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We construct an approximate renormalization for Hamiltonian systems with two degrees of freedom in order to study the break-up of invariant tori with arbitrary frequency. We derive the equation of the critical surface of the renormalization…
We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We…
We study the problem of conservation of maximal and lower-dimensional invariant tori for analytic convex quasi-integrable Hamiltonian systems. In the absence of perturbation the lower-dimensional tori are degenerate, in the sense that the…
The disadvantage of `traditional' multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis…
In this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the…
We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given…
Linear time-periodic (LTP) dynamical systems frequently appear in the modeling of phenomena related to fluid dynamics, electronic circuits, and structural mechanics via linearization centered around known periodic orbits of nonlinear…
Randomized compilation protocols have recently attracted attention as alternatives to traditional deterministic Trotter-Suzuki methods, potentially reducing circuit depth and resource overhead. These protocols determine gate application…
We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. It makes the connection…
Using discretized orthogonal systems (curvature line systems) with periodicity, created using Darboux transformations and their permutability, we have discrete and semi-discrete k-dimensional isothermic tori which are full in n-dimensional…
In this paper, we present an algorithm to computea fiberwise Lagrangian torus in quasi-periodic (QP) Hamiltonian systems, whose convergence is proved in the [CHP25]. We exhibit the algorithm with two models. The first is a Tokamak model…
We study the existence of infinite-dimensional invariant tori in a mechanical system of infinitely many rotators weakly interacting with each other. We consider explicitly interactions depending only on the angles, with the aim of…
This paper proposes a model order reduction method for a class of parametric dynamical systems. Using a temporal Fourier transform, we reformulate these systems into complex-valued elliptic equations in the frequency domain, containing…
In this paper, we analyze the convergence of several discretize-then-optimize algorithms, based on either a second-order or a fourth-order finite difference discretization, for solving elliptic PDE-constrained optimization or optimal…
Trotter and linear-combination-of-unitary (LCU) are two popular Hamiltonian simulation methods. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates…
We consider the classical problem of the construction of invariant tori exploiting suitable Hamiltonian normal forms. This kind of approach can be translated by means of the Lie series method into explicit computational algorithms, which…
We present a complexity reduction algorithm for a family of parameter-dependent linear systems when the system parameters belong to a compact semi-algebraic set. This algorithm potentially describes the underlying dynamical system with…
We introduce novel algorithms for the quantum simulation of molecular systems which are asymptotically more efficient than those based on the Trotter-Suzuki decomposition. We present the first application of a recently developed technique…
Hamiltonian matrices appear in a variety or problems in physics and engineering, mostly related to the time evolution of linear dynamical systems as for instance in ion beam optics. The time evolution is given by symplectic transfer…
We review a method of control for Hamiltonian systems which is able to create smooth invariant tori. This method of control is based on an apt modification of the perturbation which is small and localized in phase space.