arXiv subjects

Pavao Mardešić

Publications and source records attributed to Pavao Mardešić.

7 recordsLinked to original sources

arXiv2026

Hamiltonian Monodromy in a Tavis-Cummings System with an $A_2$ Singularity

Singular Lagrangian fibrations arising from three-degree-of-freedom integrable Hamiltonian systems remain largely unexplored. While several results describe the global structure of large classes of systems with two degrees of freedom, only a few examples are understood in higher dimensions. We present a three-degree-of-freedom system derived from the two-spin Tavis-Cummings model whose singular Lagrangian fibration exhibits a topology that has not been observed in other physical models. We show that the most degenerate singular fiber is homeomorphic to $\mathbf{S}^2\times\mathbf{S}^1$ with a singularity of $A_2$ type. We further describe the bifurcation diagram and the global topology of the fibration, and we compute its Hamiltonian monodromy.

math-ph↗

arXiv2024

Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits

We consider generic 1-parameter unfoldings of parabolic vector fields. It is known that the box dimension of orbits of their time-one maps is discontinuous at the bifurcation value. Here, we expand asymptotically the Lebesgue measure of the epsilon-neighborhoods of orbits of the time-one maps in a Chebyshev scale, uniformly with respect to the bifurcation parameter. We use the so-called Ecalle-Roussarie-type compensators. We read from the expansion the number of hyperbolic points born in the unfolding of the parabolic point (i.e. the codimension of the bifurcation). We consider generic analytic 1-parameter unfoldings of saddle-node germs of analytic vector fields on the real line, their time-one maps and the Lebesgue measure of $\varepsilon$-neighborhoods of the orbits of these time-one maps.The box dimension of an orbit gives the asymptotics of the principal term of this Lebesgue measure and it is known that it is discontinuous at bifurcation parameters. In order to recover continuous dependence of the asymptotics on the parameter, here we expand asymptotically the Lebesgue measure of $\varepsilon$-neighborhoods of orbits of time-one maps in a Chebyshev system, uniformly with respect to the bifurcation parameter. We use Écalle-Roussarie-type compensators. We show how the number of fixed points of the time-one map born in the universal analytic unfolding of the parabolic point corresponds to the number of terms vanishing in this uniform expansion of the Lebesgue measure of $\varepsilon$-neighborhoods of orbits.

math.DS↗

arXiv2024

Nilpotence of Orbits under Monodromy and the Length of Melnikov Functions

Let $F\in\mathbb{C}[x,y]$ be a polynomial, $γ(z)\in π_1(F^{-1}(z))$ a non-trivial cycle in a generic fiber of $F$ and let $ω$ be a polynomial $1$-form, thus defining a polynomial deformation $dF+εω=0$ of the integrable foliation given by $F$. We study different invariants: the orbit depth $k$, the nilpotence class}$n$, the derivative length}$d$ associated with the couple $(F,γ)$. These invariants bound the length $\ell$ of the first nonzero Melnikov function of the deformation $dF+εω$ along $γ$. We study in detail a simple example of a polynomial $F$ given as product of four lines. We show how these invariants vary depending on the relative position of the four lines and relate it also to the length of the corresponding Godbillon-Vey sequence. We formulate a conjecture motivated by the study of this example.

math.CA↗

arXiv2022

Fractal zeta functions of orbits of parabolic diffeomorphisms

In this paper, we prove that fractal zeta functions of orbits of parabolic germs of diffeomorphisms can be meromorphically extended to the whole complex plane. We describe their set of poles (i.e. their complex dimensions) and their principal parts which can be understood as their fractal footprint. We study the fractal footprint of one orbit of a parabolic germ f and extract intrinsic information about the germ f from it, in particular, its formal class. Moreover, we relate complex dimensions to the generalized asymptotic expansion of the tube function of orbits with oscillatory "coefficients" as well as to the asymptotic expansion of their dynamically regularized tube function. Interestingly, parabolic orbits provide a first example of sets that have nontrivial Minkowski (or box) dimension and their tube function possesses higher order oscillatory terms, however, they do not posses non-real complex dimensions and are therefore not called fractal in the sense of Lapidus.

math.DS↗

arXiv2020

Realization of analytic moduli for parabolic Dulac germs

In a previous paper we have determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, that appear as first return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli.

math.DS↗

arXiv2020

Analytic moduli for parabolic Dulac germs

In this paper we give moduli of analytic classification for parabolic Dulac i.e. almost regular germs. Dulac germs appear as first return maps of hyperbolic polycycles. Their moduli are given by a sequence of Ecalle-Voronin-like germs of analytic diffeomorphisms. We state the result in a bigger class of parabolic generalized Dulac germs having power-logarithmic asymptotic expansions.

math.DS↗

arXiv2008

Monodromy and Tangential Center Problems

We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center focus problem asks for the conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both these questions for the case when the Hamiltonian is hyperelliptic. As a side-product, we solve the corresponding problems for the "0-dimensional Abelian integrals" defined by Gavrilov and Movasati.

math.DS↗