±oo (e) lim Ix(t)1 t---+oo (f) = lim Ix(t)1 t---+-oo = 00; 00, lim x(t) does not exist; t---+-oo = 00, lim x(t) does not exist. t-+oo Hint: See Problem 5 in Problem Set 9. 9. 9 Stability Theory In this section we define the stable, unstable and center subspaces, ES, EU and E C respectively, of a linear system x=Ax.

En] where the 2 x 2 blocks Ej for j = k + 1, ... , n. ~] We illustrate this result with an example. Example. 6. Complex Eigenvalues has eigenvalues Al eigenvectors VI Thus = = -3, A2 = 2 + i (and m ~2 = 2 - i). The corresponding and w, ~ u, + iv, ~ [1 0 1 0 p~ [~ and l p-l= P-'AP~ n [~ 0 1 0 r i ]. l ~ -~] The solution of the initial value problem (1) is given by e- 3t x(t) = P [ 0 o e- 3t = [ 0 o 0 e2t cos t e 2t sin t 0] p-lxo _e 2t sin t e 2t cos t 0 e 2t(cos t + sin t) e 2t sin t 0 _2e 2t sin t e 2t ( cos t - sin t) 1Xo.

### Nonlinear systems analysis by Vidyasagar M.

by Mark

4.3