Nonlinear Phenomena in the Dynamics of a Class of Rolling Pendulums: A Trigger of Coupled Singularities: Plenary Review Lecture

Abstract

In the introductory part of the plenary lecture, an overview of nonlinear differential equations of heavy ball and heavy thin dick rolling along curvilinear paths and surfaces of different shapes were presented. This is reason that this content is omitted from present review paper. In the introductory part of this article, we will present nonlinear phenomena of motion of a heavy material point moving along a rotating, smooth circle around a vertical, central or eccentric axis, as well as around an eccentric oblique axis relative to the vertical, at a constant angular velocity. Using linear and nonlinear approximations of the nonlinear differential equation in the vicinity of singular points of the observed dynamics, the analysis of the local dynamics of the heavy material point system along the rotating circle around the oblique axis is given. A mathematical analogy between this model and the model of the dynamics of a thin heavy disk rolling in a rotating circle around an eccentric-centric oblique axis is pointed out. Using linear and nonlinear approximations of the nonlinear differential equation in the vicinity of the singular points of the observed dynamics, the analysis of the local dynamics of the heavy material point system along the rotating circle around the oblique axisat a constant angular velocity is given. A mathematical analogy between this model and the dynamics model of a thin heavy disk rolling in a rotating circle around an eccentric-centric vertical-oblique axis is pointed out. The central and main subject of the paper is the identification and presentation of nonlinear phenomena in the nonlinear dynamics of a class of generalized rolling pendulums, whose heavy bodies roll along curvilinear paths, lying in a vertical plane, rotating around a vertical axis, at a constant angular velocity. The bifurcation parameter of coupled rotations is identified. The bifurcation of the position of stable equilibrium of the generalized rolling pendulum and the corresponding representative singular points of the type of the stable center is described, as well as the stratification and transformation of phase trajectories in the phase portrait of nonlinear dynamics of the generalized rolling pendulum in the Earth's gravitational field, and along curvilinear route in rotate vertical plane around vertical axis at a constant angular velocity. Additionally a theorem of trigger of coupled singularities and a homoclinic orbit in the form of the number “eight” is graphically proofed. A series of graphs of characteristic equation oh nonlinear dynamics as well as series of phase portraits for different coefficients of curvilinear paths described by parabola, bi-quadratic parabola or polynomials of the eighth degree is presented and sets of transformed phase trajectories and homoclinic orbits in the form of the number “eight” are presented, which include one or more triggers of coupled singular points in nonlinear dynamics of relative rolling thin heavy disk along these curvilinear trace in rotate vertical plane around vertical axis at a constant angular velocity.