Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

Abstract

In 1985 Franz Rothe [16] found, by means of the thermodynamical equilibrium theory, an asymptotic estimate of period of solutions of Ordinary Differential Equations originated by predator -prey Volterra -Lotka model. We extend some of Rothe's ideas to more general systems:and succeed in calculating the period's asymptotic analytic expression as a function of the energy level. We finally check our result re-obtaining classical period's estimation of some popular Hamiltonian systems. We apply our technique also to a nonlinear Hamiltonian system whose period is not available in the literature.KEYWORD: Hamiltonian systems, series reversion, period, asymptotic expansion NOTE: this is the post print version of the paper

The local period function for Hamiltonian systems with applications

“…with a non-degenerated center at the origin (that without loss of generality we will associated to h = 0 and then I = [0, h 1 ) ⊂ R) it is known that T (h), in a neighborhood of h = 0, is an analytic function of the energy h and it is given by the derivative with respect h of the area function A(h), see [33]. There are several authors that compute the Taylor series of T at h = 0 for particular Hamiltonian systems but, to the best of our knowledge, most examples deal with Hamiltonian functions with separated variables H(x, y) = F (x) + G(y), see for instance [5,16] and their references. Our first result provides a systematic constructive approach for finding this Taylor series up to any order for any Hamiltonian system.…”

“…(iv) The result follows by using induction on n and equalities (16) taking in each step blocks of three integrals I 3n , I 3n+1 and I 3n+2 . For instance,…”

The Goodwin cycle improved with generalized wages: phase portrait, periodic behaviour

“…Hereinafter, founding upon a theorem proved in [9], we can provide an asymptotic approximation of its Period function even not going into any Phillips curve linearization. Rothe introduced the concept of Period function for Downloaded by [MUSC Library] at 22:00 13 June 2016 the Volterra-Lotka system, where F(p) = c(e p − p − 1) and G(q) = a(e q − q − 1) , establishing a celebrated asymptotic expansion theorem concerning its Period function, Lemma 3, pp.…”

“…133 [23] for small oscillations performed at low energy levels. In [9] such result was generalized: we quote the theorem which covers all the systems like (4.1).…”

“…Based on Rebelo's article [22], we are able to discard the Lipschitz condition for uniqueness of the solution which, in our context, is ensured only by continuity assumption. Afterwards, an asymptotic approximation for the period-energy function using a result from [9] is supplied. Finally a sufficient condition ensuring its monotonicity is formulated following [4,5,23].…”