arXiv subjects

Stavros Anastassiou

Publications and source records attributed to Stavros Anastassiou.

10 recordsLinked to original sources

arXiv2026

Detecting invariant manifolds of dynamical systems using persistent homology

We use methods of Persistent Homology Theory to study invariant manifolds of dynamical systems. We first establish connections between the persistence diagrams of two sets which are close to each other, with respect to the Hausdorff distance. We then apply these results to study properties of limit sets of specific dynamical systems, by using the persistence diagram of a numerically obtained sample set. Under mild assumptions, we show how to use numerical data to state analytical results concerning the geometry of the limit sets.

math.DS↗

arXiv2024

Planar vector fields in the kernel of a 1--form

We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the form defines a, possibly singular, symplectic form. In every case, we provide a fairly complete list of local models for such fields and construct their transversal unfoldings. Thus, the local bifurcations of vector fields of interest can be studied, among them being the integrable fields of the plane.

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arXiv2024

Singularities of 3d vector fields preserving the form of Martinet

We study the local structure of vector fields on $\mathbb{R}^3$ which preserve the Martinet $1$-form $α=(1+x)dy\pm zdz$. We present the classification of their singularities, up to diffeomorphisms preserving the form $α$, as well as their transversal unfoldings. We are thus able to provide a fairly complete list of the bifurcations such vector fields undergo.

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arXiv2024

Local models for smooth vector fields of the line

We present the local classification of singularities of smooth vector fields on the line, with respect to the equivalence relation of $C^1$--conjugacy. Along the way, we recall the analogous classification, up to $C^0$ and $C^{\infty}$ conjugacy. We also give the transversal unfoldings of the corresponding normal forms and treat the case where the changes of coordinates are tangent to the identity. Thus, a fairly complete description of the $1$--d case is achieved.

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arXiv2024

Stationary Solitons in discrete NLS with non-nearest neighbour interactions

The aim of this paper is to provide a construction of stationary discrete solitons in an extended one-dimensional Discrete NLS model with non-nearest neighbour interactions. These models, models of the type with long-range interactions were studied in various other contexts. In particular, it was shown that, if the interaction strength decays sufficiently slowly as a function of distance, it gives rise to bistability of solitons, which may find applications in their controllable switching. Dynamical lattices with long-range interactions also serve as models for energy and charge transport in biological molecules. Using a dynamical systems method we are able to construct, with great accuracy, stationary discrete solitons for our model, for a large region of the parameter space.

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arXiv2021

Ancient solutions of the homogeneous Ricci flow on flag manifolds

For any flag manifold $M=G/K$ of a compact simple Lie group $G$ we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on $M$, and by [BöLS17] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold $M=G/K$ with second Betti number $b_{2}(M)=1$, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose $α$-limit set consists of fixed points at infinity of ${\mathscr{M}}^G$. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system's phase space.

math.DG↗

arXiv2018

Darboux polynomials and global phase portraits for the D_2 vector field

We study a vector field of R^3 equivariant under the D_2 symmetry group, called "the D_2 field" in the literature. We construct the complete list of Darboux polynomials for it, solving the partial differential equation defining them. We also use these polynomials to comment on its global qualitative behaviour. This is meant to be a first step towards the comparison of vector fields based on the module generated by their Darboux polynomials.

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arXiv2017

Homoclinic points of 2-D and 4-D maps via the Parametrization Method

An interesting problem in solid state physics is to compute discrete breather solutions in $\mathcal{N}$ coupled 1--dimensional Hamiltonian particle chains and investigate the richness of their interactions. One way to do this is to compute the homoclinic intersections of invariant manifolds of a saddle point located at the origin of a class of $2\mathcal{N}$--dimensional invertible maps. In this paper we apply the parametrization method to express these manifolds analytically as series expansions and compute their intersections numerically to high precision. We first carry out this procedure for a 2--dimensional (2--D) family of generalized Henon maps ($\mathcal{N}$=1), prove the existence of a hyperbolic set in the non-dissipative case and show that it is directly connected to the existence of a homoclinic orbit at the origin. Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond which the homoclinic intersection disappears. Proceeding to $\mathcal{N}=2$, we use the same approach to determine the homoclinic intersections of the invariant manifolds of a saddle point at the origin of a 4--D map consisting of two coupled 2--D cubic Hénon maps. In dependence of the coupling the homoclinic intersection is determined, which ceases to exist once a certain amount of dissipation is present. We discuss an application of our results to the study of discrete breathers in two linearly coupled 1--dimensional particle chains with nearest--neighbor interactions and a Klein--Gordon on site potential.

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arXiv2016

Dynamical systems on the Liouville plane and the related strictly contact systems

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. We discuss, in addition, their relation with strictly contact vector fields in dimension three. Analogous results for diffeomorphisms are also given.

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arXiv2010

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

We give the global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces descents to a parameter depending system of two or three ordinary differential equations, respectively. We present here the qualitative study of these system's global phase portrait, by using techniques of Dynamical Systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds, and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.

math.DG↗