0000000000149614
AUTHOR
Pavao Mardešić
showing 23 related works from this author
A generalization of Françoise's algorithm for calculating higher order Melnikov functions
2002
Abstract In [J. Differential Equations 146 (2) (1998) 320–335], Francoise gives an algorithm for calculating the first nonvanishing Melnikov function Ml of a small polynomial perturbation of a Hamiltonian vector field and shows that Ml is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form ω on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Francoise's condition is not verified. We generalize Francoise's algorithm to this case and we show that Ml belongs to the C [ log t,t,1/t] module above the Abelian integrals. We also establish the linear differentia…
Pseudo-abelian integrals: Unfolding generic exponential case
2009
The search for bounds on the number of zeroes of Abelian integrals is motivated, for instance, by a weak version of Hilbert's 16th problem (second part). In that case one considers planar polynomial Hamiltonian perturbations of a suitable polynomial Hamiltonian system, having a closed separatrix bounding an area filled by closed orbits and an equilibrium. Abelian integrals arise as the first derivative of the displacement function with respect to the energy level. The existence of a bound on the number of zeroes of these integrals has been obtained by A. N. Varchenko [Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 14–25 ; and A. G. Khovanskii [Funktsional. Anal. i Prilozhen. 18 (1984), n…
Darboux Linearization and Isochronous Centers with a Rational First Integral
1997
Abstract In this paper we study isochronous centers of polynomial systems. It is known that a center is isochronous if and only if it is linearizable. We introduce the notion of Darboux linearizability of a center and give an effective criterion for verifying Darboux linearizability. If a center is Darboux linearizable, the method produces a linearizing change of coordinates. Most of the known polynomial isochronous centers are Darboux linearizable. Moreover, using this criterion we find a new two-parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family. We also determine all Hamiltonian cubic systems which are Darbou…
Rotation Forms and Local Hamiltonian Monodromy
2017
International audience; The monodromy of torus bundles associated with completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article, we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non-degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach …
Multiplicity of fixed points and growth of ε-neighborhoods of orbits
2012
We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity o…
Unfolding of saddle-nodes and their Dulac time
2016
Altres ajuts: UNAB10-4E-378, co-funded by ERDF "A way to build Europe" and by the French ANR-11-BS01-0009 STAAVF. In this paper we study unfoldings of saddle-nodes and their Dulac time. By unfolding a saddle-node, saddles and nodes appear. In the first result (Theorem A) we give a uniform asymptotic expansion of the trajectories arriving at the node. Uniformity is with respect to all parameters including the unfolding parameter bringing the node to a saddle-node and a parameter belonging to a space of functions. In the second part, we apply this first result for proving a regularity result (Theorem B) on the Dulac time (time of Dulac map) of an unfolding of a saddle-node. This result is a b…
Geometric Origin of the Tennis Racket Effect
2020
The tennis racket effect is a geometric phenomenon which occurs in a free rotation of a three-dimensional rigid body. In a complex phase space, we show that this effect originates from a pole of a Riemann surface and can be viewed as a result of the Picard-Lefschetz formula. We prove that a perfect twist of the racket is achieved in the limit of an ideal asymmetric object. We give upper and lower bounds to the twist defect for any rigid body, which reveals the robustness of the effect. A similar approach describes the Dzhanibekov effect in which a wing nut, spinning around its central axis, suddenly makes a half-turn flip around a perpendicular axis and the Monster flip, an almost impossibl…
ON THE INDEX OF VECTOR FIELDS TANGENT TO HYPERSURFACES WITH NON-ISOLATED SINGULARITIES
2002
Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kahler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field $X:=\tilde{X}|_V$ on $V$ : \begin{equa…
Infinite orbit depth and length of Melnikov functions
2019
Abstract In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along…
The Fatou coordinate for parabolic Dulac germs
2017
We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated log scale.
Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles
2007
We call Poincare time the time associated to the Poincar6 (or first return) map of a vector field. In this paper we prove the non-accumulation of isolated critical points of the Poincare time T on hyperbolic polycycles of polynomial vector fields. The result is obtained by proving that the Poincare time of a hyperbolic polycycle either has an unbounded principal part or is an almost regular function. The result relies heavily on the proof of Il'yashenko's theorem on non-accumulation of limit cycles on hyperbolic polycycles.
