Related papers: Typical dynamics of Newton's method

In this paper, we present a Newton-like method based on model reduction techniques, which can be used in implicit numerical methods for approximating the solution to ordinary differential equations. In each iteration, the Newton-like method…

We introduce new classes of hydrodynamic theories inspired by the recently discovered fracton phases of quantum matter. Fracton phases are characterized by elementary excitations (fractons) with restricted mobility. The hydrodynamic…

We present a one-parameter family $F_\lambda$ of transcendental entire functions with zeros, whose Newton's method yields wandering domains, coexisting with the basins of the roots of $F_\lambda$. Wandering domains for Newton maps of…

We investigate the trajectory of an arbitrary $(2,1)$-rational $p$-adic dynamical system in a complex $p$-adic field $\C_p$. (i) In the case where there is no fixed point we show that the $p$-adic dynamical system has a 2-periodic cycle…

Newtonian dynamical systems accepting the normal shift on an arbitrary Riemannian manifold are considered. Partial differential equations forming the weak and additional normality conditions for them are reported.

Motivated by the growing interest in the nonmetricity-matter couplings, we develop the scalar-tensor formulation of recently introduced $f(Q,T)$ gravity, where $Q$ is the nonmetricity and $T$ is the trace of the energy-momentum tensor. The…

Invariance and stability are essential notions in dynamical systems study, and thus it is of great interest to learn a dynamics model with a stable invariant set. However, existing methods can only handle the stability of an equilibrium. In…

We survey the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. These are results on global attraction to stationary states, to solitons and to stationary orbits, on adiabatic…

For more than three decades, clear discrepancies have existed between spin densities in momentum space revealed by Magnetic Compton scattering experiments and theoretical calculations based on density functional theory (DFT). Here by making…

We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit…

A stochastic version of the Noether Theorem is derived for systems under the action of external random forces. The concept of moment generating functional is employed to describe the symmetry of the stochastic forces. The theorem is applied…

In the first half of the paper, some recent advances in coupled dynamical systems, in particular, a globally coupled map are surveyed. First, dominance of Milnor attractors in partially ordered phase is demonstrated. Second, chaotic…

In continuum mechanics, the equations of motion for mixtures are derived through the use of Hamilton's extended principle which regards the mixture as a collection of distinct continua. The internal energy is assumed to be a function of…

We present a formalism for Newtonian multi-fluid hydrodynamics derived from an unconstrained variational principle. This approach provides a natural way of obtaining the general equations of motion for a wide range of hydrodynamic systems…

Variational principle for a solid in classical mechanics is formulated in terms of a thin elastic 4D bar strain in Minkowsky events space of special relativity. It is shown, that the sum of elastic 4-energies of weak twist and bending under…

This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions,…

Quantum state wave functionals are constructed in exact form for the graviton-like field theory obtained by breaking down the topological symmetry of the string action related with the Euler characteristic of the world-surface; their…

In this work the issue of whether key energetic properties (nonlinear, exponential-type dissipation in the abscence of forcing and long-term stability under conditions of time dependent loading) are automatically inherited by deforming…

This article studies typical dynamics and fluctuations for a slow-fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we…

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental…