Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium

Abstract

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium The paper deals with the problem of existence of a convergent "strong" normal form in the neighbourhood of an equilibrium, for a finite dimensional system of differential equations with analytic and time-dependent non-linear terms. The problem can be solved either under some non-resonance hypotheses on the spectrum of the linear part or if the non-linear term is assumed to be (slowly) decaying in time. This paper "completes" a pioneering work of Pustyl'nikov in which, despite under weaker non-resonance hypotheses, the nonlinearity is required to be asymptotically autonomous. The result is obtained as a consequence of the existence of a strong normal form for a suitable class of real-analytic Hamiltonians with non-autonomous perturbations. Published by AIP Publishing. [http://dx

Free propagation of resonant waves in nonlinear dissipative metamaterials

“…The procedure outlined in [29] reduces the problem to solving the homological equation {χ , ν play the role of unknowns. According to [35], it is possible to prove that this particular form for χ is the most general choice for Hamiltonians defined as in (3.8).…”

Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

“…Given the structure of the Hamiltonian, by following the lines of [59], a generating function of the form…”

“…In other words, one can "forget to have used the Hamiltonian formalism" at this point, as the Y variables will never appear in the transformed system. See [59] for the proof.…”

“…In parallel to asymptotic techniques applied directly to the ODEs or PDEs governing the wave propaga-tion problem, other works [59] and [60] showed how asymptotic results involving systems of ODEs close to integrability (either autonomous or non-autonomous) can be obtained by using the extensively developed tools of Hamiltonian Perturbation Theory. The key step consists in the definition of an equivalent Hamiltonian system in a suitably extended phase space.…”

“…Another remarkable advantage of the Hamiltonian formulation is that the problem can be studied up to an arbitrary high order of normalization by using constructive and explicit algorithms such as the mentioned Lie Transform method. It is interesting to point out that either the Lie series or the Lie transform method, in the formalism developed by Giorgilli [70], turned out to be successful in the generalization to the case of nearly integrable non autonomous systems with a general (i.e., aperiodic) time dependence [59,[71][72][73].…”