crude averaged system: Example: Van der Pol Equation Van der Pol was concerned with obtaining approximate solution for equations of the type where following the previous notation. This system is named Van der Pol oscillator. If we apply periodic averaging to this nonlinear oscillator, this give us qualitative knowledge of the phase space without solving explicitly the system. The averaged system is and we can analyze the fixed points and their stability. There is an unstable fixed point at the origin and a stable limit cycle represented by . The existence of such stable limit-cycle can be stated as a theorem. Theorem (Existence of a periodic orbit): If is a hyperbolic fixed point of Then there exists such that for all , has a unique hyperbolic periodic orbit of the same stability type as . The proof can be found at Guckenheimer and Holmes, Sanders et al. and for the angle case in Chicone. Example: Restricting the time interval The average theorem assumes existence of a connected and bounded region which affects the time interval of the result validity. The following example points it out. Consider the where . The averaged system consists of which under this initial condition indicates that the original solution behaves like where it holds on a bounded region over . Damped Pendulum Consider a damped pendulum whose point of suspension is vibrated vertically by a small amplitude, high frequency signal (this is usually known as dithering). The equation of motion for such a pendulum is given by where describes the motion of the suspension point, describes the damping of the pendulum, and is the angle made by the pendulum with the vertical. The phase space form of this equation is given by where we have introduced the variable and written the system as an autonomous, first-order system in -space. Suppose that the angular frequency of the vertical vibrations, , is much greater than the natural frequency of the pendulum, . Suppose also that the amplitude of the vertical vibrations, , is much less than the length of the pendulum. The pendulum's trajectory in phase space will trace out a spiral around a curve , moving along at the slow rate but moving around it at the fast rate . The radius of the spiral around will be small and proportional to . The average behaviour of the trajectory, over a timescale much larger than ,