Related papers: Fractional Zaslavsky and Henon Discrete Maps
We present a novel numerical framework for studying nonlinear dispersive equations in higher-dimensional settings, specifically designed for solutions featuring traveling waves along a preferred axis (or field-aligned traveling waves).…
Systems with predetermined Lyapunov functions play an important role in many areas of applied mathematics, physics and engineering: dynamic optimization methods (objective functions and their modifications), machine learning (loss…
This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet…
In this paper we consider a generalized classical mechanics with fractional derivatives. The generalization is based on the time-clock randomization of momenta and coordinates taken from the conventional phase space. The fractional…
In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also…
Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems.…
Disordered and amorphous materials often retain memories of perturbations they have experienced since preparation. Studying such memories is a gateway to understanding this challenging class of systems, yet it often requires the ability to…
Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…
We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a…
The main result of this paper is the discretization of Hamiltonian systems of the form $\ddot x = -K \nabla W(x)$, where $K$ is a constant symmetric matrix and $W\colon\mathbb{R}^n\to \mathbb{R}$ is a polynomial of degree $d\le 4$ in any…
The space of images can be equipped with a Riemannian metric measuring both the cost of transport of image intensities and the variation of image intensities along motion lines. The resulting metamorphosis model was introduced and analyzed…
We propose and analyze a class of second-order dynamical systems for continuous-time optimization that incorporate fractional-order gradient terms. The system is given by \begin{equation} \ddot{x}(t) + \frac{\alpha}{t}\dot{x}(t) +…
With the emergence of powerful representations of continuous data in the form of neural fields, there is a need for discretization invariant learning: an approach for learning maps between functions on continuous domains without being…
A theory of time and space with fractional dimensions (FD) of time and space ($d_{\alpha}, \alpha=t,{\bf r})$ defined on multifractal sets is proposed. The FD is determined (using principle of minimum the functionals of FD) by the energy…
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$…
The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…
We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour…
Fractional dissipation is a powerful tool to study non-local physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational…
In this article, I have outlined how an accomplished researcher like Robert Ziff has influenced a new generation of researchers across the globe like gravity as an action-at-a-distance. In the 80s Ziff made significant contributions to the…
We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the…