Abstract
M.Sc. (Applied Mathematics)
Amongst the very interesting, exciting and beautiful topics in mathematics
we have dynamical systems which can be defined as the branch of mathematics
that attempts to understand processes in motion. Since such systems
are constantly changing in time, predicting where the system is heading or
where it will ultimately go is a major concern. Some dynamical systems
are predictable while others are not. The culprit, the cause of this unpredictable
behavior, has been called "chaos" by mathematicians [18]. This
piece of work attempts to analyze the behavior of an excellent example of a
chaotic dynamical system known as the double pendulum [19].
The study involved a frictionless rigid pendulum of mass m2 and length
l2, attached to the bottom of another frictionless rigid pendulum of mass
m1 and length l1. The term rigid pendulum implies a mass on the end of
a mass-less rigid rod. After obtaining a suitable mathematical model that
could describe the motion of our rigid pendulum, firstly, some assumptions
were made to analyze its motion in the case of small oscillations. Secondly, in
the case of large oscillations, various numerical simulations were conducted
using the Fourth-order implicit Runge-Kutta method. Some existing C++
codes were useful to characterize different states of this dynamical system:
from the calculation of Lyapunov exponents to implementing the energy of
the system as a bifurcation parameter.
Interestingly, different results were obtained in the two cases of oscillation.
In the first case namely when released at small oscillations, the double
pendulum exhibits a regular and repeated motion, while in the second case,
strange and unpredictable behavior was observed. This means the system
exhibits a chaotic motion when released at large initial angles. Furthermore,
no exact energy level was obtained at which above or below it, a regular or
chaotic motion was exhibited by the system independent of initial conditions.
Hence, it can be said that energy is not a bifurcation parameter,
that is there is no energy at which motion changes topology, from regular
to chaotic motion.