Pavao Mardešić

A generalization of Françoise's algorithm for calculating higher order Melnikov functions

Abstract In [J. Differential Equationsxa0146 (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 Francoises condition is not verified. We generalize Francoises 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 differential system verified by these Melnikov functions Ml(t).

On the time function of the Dulac map for families of meromorphic vector fields

Given an analytic family of vector fields in 2 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.

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

Abstract We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on e of the length of e-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered in Elezovic, Žubrinic and Županovic (2007) [5] 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 of fixed points and the dependence on e of the length of e-neighborhoods of orbits in non-differentiable cases. Applications include in particular Poincare maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the e-neighborhood of one orbit of the Poincare map (for example numerically), and by comparing it to the appropriate scale.

The tennis racket effect in a three-dimensional rigid body

Abstract 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 ( π - rotation) of the head of the racket when a full ( 2 π ) 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. Different examples are discussed.

Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond

The design of efficient and robust pulse sequences is a fundamental requirement in quantum control. Numerical methods can be used for this purpose, but with relatively little insight into the control mechanism. Here, we show that the free rotation of a classical rigid body plays a fundamental role in the control of two-level quantum systems by means of external electromagnetic pulses. For a state to state transfer, we derive a family of control fields depending upon two free parameters, which allow us to adjust the efficiency, the time and the robustness of the control process. As an illustrative example, we consider the quantum analog of the tennis racket effect, which is a geometric property of any classical rigid body. This effect is demonstrated experimentally for the control of a spin 1/2 particle by using techniques of Nuclear Magnetic Resonance. We also show that the dynamics of a rigid body can be used to implement one-qubit quantum gates. In particular, non-adiabatic geometric quantum phase gates can be realized based on the Montgomery phase of a rigid body. The robustness issue of the gates is discussed.

Vanishing Abelian integrals on zero-dimensional cycles

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 zerosum integers {; ; ; ni}; ; ; such that the function t 7→ Pnig(zi(t)) vanishes identically. Here zi(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 process. We also solve the problem on some balanced cycles. The main motivation for our study is the problem of vanishing of Abelian integrals on single families of one- dimensional cycles. We show that our problem and our main result are sufficiently rich to include some related problems, as hyper-elliptic integrals on one-cycles, some applications to slow-fast planar systems, and the polynomial (and trigonometric) moment problem for Abel equation. This last problem was recently solved by Pakovich and Muzychuk. Our approach is largely inspired by their work, thought we provide examples of vanishing Abelian integrals on zero-cycles which are not given as a sum of composition terms contrary to the situation in the solution of the polynomial moment problem

Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles

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 Ilyashenkos theorem on non-accumulation of limit cycles on hyperbolic polycycles.

Abelian Integrals: From the Tangential 16th Hilbert Problem to the Spherical Pendulum

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 in particular the simplest example presenting non-trivial Hamiltonian monodromy: the spherical pendulum.

Index of singularities of real vector fields on singular hypersurfaces

Gomez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincare-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gomez-Mont and Giraldo about calculating the GSV-index

The period function of reversible quadratic centers

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