The period function of reversible quadratic centers
2006
Abstract In this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers, namely the dehomogenized Loud's systems. The local bifurcation diagram of the period function at the center is fully understood using the results of Chicone and Jacobs [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486]. Most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center. The techniques that we use here are different from the ones in [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vecto…
Vanishing Abelian integrals on zero-dimensional cycles
2011
In this paper we study conditions for the vanishing of Abelian integrals on families of zero-dimensional cycles. That is, for any rational function $f(z)$, characterize all rational functions $g(z)$ and zero-sum integers $\{n_i\}$ such that the function $t\mapsto\sum n_ig(z_i(t))$ vanishes identically. Here $z_i(t)$ are continuously depending roots of $f(z)-t$. We introduce a notion of (un)balanced cycles. Our main result is an inductive solution of the problem of vanishing of Abelian integrals when $f,g$ are polynomials on a family of zero-dimensional cycles under the assumption that the family of cycles we consider is unbalanced as well as all the cycles encountered in the inductive proce…
Normalizability, Synchronicity, and Relative Exactness for Vector Fields in C2
2004
In this paper, we study the necessary and su.cient condition under which an orbitally normalizable vector field of saddle or saddle-node type in C2 is analytically conjugate to its formal normal form (i.e., normalizable) by a transformation fixing the leaves of the foliation locally. First, we express this condition in terms of the relative exactness of a certain 1-form derived from comparing the time-form of the vector field with the time-form of the normal form. Then we show that this condition is equivalent to a synchronicity condition: the vanishing of the integral of this 1-form along certain asymptotic cycles de.ned by the vector field. This can be seen as a generalization of the clas…
The tennis racket effect in a three-dimensional rigid body
2017
We propose a complete theoretical description of the tennis racket effect, which occurs in the free rotation of a three-dimensional rigid body. This effect is characterized by a flip ($\pi$- rotation) of the head of the racket when a full ($2\pi$) rotation around the unstable inertia axis is considered. We describe the asymptotics of the phenomenon and conclude about the robustness of this effect with respect to the values of the moments of inertia and the initial conditions of the dynamics. This shows the generality of this geometric property which can be found in a variety of rigid bodies. A simple analytical formula is derived to estimate the twisting effect in the general case. Differen…
A note on a generalization of Françoise's algorithm for calculating higher order Melnikov functions
2004
In [J. Differential Equations 146 (2) (1998) 320–335], Françoise gives an algorithm for calculating the first nonvanishing Melnikov function M of a small polynomial perturbation of a Hamiltonian vector field and shows that M is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form ω on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Françoise’s condition is not verified. We generalize Françoise’s algorithm to this case and we show that M belongs to the C[log t, t, 1/t] module above the Abelian integrals. We also establish the linear differential system ver…
Normal forms and embeddings for power-log transseries
2016
First return maps in the neighborhood of hyperbolic polycycles have their asymptotic expansion as Dulac series, which are series with power-logarithm monomials. We extend the class of Dulac series to an algebra of power-logarithm transseries. Inside this new algebra, we provide formal normal forms of power-log transseries and a formal embedding theorem. The questions of classifications and of embeddings of germs into flows of vector fields are common problems in dynamical systems. Aside from that, our motivation for this work comes from fractal analysis of orbits of first return maps around hyperbolic polycycles. This is a joint work with Pavao Mardešić, Jean-Philippe Rolin and Vesna Župano…
Perturbations of symmetric elliptic Hamiltonians of degree four
2006
AbstractIn this paper four-parameter unfoldings Xλ of symmetric elliptic Hamiltonians of degree four are studied. We prove that in a compact region of the period annulus of X0 the displacement function of Xλ is sign equivalent to its principal part, which is given by a family induced by a Chebychev system; and we describe the bifurcation diagram of Xλ in a full neighborhood of the origin in the parameter space, where at most two limit cycles can exist for the corresponding systems.
Godbillon–Vey sequence and Françoise algorithm
2019
Abstract We consider foliations given by deformations d F + ϵ ω of exact forms dF in C 2 in a neighborhood of a family of cycles γ ( t ) ⊂ F − 1 ( t ) . In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function Δ along γ of such deformations. This algorithm recalls the well-known Godbillon–Vey sequences discovered in 1971 for investigation of integrability of a form ω. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon–Vey sequences to the Francoise algorithm settings.
Pseudo-Abelian integrals along Darboux cycles
2008
We study polynomial perturbations of integrable, non-Hamiltonian system with first integral of Darboux-type with positive exponents. We assume that the unperturbed system admits a period annulus. The linear part of the Poincare return map is given by pseudo-Abelian integrals. In this paper we investigate analytic properties of these integrals. We prove that iterated variations of these integrals vanish identically. Using this relation we prove that the number of zeros of these integrals is locally uniformly bounded under generic hypothesis. This is a generic analog of the Varchenko-Khovanskii theorem for pseudo-Abelian integrals. Finally, under some arithmetic properties of exponents, the p…
Principal part of multi-parameter displacement functions
2012
This paper deals with a perturbation problem from a period annulus, for an analytic Hamiltonian system [J.-P. Françoise, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 87–96 ; L. Gavrilov, Ann. Fac. Sci. Toulouse Math. (6) 14(2005), no. 4, 663–682. The authors consider the planar polynomial multi-parameter deformations and determine the coefficients in the expansion of the displacement function generated on a transversal section to the period annulus. Their first result gives a generalization to the Françoise algorithm for a one-parameter family, following [J.-P. Françoise and M. Pelletier, J. Dyn. Control Syst. 12 (2006), no. 3, 357–369. The second result expresses the principal terms in …
On the time function of the Dulac map for families of meromorphic vector fields
2003
Given an analytic family of vector fields in Bbb R2 having a saddle point, we study the asymptotic development of the time function along the union of the two separatrices. We obtain a result (depending uniformly on the parameters) which we apply to investigate the bifurcation of critical periods of quadratic centres.
Abelian Integrals: From the Tangential 16th Hilbert Problem to the Spherical Pendulum
2016
In this chapter we deal with abelian integrals. They play a key role in the infinitesimal version of the 16th Hilbert problem. Recall that 16th Hilbert problem and its ramifications is one of the principal research subject of Christiane Rousseau and of the first author. We recall briefly the definition and explain the role of abelian integrals in 16th Hilbert problem. We also give a simple well-known proof of a property of abelian integrals. The reason for presenting it here is that it serves as a model for more complicated and more original treatment of abelian integrals in the study of Hamiltonian monodromy of fully integrable systems, which is the main subject of this chapter. We treat i…