

math.SG↗

arXiv2013

Singularities of Functions on the Martinet Plane, Constrained Hamiltonian Systems and Singular Lagrangians

We consider here the analytic classification of pairs $(ω,f)$ where $ω$ is a germ of a 2-form on the plane and $f$ is a quasihomogeneous function germ with isolated singularities. We consider only the case where $ω$ is singular, i.e. it vanishes non-degenerately along a smooth line $H(ω)$ (Martinet case) and the function $f$ is such that the pair $(f,H(ω))$ defines an isolated boundary singularity. In analogy with the ordinary case (for symplectic forms on the plane) we show that the moduli in the classification problem are analytic functions of 1-variable and that their number is exactly equal to the Milnor number of the corresponding boundary singularity. Moreover we derive a normal form for the pair $(ω,f)$ involving exactly these functional invariants. Finally we give an application of the results in the theory of constrained Hamiltonian systems, related to the motion of charged particles in the quantisation limit in an electromagnetic field, which in turn leads to a list of normal forms of generic singular Lagrangians (of first order in the velocities) on the plane.

math.DS↗