The method of averaging

Techniques of averaging in the theory of differential equations have interested me for a long time. It happens that when determining the asymptotics of certain solutions it is important to show that certain integrals are finite although these integrals are not absolutely convergent. For this there must be a suitable cancellation of positive and negative contributions. Back in 2007 I published a paper (Class. Quantum Grav. 24, 667) where I studied this kind of phenomenon in certain inflationary cosmological models. There I did not use any general techniques but instead I just derived estimates by hand. More recently I have spent some time learning what general techniques there are. One famous method is the Krylov-Bogoliubov averaging method. Next semester I will organize a seminar on the subject of the method of averaging.

An iconic example which is a good starting point for discussing this subject is the Kapitza pendulum. Suppose that a rigid rod is attached to a support about which it can rotate freely in a vertical plane. If the support is fixed we get an ordinary pendulum. The steady state where the rod is vertically above the support is obviously unstable. The Kapitza pendulum is obtained by supposing that instead the support undergoes oscillations in the vertical direction with small amplitude and large frequency. For suitable choices of the parameters this stabilizes the unstable steady state of the ordinary pendulum.

How can the situation just described be understood mathematically? I follow here the discussion in Hale’s book on ordinary differential equations. The basic equation of motion is second order. If friction is ignored the equations can be put in Hamiltonian form which means reducing them to a system of two first order equations in a certain way. Introducing a rescaled time coordinate leads to a system of the form x'=\epsilon f(t,x)+\epsilon h(\epsilon t,x) which is a standard form for the method of averaging. The system contains two time scales t and \epsilon t. We now average over the fast time t, defining f_0(x)=\frac{1}{T}\int_0^T f(t,x) dt. Then the original equation is replaced by the averaged equation x'=\epsilon f_0(x)+\epsilon h(\epsilon t,x). Here T is the period of the oscillation. The definition of f_0 given by Hale is more complicated since he wants to allow more general motions of the support which might be only almost periodic. The question is now to what extent solutions of the original equation can be approximated by solutions of the averaged equation for \epsilon small. Write the right hand side of the averaged equation in the form \epsilon G(\epsilon t,x). Suppose that the averaged equation has a periodic solution. We linearize the averaged equation about that solution. This gives rise to characteristic exponents, which are the exponential growth rates of linearized perturbations. If none of these characteristic exponents is purely imaginary then we obtain an existence theorem for solutions of the original equation which are small perturbations of the periodic solutions of the averaged equation. If in addition all characteristic exponents have negative real parts then the perturbed solution is asymptotically stable and if there is a characteristic exponent with positive real part it is unstable. It is a separate problem to prove the existence of a periodic solution of the averaged equation.


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