of the averaging technique, there is class of system extensively studied which give us meaningful examples we will discuss below. The class of system is given by: where is smooth. This system is similar to a linear system with a small nonlinear perturbation given by : differing from the standard form. Hence there is a necessity to perform a transformation to make it in the standard form explicitly. We are able to change coordinates using variation of constants method. We look at the unperturbed system, i.e. , given by which has the fundamental solution corresponding to a rotation. Then the time-dependent change of coordinates is where is the coordinates respective to the standard form. If we take the time derivative in both sides and invert the fundamental matrix we obtain Remarks The same can be done to time-dependent linear parts. Although the fundamental solution may be non-trivial to write down explicitly, the procedure is similar. See Sanders et al. for further details. If the eigenvalues of are not all purely imaginary this is called hyperbolicity condition. For this occasion, the perturbation equation may present some serious problems even whether is bounded, since the solution grows exponentially fast. However, qualitatively, we may be able to know the asymptotic solution, such as Hartman-Grobman results and more. Occasionally in order to obtain standard forms which are easier to work on, we may choose a rotating reference frame set of coordinates - polar coordinates - given by which determines the initial condition as well, and defines system: If we average it as long as a neighborhood of the origin is excluded (since the polar coordinates fail) yields: where the averaged system is Example: Misleading averaging results The method contains some assumptions and restrictions. These limitations play important role when we average the original equation which is not into the standard form, and we can discuss counterexample of it. The following example in order to discourage this hurried averaging: where we put following the previous notation. This systems corresponds to a damped harmonic oscillator where the damping term oscillates between and . Averaging the friction term over one cycle of yields the equation: The solution is which the convergence rate to the origin is . The averaged system obtained from the standard form yields: which in the rectangular coordinate shows explicitly that indeed the rate of convergence to the origin is differing from the previous