Hilbert's 16th problem for classical Liénard equations of even degree

AbstractClassical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations x¨+f(x)x˙+x=0. In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio

Absolute cyclicity, Lyapunov quantities and center conditions

AbstractIn this paper we consider analytic vector fields X0 having a non-degenerate center point e. We estimate the maximum number of small amplitude limit cycles, i.e., limit cycles that arise after small perturbations of X0 from e. When the perturbation (Xλ) is fixed, this number is referred to as the cyclicity of Xλ at e for λ near 0. In this paper, we study the so-called absolute cyclicity; i.e., an upper bound for the cyclicity of any perturbation Xλ for which the set defined by the center conditions is a fixed linear variety. It is known that the zero-set of the Lyapunov quantities correspond to the center conditions (Caubergh and Dumortier (2004) [6]). If the ideal generated by the Lyapunov quantities is regular, then the absolute cyclicity is the dimension of this so-called Lyapunov ideal minus 1. Here we study the absolute cyclicity in case that the Lyapunov ideal is not regular

Measure under pressure : calibration of pressure measurement

Piston-cylinder assemblies are used to create a calculable pressure in a container, which can then be used for calibration of other instruments. For this purpose one needs to calculate the pressure in the container so accurately that both imperfections in the piston, and the leakage of fluid or gas through the small space between cylinder and piston have to be taken into account. Because of these effects, the piston behaves as if its area was slightly larger than it actually is. This slightly larger area is called the effective area of the piston-cylinder assembly, and its computation is the subject of this report. We derive a formula for this effective area, which under some simplifications leads to the formula used by four European metrological institutes. The formula used by NMi is based on a further simplification. We conclude with some recommendations to NMi concerning which formula to use and how to compute the uncertainty in the results. Keywords: effective area, piston-cylinder assemblies, pressure balance, thin film approximation