Related papers: Dynamical systems and \sigma-symmetries
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
Recent progress concerning regularization of supersymmetric theories is reviewed. Dimensional reduction is reformulated in a mathematically consistent way, and an elegant and general method is presented that allows to study the…
Spectral decompositions for the evolution operator on an energy shell in phase space are constructed for the free motion on compact 2D surfaces of constant negative curvature. Applications to quantum chaos and in particular to the recently…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
Several interesting physical systems, such as the Lovelock extension of General Relativity in higher dimensions, classical time crystals, k-essence fields, Horndeski theories, compressible fluids, and nonlinear electrodynamics, have…
Optimizing problems in a distributed manner is critical for systems involving multiple agents with private data. Despite substantial interest, a unified method for analyzing the convergence rates of distributed optimization algorithms is…
Given a dynamical system with $m$ independent conserved quantities, we construct a multi-parameter family of new systems in which these quantities evolve monotonically and proportionally, and are replaced by $m-1$ conserved linear…
The study of the behavior of solutions of ODEs often benefits from deciding on a convenient choice of coordinates. This choice of coordinates may be used to "simplify" the functional expressions that appear in the vector field in order that…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
Conservation of energy is at the core of many physical phenomena and dynamical systems. There have been a significant number of works in the past few years aimed at predicting the trajectory of motion of dynamical systems using neural…
Despite the advancements in learning governing differential equations from observations of dynamical systems, data-driven methods are often unaware of fundamental physical laws, such as frame invariance. As a result, these algorithms may…
It is shown that the presence of Lie-point-symmetries of (non-Hamiltonian) dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical sytem (or vector field) into Poincar\'e-Dulac normal form.
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an…
Using the procedure initiated in \cite{Ma2013}, we deform Lax-type equations though a scaling of the time parameter. This gives an equivalent (deformed) equation which is integrable in terms of power series of the scaling parameter. We then…
Classical Bianchi-Lie, Backlund and Darboux transformations are considered. Their generalizations for the dynamical systems are discussed. For the transformation being the generalization of the normal shift the special class of dynamical…
The paper proposes an algorithm which could identify a general class of pdes describing dynamical systems with similar symmetries. The way that will be followed starts from a given group of symmetries, the determination of the invariants…
We investigate higher-dimensional $\Delta$-systems indexed by finite sets of ordinals, isolating a particular definition thereof and proving a higher-dimensional version of the classical $\Delta$-system lemma. We focus in particular on…
We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.
We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an…
We present two types of meta-algorithm that can greatly improve the accuracy of existing algorithms for integrating the equations of motion of dynamical systems. The first meta-algorithm takes an integrator that is time-symmetric only for…