Nonlinear dynamical models and jerky motion

Abstract

We investigate the connection between one-dimensional Newtonian jerky dynamics and nonlinear dynamical systems in a three-dimensional phase space. With exact transformations, we show that the Rössler model, as well as the Lorenz model, can be interpreted as jerky motion and discuss whether they are Newtonian or not. Moreover, Sprott’s model R is identified as one of the simplest Newtonian jerky dynamics that can lead to chaos. Using a wide class of jerk functions, we derive a criterion for being Newtonian jerky.

New Concepts in Electromagnetic Jerky Dynamics and Their Applications in Transient Processes of Electric Circuit

“…SLG jerky dynamics has transcend traditional physics. It generalized the classical mechanical concept (jerk) into the mathematical jerk function or jerk equation and discussed the regular [17,18], chaotic [3,4] solutions and the relative rules [11,19]. Some summarizations and remarks have been given in Ref.…”

“…The concept is a natural extension of a line of thinking that originated with Galileo. In 1997, Linz [3] and Sprott [4] generalize the conception of jerk into the general jerk function (the third-order time derivative of any independent variable). The jerk functions that reflect jerky motion are determined by third-order differential equations.…”

A Simple Jerky Dynamics, Genesio System

“…The term jerk [1], i.e., the third derivative of displacement, x  , has attracted some attention because of its relevance to the theory of chaos [2][3][4][5][6][7][8][9][10][11]. Some papers appeared in response to a question [2] posed by Gottlieb concerning simple jerk functions which may lead to chaotic phenomena.…”

“…Sprott [3,4] found several simple nonlinear jerk functions which gave strange attractor for appropriate choices of equation parameters and initial conditions. Linz [5,6] introduced the idea and conditions for Newtonian jerky dynamics, derivable by differentiation of a (one-space dimension) Newtonian equation of motion for x  , and analyzed the jerky dynamics for onevariable obtained from several familiar autonomous systems of three simultaneous first-order ordinary differential equations which are known to have chaotic solutions. He also allowed for the possibility of a memory or temporal history integral term in the force function.…”

Chaotic and Hyperchaotic Complex Jerk Equations

“…By "jerk function" he means a function j such that the third-order ODE can be written in the form The jerk equations describe various phenomena in engineering and physics, for example, electrical circuits, laser physics, mechanical oscillators, damped harmonic oscillators, and biological systems. In the literature some jerk equations are introduced and studied [7][8][9][10][11][12][13]. Kocić et al [14] considered and studied two modifications of a 3-dimensional dynamic flow known as jerk dynamical systems of Sprott [13].…”