Next, let us consider the quasi-periodically forced Duffing oscillator By means of the variables u$_{1}$ = u and u$_{2}$ = u ′ one can reduce (5.12) to the system We perturb (5.13) with the solutions of (5.11), and set up the following system, System (5.14) is in the form of (1.1) with and The eigenvalues of the matrix A are − 3 4 ± $_{i}$√$_{23}$ 4 , and the function G ( t, y$_{1}$, y$_{2}$ ) is quasi-periodic in t uniformly for ( y$_{1}$, y$_{2}$ ) ∈ R $^{2}$. One can confirm that where Therefore, ∥ e At ∥ ≤ Ne − $^{ωt}$, t ≥ 0 , where N = ‖ P ‖ ∥ P − 1 ∥ ≈ 2 . 0029 and ω = 3 / 4 . In order to show that system (5.13) possesses an asymptotically stable quasi-periodic solution, let us 11 u ′′ + 3 2 u ′ + 2 u + 0 . 0001 u 3 = 3 2 sin ( t √$_{2}$ ) + 1 2 sin( πt ) . (5.12) u ′ 1 = u$_{2}$ u ′ 2 = − 2 u$_{1}$ − 3 2 u$_{2}$ − 0 . 0001 u 3 $_{1}$+ 3 2 sin ( t √$_{2}$ ) + 1 2 sin( πt ) . (5.13) y ′ 1 = y$_{2}$ + 6 arctan( x$_{1}$ ( t )) y ′ 2 = − 2 y$_{1}$ − 3 2 y$_{2}$ − 0 . 0001 y 3 1 + 3 2 sin ( t √$_{2}$ ) + 1 2 sin( πt ) + 2 x$_{2}$ ( t ) . (5.14) A =    0 1 − 2 − 3 2    , G ( t, y$_{1}$, y$_{2}$ ) =    0 − 0 . 0001 y 3 1 + 3 2 sin ( t √$_{2}$ ) + 1 2 sin( πt )    , H ( x$_{1}$, x$_{2}$ ) =    6 arctan( x$_{1}$ ) 2 x$_{2}$    . e At = e − 3 t/ $^{4}$P    cos ( √$_{23}$ 4 t ) − sin ( √$_{23}$ 4 t ) sin ( √$_{23}$ 4 t ) cos ( √$_{23}$ 4 t )    P − $^{1}$, P =    0 1 √$_{23}$ 4 − 3 4    .