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Our aim is to determine positive numbers ϵ$_{1}$, Δ and an interval J 1 ⊂ [0 , ∞ ) with length Δ such that the inequality ‖ y ( t ) − y ( t ) ‖ > ϵ$_{1}$ holds for all t ∈ J $^{1}$. 

It is clear that the collection of chaotic solutions of system (2.1) is an equicontinuous family on [0 , ∞ ) . Making use of the uniform continuity of the function g : R 3 × R 3 → R $^{3}$, defined as g ( ν$_{1}$, ν$_{2}$ ) = g ( ν$_{1}$ ) − g ( ν$_{2}$ ) , on the compact region R = { ( ν$_{1}$, ν$_{2}$ ) ∈ R 3 × R 3 : ‖ ν$_{1}$ ‖ ≤ H$_{0}$, ‖ ν$_{2}$ ‖ ≤ H$_{0}$ } together with the equicontinuity of the collection of chaotic solutions of (2.1), one can verify that the collection F consisting of the functions of the form g$_{j}$ ( x$_{1}$ ( t )) − g$_{j}$ ( x$_{2}$ ( t )) , j = 1 , 2 , 3 , where x$_{1}$ ( t ) and x$_{2}$ ( t ) are chaotic solutions of system (2.1), is an equicontinuous family on [0 , ∞ ) . 

According to the equicontinuity of the family F , one can find a positive number τ < Δ , which is independent of x ( t ) and x ( t ) , such that for any t$_{1}$, t$_{2}$ ∈ [0 , ∞ ) with | t$_{1}$ − t$_{2}$ | < τ, the inequality 

<img src='content_image/325155.jpg'>

holds for all j = 1 , 2 , 3 . 

On the other hand, for each t ∈ J, there exists an integer j$_{0}$, 1 ≤ j$_{0}$ ≤ 3 , which possibly depends on t, such that 

Otherwise, if there exists s ∈ J such that the inequality 

holds for each j = 1 , 2 , 3 , then one encounters with a contradiction since 

<img src='content_image/325158.jpg'>

<img src='content_image/325156.jpg'>

<img src='content_image/325157.jpg'>

Denote by s$_{0}$ the midpoint of the interval J, and let θ = s$_{0}$ − τ/ 2 . There exists an integer j$_{0}$, 1 ≤ j$_{0}$ ≤ 3 , such that 

<img src='content_image/325159.jpg'>

Using the inequality (6.6) it can be verified for all t ∈ [ θ, θ + τ ] that 

<img src='content_image/325160.jpg'>

