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where z ( t ) = ( z$_{1}$ ( t ) , z$_{2}$ ( t ) , z$_{3}$ ( t )) ∈ R $^{3}$, B =    − 3 2 − 1 0 − 5 / 2 0 2 − 11 / 2 − 1   $^{,}$ and ψ ( t ) is the unpredictable solution of (6.11). In system (6.15), the function f : R 3 → R $^{3}$, f ( u ) = ( f$_{1}$ ( u ) , f$_{2}$ ( u ) , f$_{3}$ ( u )) , is defined as f$_{1}$ ( u ) = 0 . 03 sin u$_{1}$, f$_{2}$ ( u ) = 0 . 04 u 2 for | u$_{2}$ | ≤ 1 , f$_{2}$ ( u ) = 0 . 04 for | u$_{2}$ | > 1 , f$_{3}$ ( u ) = 0 . 06 tanh u$_{3}$, and the function h : R 3 → R $^{3}$, h ( u ) = ( h$_{1}$ ( u ) , h$_{2}$ ( u ) , h$_{3}$ ( u )) , is defined as h$_{1}$ ( u ) = 2 arctan( u$_{1}$ ) , h$_{2}$ ( u ) = u$_{3}$ +0 . 1 u 3 $_{3}$, h$_{3}$ ( u ) = 0 . 5 u 2 $_{2}$, where u = ( u$_{1}$, u$_{2}$, u$_{3}$ ) . 

The eigenvalues of the matrix B are − 5 / 2 , − 2 ± i, and e Bt = P e $^{Jt}$P − $^{1}$, where J =    − 5 / 2 0 0 0 − 2 − 1 0 1 − 2    and P =    1 0 − 1 1 / 2 0 0 1 / 2 1 1   $^{.}$ One can verify that ∥ e Bt ∥ ≤ K$_{0}$e − $^{ωt}$, t ≥ 0 , with K$_{0}$ = ‖ P ‖ ∥ P − 1 ∥ ≈ 7 . 103 and ω = 2 . The function f ( u ) is bounded and it satisfies the Lipschitz condition ‖ f ( u ) − f ( u ) ‖ ≤ L$_{f}$ ‖ u − u ‖ , u, u ∈ R $^{3}$, with L$_{f}$ = 0 . 08 such that the inequality K$_{0}$L$_{f}$ − ω < 0 is valid. 

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On the other hand, the function h ( u ) satisfies the conditions of Theorem 5.2 with L$_{1}$ = 0 . 3 and L$_{2}$ = 2 . 083 inside the region H defined by (6.14) so that h ( x ( t )) is an unpredictable function if x ( t ) is an unpredictable function. Therefore, according to Theorem 2.2, system (6.15) possesses a unique uniformly exponentially stable unpredictable solution. 

Let us denote by θ ( t ) the solution of (6.11) whose trajectory is shown in Figure 3. To demonstrate the extension of Poincaré chaos, we take into account the system 

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