to plasma physics, is the book by Sagdeev et al. [21]. Also, in problems related to engineering and control, the study of
the pendulum is of much importance. For a textbook in which the basic characteristics of chaos are introduced with the
help of the simple pendulum (see [22]).
The present work attempts to give a unified view of different known aspects of the pendulum when suffering distinct
external perturbations. In particular we are interested in considering the general case of the plane simple pendulum,
whose pivot is subjected to having different motions on the plane. For the case of the spherical pendulum, that is not
included in this work (see [23–25]). In principle, our main interest lies in the possibility to integrating, in one single
model, several known particular situations, such as the familiar forced pendulum, the harmonically forced pendulum
[26–28], the vertically forced pendulum and combinations of them, such as the rotating pendulum. Moreover, we introduce
in the expression of the generalized pendulum nonlinear damping terms, which in spite of its interest for many
practical purposes, they are seldom used. All these models have been studied separately by many authors, but a general
scheme as the one we are offering here is clearly lacking. Besides the forced pendulum, perhaps the case which has
deserved more attention in the literature is the pendulum with a vertically oscillating pivot. This system was apparently
first studied by Stephenson in 1908 [29] and somewhat later, in the twenties, by van der Pol and Strutt [30]. A good
treatment of the inverted pendulum may be found in [31,32]. One outstanding interest in it relies upon its stability
properties (see for instance [33,34] and the many references therein).
Once this generalized expression of the equations of motion of the simple pendulum, which we call a generalized
perturbed pendulum, is obtained, different possible avenues of further study may be open. The strategy that we have
followed here is based on the approach given by the Melnikov method, which typically applies to continuous dynamical
systems. This method gives some conditions for the chaotic motion of these dynamical systems, which basically are
related to the topological behavior of the invariant manifolds associated to hyperbolic saddle points in phase space. As
a natural consequence of the use of this method to the generalized perturbed pendulum, some formulae related to the
chaotic behavior (homoclinic chaos) of the pendulum are given, which comprise most particular cases that one may
have taken into consideration.
2. Equations of motion of the generalized perturbed pendulum
Here we attempt to give a general formulation of the simple pendulum, where different forcing and damping terms
are included in a single expression, with the aim of offering an overview of various situations that a pendulum may have
and portrait all of them in a common framework. From this perspective, several familiar cases including external
perturbations appear in a natural way, as particular cases of this generalized equation.
A simple mathematical pendulum is modelled by a bob of mass m, hanging at the end of a wire of length l and fixed
to a supporting point O (see Fig. 1), swinging to and fro in a vertical plane.
The equations of motion are straightforward to obtain using Lagrangian or Newtonian methods. For its simplicity,
we show here the pendulum equations using Newtonian methods. In this framework it is much more intuitive to visualize
the forces acting on the system, providing a more clearer physical picture of the dynamics of the pendulum, even
though other general formulations are possible. In this context Fig. 1 shows the force diagram of the simple pendulum,