Perturbed damped pendulum: finding periodic solutions

Details Perturbed damped pendulum: finding periodic solutions Perturbed damped pendulum: finding periodic solutions

2011-12-09

arxiv.org/abs/1112.2129v1

Mathematics

Dynamical Systems

Scientific paper

Using the equation of motion of the damped pendulum, we introduce the averaging method on the study of periodic orbits of dynamical systems with small perturbation. We provide sufficient conditions for the existence of periodic solutions of the perturbed damped pendulum with small oscillations having equations of motion \[ \ddot{\T}=-a\T-b\dot{\T}+\e f(t,\T,\dot{\T}), \] where a>0, b>0 and $\e$ are real parameters, with a=g/l, g the acceleration of the gravity, l the length of the rod and b the damped coefficient. Here the parameter $\e$ is small and the smooth function f is T-periodic in t. The averaging theory provides a useful means to study dynamical systems, accessible to Master and PhD students.

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