content_image/302539.jpg content_image/302540.jpg content_image/302541.jpg content_image/302543.jpg content_image/302542.jpg content_image/302544.jpg content_image/302545.jpg content_image/302546.jpg
Next, let us consider the quasi-periodically forced Duffing oscillator 

<img src='content_image/302539.jpg'>

By means of the variables u$_{1}$ = u and u$_{2}$ = u ′ one can reduce (5.12) to the system 

<img src='content_image/302540.jpg'>

We perturb (5.13) with the solutions of (5.11), and set up the following system, 

<img src='content_image/302541.jpg'>

System (5.14) is in the form of (1.1) with 

and 

<img src='content_image/302543.jpg'>

<img src='content_image/302542.jpg'>

<img src='content_image/302544.jpg'>

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 

<img src='content_image/302545.jpg'>

<img src='content_image/302546.jpg'>

