##### From Chaos, Bifurcations And Fractals Around Us: A Brief Introduction, Series A, Vol. 47

## 3.1. Equation of Motion, Linear and Weakly Nonlinear Oscillations

Consider a physical model exemplified by the plane pendulum depicted in Figure 3.1. The pendulum consists of a heavy, small-diameter ball with mass *m* suspended on a rigid and very light rod of length *l*. The rod can rotate around the horizontal axis *O*. It follows that the ball can move along a circle in a vertical plane, and its position is determined by a single coordinate, for instance, by the angular displacement denoted as *x* in Figure 3.1. The motion of the ball is ruled by the gravity force *mg,* the damping force *P* * _{t}*, and the moment of external periodic forces applied to the axis of rotation,

*M( ?)*. The considered physical model is often regarded as a satisfactory approximation of many technical devices.

Fig. 3.1: Mechanical model of the forced pendulum.

A physical experimental investigation of motion of the pendulum and, in particular, measurements of the sought position *x* and the velocity *v=dx/dt* of the ball, is not a convenient tool in the study of chaotic phenomena. Instead, we may apply a numerical approach, as the computer simulation enables us to find the output of the system. Numerical procedures also allow us to obtain unstable solutions , that is the solutions that, although unrealizable in any physical experiment, play an important role in the analysis of system behavior and the related concepts. This will lead us to the discussion...

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