Claudia Valls

A formal approximation of the splitting of separatrices in the classical Arnold's example of diffusion with two equal parameters

We consider the classical Arnold example of diffusion with two equal parameters. Such a system has two-dimensional partially hyperbolic invariant tori. We mainly focus on the tori whose ratio of frequencies is the golden mean. We present formal approximations of the three-dimensional invariant manifolds associated with this torus and numerical globalization of these manifolds. This allows one to obtain the splitting (of separatrices) vector and to compute its Fourier components. It is apparent that the Melnikov vector provides the dominant order of the splitting provided the contribution of each harmonic is computed after a suitable number of averaging steps, depending on the harmonic. We carry out the first-order analysis of the splitting based on that approach, mainly looking for bifurcations of the zero-level curves of the components of the splitting vector and of the homoclinic points.

On the number of limit cycles of a class of polynomial differential systems

We study the number of limit cycles of polynomial differential systems of the form where g1,f1,g2 and f2 are polynomials of a given degree. Note that when g1(x)=f1(x)=0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre , using the averaging theory of first and second order.

Phase portraits of the two-body problem with Manev potential

The Manev systems are two-body problems defined by a potential of the form a/r + b/r2, where r is the distance between the two particles, and a and b are arbitrary constants. The Hamiltonian H = (pr2 + pθ2/r2)/2 + a/r + b/r2 and the angular momentum pθ = r2 associated with Manev systems are two first integrals, which are independent and in involution. Let Ih (respectively Ic) be the set of points of the phase space on which H (respectively pθ) takes the value h (respectively c). Since H and pθ are first integrals, the sets Ih, Ic and Ihc = IhIc are invariant under the flow of the Manev systems. We characterize the global flow of these systems when a and b vary. Thus we describe the foliation of the phase space by the invariant sets Ih and the foliation of Ih by the invariant sets Ihc.

The global flow of the quasihomogeneous potentials of Manev- Schwarchild type

Abstract In this paper, we study two-body problems defined by a potential of the form V(r)=a/r+b/r2+c/r3, where r is the distance between the two particles and a, b, c are arbitrarily chosen constants. The Hamiltonian H(r,p r ,θ,p θ )= 1 2 p r 2 + p θ 2 r 2 + a r + b r 2 + c r 3 and the angular momentum p θ =r 2 θ are two first integrals, independent and in involution. Let Ih (respectively Im) be the set of points on the phase space on which H (respectively pθ) takes the value h (respectively m). Since H and pθ are first integrals, the sets Ih, Im and Ihm=Ih∩Im are invariant under the systems associated to the Hamiltonian H. We characterize the global flow of the systems when a, b, and c vary, describing the foliation of the phase space by the invariant sets Ih, the foliation of Ih by the invariant sets Ihm and the movement of the flow over Ihm.

A version of a theorem of Pliss for non-uniform and non-invertible dichotomies

For a dynamics on the whole line, for both discrete and continuous time, we extend a result of Pliss that gives a characterization of the notion of a trichotomy in various directions. More precisely, the result gives a characterization in terms of an admissibility property in the whole line (namely, the existence of bounded solutions of a linear dynamics under any nonlinear bounded perturbation) of the existence of a trichotomy, i.e. of exponential dichotomies in the future and in the past, together with a certain transversality condition at time zero. In particular, we consider arbitrary linear operators acting on a Banach space as well as sequences of norms instead of a single norm, which allows us to consider the general case of non-uniform exponential behaviour.