Analytic solution

(83) |

The linearized equations of motion of the pendulum take the form:

Suppose that the pendulum's position, , and velocity, , are specified at time . As is well-known, in this case, the above equations of motion can be solved analytically to give:

Here,

(88) |

It is often convenient to *visualize* the motion of a dynamical system as an orbit, or trajectory, in
*phase-space*, which is defined as the space of all of the dynamical variables required to
specify the instantaneous state of the system. For the case in hand, there are two dynamical
variables, and , and so phase-space corresponds to the - plane. Note that
each different point in this plane corresponds to a unique instantaneous state of the pendulum.
[Strictly speaking, we should also consider to be a dynamical variable, since it
appears explicitly on the right-hand side of Eq. (82).]

It is clear, from Eqs. (86) and (87), that if we wait long enough for all
of the transients to decay away then the motion of the pendulum settles down to the
following simple orbit in phase-space:

(89) | |||

(90) |

This orbit traces out the closed curve

(91) |

As illustrated in Fig. 22, this curve is an

(93) |

The phase-space curve shown in Fig. 22 is called a *periodic attractor*. It is
termed an ``attractor'' because,
irrespective of the initial conditions, the trajectory of the system in phase-space tends
asymptotically to--in other words, is attracted to--this curve as
. This
gravitation of phase-space trajectories towards the attractor is illustrated in Figs. 24 and
25. Of course, the attractor is termed ``periodic'' because it corresponds to motion which is
periodic in time.

Let us summarize our findings, so far. We have discovered that if a damped pendulum is
subject to a low amplitude, periodic, drive then its *time-asymptotic* response (*i.e.*,
its response after any transients have died away) is *periodic*, with the same period
as the driving torque. Moreover, the response exhibits *resonant* behaviour as the driving
frequency approaches the natural frequency of oscillation of the pendulum. The amplitude
of the resonant response, as well as the width of the resonant window, is governed by the
amount of damping in the system. After a little reflection, we can easily appreciate that
all of these results are a direct consequence of the *linearity* of the pendulum's
equations of motion in the low amplitude limit. In fact,
it is easily demonstrated that the time-asymptotic response of *any* intrinsically
stable linear system (with a discrete spectrum of normal modes) to a periodic drive is periodic,
with the same period as the drive. Moreover, if the driving frequency approaches
one of the natural frequencies of oscillation of the system then the response exhibits
resonant behaviour. But, is this the only allowable time-asymptotic response of a dynamical system to a
periodic drive? Most undergraduate students might be forgiven for answering this question
in the affirmative. After all, the majority of undergraduate classical dynamics courses focus almost
exclusively on linear systems. The correct answer, as we shall see, is no. The response of
a *non-linear* system to a periodic drive is generally far more rich and
diverse than simple periodic motion. Since the majority of naturally
occurring dynamical systems are non-linear, it is clearly important that we gain a
basic understanding of this phenomenon. Unfortunately, we cannot achieve this goal via
a standard analytic approach--non-linear equations of motion generally do not possess
simple analytic solutions. Instead, we must use *computers*. As an example, let us
investigate the dynamics of a damped pendulum, subject to a periodic drive, with *no
restrictions* on the amplitude of the pendulum's motion.