Related papers: A Dual-Radix Modular Division Algorithm for Comput…
We construct an algorithm for computing the cycle classes of the spin components of a stratum of differentials in the moduli space of stable curves $\overline{\mathcal{M}}_{g,n}$. In addition, we implement it within the Sage package…
The search of high-order periodic orbits has been typically restricted to problems with symmetries that help to reduce the dimension of the search space. Well-known examples include reversible maps with symmetry lines. The present work…
An analytical approximation to periodic orbits in the circular restricted three-body problem is provided. The formulation given in this work is based in calculations known from classical mechanics, but with the addition of the necessary…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
An alternative numerical method is developed to find stable and unstable periodic orbits of nonlinear dynamical systems. The method exploits the high-efficiency of the Levenberg-Marquardt algorithm for medium-sized problems and has the…
We extensively describe our recently established "divide-and-conquer" semiclassical method [M. Ceotto, G. Di Liberto and R. Conte, Phys. Rev. Lett. 119, 010401 (2017)] and propose a new implementation of it to increase the accuracy of…
We study the resonant dynamics in a simple one degree of freedom, time dependent Hamiltonian model describing spin-orbit interactions. The equations of motion admit periodic solutions associated with resonant motions, the most important…
This paper extends our previous work~(Szumi\'nski and Maciejewski, 2024), where we explored the dynamics and integrability of the double-spring pendulum. Here, we investigate the variable-length double pendulum, a three-degree-of-freedom…
We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the…
We survey dynamic logics for specifying and verifying properties of dynamical systems, including hybrid systems, distributed hybrid systems, and stochastic hybrid systems. A dynamic logic is a first-order modal logic with a pair of…
We introduce a new coherent state expansion of the exponential representation of the S-matrix for the classical gravitational two-body problem. By combining the Kosower-Maybee-O'Connell (KMOC) formalism with the Dirac bracket structure…
Periodic solutions of the three body problem are very important for understanding its dynamics either in a theoretical framework or in various applications in celestial mechanics. In this paper we discuss the computation and continuation of…
Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose a method for semiclassical quantization based upon the Pade approximant to the periodic orbit sums. The Pade…
By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain…
We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a…
A 1:2 internally resonant mechanical system can undergo secondary Hopf (Neimark-Sacker) bifurcations, resulting in a quasi-periodic response when the system is subject to harmonic excitation. While these quasi-periodic orbits have been…
We consider methods based on the topological degree theory to compute periodic orbits of area preserving maps. Numerical approximations of the Kronecker integral and the application of Stenger's method allows us to compute the value of the…
A set of algorithms is presented for efficient numerical calculation of the time evolution of classical dynamical systems. Starting with a first approximation for solving the differential equations that has a "reversible" character, we show…
By double ideal quotient, we mean $(I:(I:J))$ where ideals $I$ and $J$. In our previous work [11], double ideal quotient and its variants are shown to be very useful for checking prime divisor and generating primary component. Combining…
We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational…