Related papers: Dynamics of four families of methods with the same…
In this series of eight papers we present the applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. In this part we consider a model for spin-orbital motion: orbital…
An iterative formula based on Newton Method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method…
A nonlinear operator equation $F(x)=0$, $F:H\to H,$ in a Hilbert space is considered. Continuous Newton's-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point $x_0$ and becoming…
This work investigates a dynamical system functioning as a nonsmooth adaptation of the continuous Newton method, aimed at minimizing the sum of a primal lower-regular and a locally Lipschitz function, both potentially nonsmooth. The…
We focus on the solutions of second-order stable linear difference equations and demonstrate that their behavior can be non-monotone and exhibit peak effects depending on initial conditions. The results are applied to the analysis of the…
It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a…
We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes…
In this paper, we propose a novel, unified, general approach to investigate sufficient and necessary conditions under which four types of convex sets, polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets for a linear…
An inexact Newton type method for numerical minimization of convex piecewise quadratic functions is considered and its convergence is analyzed. Earlier, a similar method was successfully applied to optimizaton problems arising in numerical…
The object of the present paper is to extend the third-order iterative method for solving nonlinear equations into systems of nonlinear equations. Since our motive is to develop the method which improve the order of convergence of Newton's…
This article describes a method for constructing approximations to periodic solutions of dynamic Lorenz system with classical values of the system parameters. The author obtained a system of nonlinear algebraic equations in general form…
The nonlinear dynamics of a system with periodic structure can be analyzed using a square matrix. We show that because the special property of the square matrix constructed for nonlinear dynamics, we can reduce the dimension of the matrix…
We propose a novel and efficient lifting approach for the optimal control of rigid-body systems with contacts to improve the convergence properties of Newton-type methods. To relax the high nonlinearity, we consider the state, acceleration,…
This paper explores the analytical approach for obtaining the multiple solutions of three-wave interacting system in (1+1) dimensions. We present a novel approach by expressing the wave solutions in terms of Jacobi elliptic functions and…
Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends…
To develop a minimal model for a cell moving in a crowded environment such as in tissue, we investigate the response of a liquid drop of active matter moving on a flat rigid substrate to forces applied at its boundaries. We consider two…
Multibody systems usually give rise to complex nonlinear dynamics, and the bicycle is not an exception. Even a simple model as the Two-Mass-Skate presents a long expression of the kinetic energy, making difficult to write explicitly the…
We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction…
In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton's method on…
We present here a new method which applies well ordered symbolic dynamics to find unstable periodic and non-periodic orbits in a chaotic system. The method is simple and efficient and has been successfully applied to a number of different…