A Lie transform approach to the construction of Lyapunov functions in autonomous and non-autonomous systems

Abstract

This paper deals with the well-known problem of constructing Lyapunov functions for a nonlinear system and the approximation of the basin of attraction associated with a given attractive equilibrium point. Following a paper by Spelberg-Korspeter et al., the problem is studied by means of perturbative methods, with particular focus on the time-reversed Van Der Pol model. As a difference, the theory is reformulated in terms of the Lie transform method, introduced by Giorgilli et al., which, remarkably, does not require any inverse function arguments to produce the inverse transformations during the normalization process. This will be shown to be, also in this case, a key feature in terms of concrete applications. The nonautonomous perturbation theory developed by the authors in previous works allows an effortless extension of such a construction to the (aperiodically) time-dependent case.

Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

“…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.…”