“Discrete” Oscillations and multiple attractors in kick‐excited systems

Abstract

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by nonlinear (inhomogeneous) external periodic excitation (as regards to the coordinates of excited systems) and is remarkable for its objective regularities: the phenomenon of “discrete” (“quantized”) oscillation excitation and strong self-adaptive stability. The main features of these systems are studied both numerically and analytically on the basis of a general model: a pendulum under inhomogeneous action of a periodic force which is referred to as a kicked pendulum. Multiple bifurcation diagram for the attractor set of the system under consideration is obtained and analyzed. The complex dynamics, evolution and the fractal boundaries of the multiple attractor basins in state space corresponding to energy and phase variables are obtained, traced and discussed. A two-dimensional discrete map is derived for this case. A general treatment of the class of kick-excited self-adaptive dynamical systems is made by putting it in correspondence to a general class of dissipative twist maps and showing that the latter is an immanent tool for general description of its behavior.

An experimental system for studying the plane pendulum in physics laboratory teaching

“…This effect was first observed by Béthenod [20,25] who considered a pendulum with a metallic sphere at the tip of the pendulum interacting with a current carrying coil (solenoid) oriented with its axis vertical and placed below the pendulum, and by Penner et al [18] who considered an arrangement similar to the one realised here, also known as Doubochinski's pendulum. Several other arrangements have been explored [19][20][21], showing the similar effect of large amplitude oscillations at selected frequencies.…”

“…The physical reason for the effect is a balancing of the frictional energy loss and the energy gain from the exciting force during each oscillation (see e.g. [19]). In a given steadystate motion the pendulum loses a specific amount of energy due to friction in air and at the ball bearing (see figure 6(c)) during a period of oscillation…”

“…Further, driven oscillations can be realised through a simple measurable magnetic coupling and oscillations near the natural frequency as well as chaotic motions in dependence of the forcing parameters (frequency and amplitude) can readily be observed and investigated. Also, the physics of resonances near higher odd harmonics of the natural frequency(ies) (also termed Doubochinski's pendulum [18][19][20][21]) can be demonstrated and characterised.…”

Investigation of argumental oscillations of a physical pendulum

“…In their recent paper, Cintra and Argoul [2] systematically summarizes six particular variations of magnetically driven argumental oscillations of a pendulum corresponding to the type of receptor (steel sphere or permanent magnet) and the orientation (vertical or horizontal) of the exciting solenoid. Recent analyzes of argumental oscillation of pendulums have focused on chaotic motion [6] in a kicked-system (Π-shaped excitation function), the dynamics toward the stabilization of argumental motion [2,3,7], and multi-stability [8] resulting from the complex nature of the driving torque.…”