First derivative of the period function with applications

Abstract

Given a centre of a planar differential system, we extend the use of the Lie bracket to the determination of the monotonicity character of the period function. As far as we know, there are no general methods to study this function, and the use of commutators and Lie bracket was restricted to prove isochronicity. We give several examples and a special method which simplifies the computations when a first integral is known. r

On the period function of planar systems with unknown normalizers

“…In general, computing the integral in (8) is not easy, even when W is itself a normalizer of a Hamiltonian system with separable variables [5]. Anyway, β does not change sign on a single orbit, so that if also η has constant sign, then its sign is that of ∂ W T .…”

“…We give an example for a Hamiltonian system with separable variables. In general, systems with separable variables may be better treated by following the approach of [5], provided the normalizer (11) is defined on all of a cycle. This occurs when G (x)F (y) does not vanish on a cycle.…”

“…In general, it is not known how to find a normalizer of a given system. In this paper we present an extension of Theorem 1 in [5] that allows us to replace a normalizer with a transversal vector field W . The main result is contained in Theorem 1, which gives an integral formula for the derivative of T along the integral curves of W .…”

On the reversible quadratic centers with monotonic period function

“…Thus, by using a Picard-Fuchs approach, Y. Zhao can completely describe the behaviour of the period function in two straight lines of the parameter plane, namely F = 3/2 in [24] and F = 2 in [25]. There is a number of different authors that have treated the general question of monotonicity of the period function (see [2,3,8,14,17,22] and references there in). Apart from the monotonicity problem, several other questions related to the behaviour of the period function have been extensively studied.…”

Finding Hamiltonian isochronous centers by non-canonical transformations

“…Isochronicity in more general Hamiltonian systems was also considered [1,2,3,4,5,6,9]. The underlying motivation of all such papers is the fact that a Hamiltonian system is usually related to the dynamic behaviour of some physical system.…”