Related papers: Dynamical systems and \sigma-symmetries
Learning complex trajectories from demonstrations in robotic tasks has been effectively addressed through the utilization of Dynamical Systems (DS). State-of-the-art DS learning methods ensure stability of the generated trajectories;…
In this paper, autonomous differential equations type delta of order one on integral domains are defined. For this we will use the autonomous ring defined on the Hurwitz expansion ring of exponential generating functions with coefficients…
We present an exact dimensional reduction for high-dimensional dynamical systems composed of $N$ identical dynamical units governed by quasi-linear ordinary differential equations (ODEs) of order $M$. In these systems, each unit follows a…
A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the so-called 'max-formula' is proven. The new concepts are used to characterize solutions of convex…
This paper presents methods for using zonotopes and constrained zonotopes to improve the practicality of a wide variety of set-based operations commonly used in control theory. The proposed methods extend the use of constrained zonotopes to…
We review the notion of reducibility and we introduce and discuss the notion of orbital reducibility for autonomous ordinary differential equations of first order. The relation between (orbital) reducibility and (orbital) symmetry is…
A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants.…
Uncovering the underlying ordinary differential equations (ODEs) that govern dynamic systems is crucial for advancing our understanding of complex phenomena. Traditional symbolic regression methods often struggle to capture the temporal…
We develop a formalism for performing real space renormalization group transformations of the "decimation type" using perturbation theory. The type of transformations beyond $d=1$ is nontrivial even for free theories. We check the formalism…
We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…
In this paper, we use what we call the shift operator so that general delay dynamic equations of the form \[ x^{\Delta}(t)=a(t)x(t)+b(t)x(\delta_{-}(h,t))\delta_{-}^{\Delta}% (h,t),\ \ \ t\in\lbrack t_{0},\infty)_{\mathbb{T}}% \] can be…
In this paper we consider an alternative approach to "un-reduction". This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions…
It is shown how the dimension of any arbitrary over-determined system of differential equations can be reduced, which makes the system suitable for numerical solution modeling. Specifically, over-determined equations of hydrodynamics are…
Stiff systems of ordinary differential equations (ODEs) are pervasive in many science and engineering fields, yet standard neural ODE approaches struggle to learn them. This limitation is the main barrier to the widespread adoption of…
Mechanical systems are usually modeled by second-order Ordinary Differential Equations (ODE) which take the form $\ddot{q} = f(t, q, \dot{q})$. While simulation methods tailored to these equations have been studied, using them in direct…
Many numerical methods for multiscale differential equations require a scale separation between the larger and the smaller scales to achieve accuracy and computational efficiency. In the area of multiscale dynamical systems, so-called,…
This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $\lambda$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in…
Q-conditional symmetries of the classical Lotka-Volterra system in the case of one space variable are completely described and a set of such symmetries in explicit form is constructed. The relevant non-Lie ans\"atze to reduce the classical…
Simulation of many-particle system evolution by molecular dynamics takes to decrease integration step to provide numerical scheme stability on the sufficiently large time interval. It leads to a significant increase of the volume of…
Optimization algorithms can be interpreted through the lens of dynamical systems as the interconnection of linear systems and a set of subgradient nonlinearities. This dynamical systems formulation allows for the analysis and synthesis of…