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 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 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 Using the inequality (6.6) it can be verified for all t ∈ [ θ, θ + τ ] that 10 | ( g$_{j}$ ( x ( t$_{1}$ )) − g$_{j}$ ( x ( t$_{1}$ ))) − ( g$_{j}$ ( x ( t$_{2}$ )) − g$_{j}$ ( x ( t$_{2}$ ))) | < L$_{g}$ϵ$_{0}$ 6 (6.6) | g$_{j}$$_{0}$ ( x ( t )) − g$_{j}$$_{0}$ ( x ( t )) | ≥ L$_{g}$ 3 ‖ x ( t ) − x ( t ) ‖ . | g$_{j}$ ( x ( s )) − g$_{j}$ ( x ( s )) | < L$_{g}$ 3 ‖ x ( s ) − x ( s ) ‖ ‖ g ( x ( s )) − g ( x ( s )) ‖ ≤ 3 ∑ j =1 | g$_{j}$ ( x ( s )) − g$_{j}$ ( x ( s )) | < L$_{g}$ ‖ x ( s ) − x ( s ) ‖ . | g$_{j}$$_{0}$ ( x ( s$_{0}$ )) − g$_{j}$$_{0}$ ( x ( s$_{0}$ )) | ≥ L$_{g}$ 3 ‖ x ( s$_{0}$ ) − x ( s$_{0}$ ) ‖ > L$_{g}$ϵ$_{0}$ 3 . (6.7) | g$_{j}$$_{0}$ ( x ( s$_{0}$ )) − g$_{j}$$_{0}$ ( x ( s$_{0}$ )) | − | g$_{j}$$_{0}$ ( x ( t )) − g$_{j}$$_{0}$ ( x ( t )) | ≤ | ( g$_{j}$$_{0}$ ( x ( t )) − g$_{j}$$_{0}$ ( x ( t ))) − ( g$_{j}$$_{0}$ ( x ( s$_{0}$ )) − g$_{j}$$_{0}$ ( x ( s$_{0}$ ))) |