content_image/310253.jpg content_image/310252.jpg content_image/310251.jpg content_image/310249.jpg content_image/310250.jpg
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 

<img src='content_image/310253.jpg'>

<img src='content_image/310252.jpg'>

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 

<img src='content_image/310251.jpg'>

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. 

## 6 Unpredictable solutions of differential equations 

Consider the differential equation 

where the function ν ( t ) is defined as 

<img src='content_image/310249.jpg'>

<img src='content_image/310250.jpg'>

