Simple nonlinear jerk functions with periodic solutions

Abstract

The appearances of jerk, the third time derivative of displacement, in several papers of mathematical and physical content are reviewed. Two simple nonlinear jerk functions, all of whose solutions are periodic, are introduced and solved. The first has period dependent on initial acceleration but independent of amplitude, and is related to the second-order cubic oscillator. The second, with intrinsically third-order dynamics, has period dependent also on initial displacement. A possible bearing on chaotic jerk functions is mooted.

Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations

“…Third-order, autonomous, nonlinear, ordinary differential equations are also known as jerk equations because the derivative of the acceleration with respect to time is referred to as the ''jerk'' [1,2], may have periodic or limit cycle solutions [3,4] and may exhibit chaos [5][6][7] because their phase space is three-dimensional.…”

Vibration control by nonlocal feedback and jerk dynamics

“…3, the same procedure is applied to the nonlinear oscillator (7)(8). We demonstrate that the most important difference with the van der Pol oscillator is the presence of phase-locked solutions.…”

“…Linz studied the connection between one-dimensional jerk dynamics and nonlinear dynamical systems in three-dimensional phase space [5,6]. Gottlieb [7] found periodic solutions and limit cycles [8,9] for some simple jerk equations by means of harmonic balance methods. Wu et al [10] proposed an improved harmonic balance method for nonlinear jerk equations, while Ma et al [11] and Hu [12] used a first-order harmonic balance procedure with a parameter perturbation technique.…”

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

“…(5) which can be determined iteratively and we shall be interested in obtaining approximately the periodic solutions of Eq. (5). To this end, we first add to the left-and right-hand sides of Eq.…”

“…Nonlinear, autonomous, third-order ordinary differential equations may also exhibit periodic or limit cycle behavior [5]. This behavior has been analyzed in the past by means of harmonic balance methods [6,7], linearized harmonic balance procedures [8], asymptotic perturbation techniques which combine the harmonic balance procedure and the method of multiple time scales [9], the Linstedt-Poincaré procedure [10], the method of averaging of Krylov-Bogoliubov-Mitropolskii [10][11][12], parameter-perturbation Linstedt-Poincaré techniques [13,14] which employ an artificial or book-keeping parameter and expand both the solution and some constants that appear (or are introduced) in the differential equation in terms of this parameter, artificial parameter-Linstedt-Poincaré techniques [15,16] based on the introduction of a linear term proportional to the unknown frequency of oscillation and a new independent variable and the use of either the third-order equation or a system of a first-order and a second-order ordinary differential equations, etc.…”