3 - Regular motion

Summary

The simplest motion occurs in one-dimensional systems subjected to time-independent forces. The most important characteristics of regular (non-chaotic) behaviour will be demonstrated by means of such motion, but this also provides an opportunity for us to formulate some general features. The overview starts with the investigation of the dynamics around unstable and stable equilibrium states, where the essentials already appear in a linear approximation. Outside of a small neighbourhood of the equilibrium state, however, non-linear behaviour is usually present, which manifests itself, for example, in the co-existence of several stable and unstable states, or in the emergence of such states as the parameters change. We monitor the motion in phase space and become acquainted with the geometrical structures characteristic of regular motion. The unstable states, and the curves emanated from such hyperbolic points, the stable and unstable manifolds, play the most important role since they form, so to say, the skeleton of all possible motion. In the presence of friction, trajectories converge to the attractors of the phase space. For regular motion, attractors are simple: equilibrium states and periodic oscillations, implying fixed point attractors and limit cycle attractors, respectively.

Instability and stability

Motion around an unstable state: the hyperbolic point

Let us start the analysis – contrary to the traditional approach – with the behaviour at and in the vicinity of an unstable equilibrium state.

An equilibrium state of a body at some position x* is unstable if, when released from a slightly displaced position, the body starts moving further away from x*.