Hence, lim n $_{→∞}$ρ ( ψ$_{t}$$_{n}$, ψ ) = 0 . Next, we will show the existence of a positive number ϵ$_{0}$ such that ρ ( ψ$_{t}$$_{n}$$_{+}$$_{τ}$$_{n}$, ψ$_{τ}$$_{n}$ ) ≥ ϵ$_{0}$ for each n ∈ $^{N}$. Denote β = min { 1 , L$_{1}$ } . For each k ∈ $^{N}$, we have that Thus, the inequality holds for each n ∈ $^{N}$, where ϵ$_{0}$ = βϵ$_{0}$. Consequently, the function ψ ( t ) is unpredictable. □ A corollary of Theorem 5.2 is as follows. Corollary 5.1 If φ : R → H is an unpredictable function, where H is a bounded subset of R $^{p}$, then the function ψ : $^{R}$→ R p defined as ψ ( t ) = P φ ( t ) , where P is a constant, nonsingular, p × p matrix, is also an unpredictable function. Proof. The function h : H → R p defined as h ( u ) = P u satisfies the inequality for u$_{1}$, u$_{2}$ ∈ H with L$_{1}$ = 1 / ∥ P − 1 ∥ and L$_{2}$ = ‖ P ‖ . Therefore, by Theorem 5.2, the function ψ ( t ) is unpredictable. □ In the next section, the existence of Poincaré chaos in the dynamics of differential equations will be presented. Consider the differential equation where the function ν ( t ) is defined as 11 ν ( t ) =     0 . 7 , if ζ$_{2}$$_{j}$ < t ≤ ζ$_{2}$$_{j}$$_{+1}$, j ∈ $^{Z}$, − 0 . 4 , if ζ$_{2}$$_{j}$ − 1 < t ≤ ζ$_{2}$$_{j}$, j ∈ $^{Z}$. (6.10) x $^{′}$( t ) = − 3 2 x ( t ) + ν ( t ) , (6.9) L$_{1}$ ‖ u$_{1}$ − u$_{2}$ ‖ ≤ ‖ h ( u$_{1}$ ) − h ( u$_{2}$ ) ‖ ≤ L$_{2}$ ‖ u$_{1}$ − u$_{2}$ ‖ , ρ ( ψ$_{t}$$_{n}$$_{+}$$_{τ}$$_{n}$, ψ$_{τ}$$_{n}$ ) ≥ βρ ( φ$_{t}$$_{n}$$_{+}$$_{τ}$$_{n}$, φ$_{τ}$$_{n}$ ) ≥ ϵ$_{0}$ ρ$_{k}$ ( ψ$_{t}$$_{n}$$_{+}$$_{τ}$$_{n}$, ψ$_{τ}$$_{n}$ ) = min { 1 , sup s ∈ [ − k,k ] ‖ h ( φ ( t$_{n}$ + τ$_{n}$ + s )) − h ( φ ( τ$_{n}$ + s )) ‖ } ≥ min 1 , L$_{1}$ sup s ∈ [ − k,k ] ‖ φ ( t$_{n}$ + τ$_{n}$ + s ) − φ ( τ$_{n}$ + s ) ‖ } ≥ βρ$_{k}$ ( φ$_{t}$$_{n}$$_{+}$$_{τ}$$_{n}$, φ$_{τ}$$_{n}$ ) . 6 Unpredictable solutions of differential equations