One can define a metric d on C ( R ) as [4] where h$_{1}$, h$_{2}$ belong to C ( R ) and ρ$_{k}$ ( h$_{1}$, h$_{2}$ ) = min { 1 , sup$_{s}$$_{∈}$$_{[}$$_{−}$$_{k,k}$$_{]}$ ‖ h$_{1}$ ( s ) − h$_{2}$ ( s ) ‖ } , k ∈ N . Let us define the mapping π : R + × C ( R ) → C ( R ) by π ( t, h ) = h$_{t}$, where h$_{t}$ ( s ) = h ( t + s ) . The mapping π defines a semi-flow on C ( R ) and it is called the Bebutov dynamical system [4]. We describe an unpredictable function as follows. Definition 3.1 An unpredictable function is an unpredictable point of the Bebutov dynamical system. According to Theorem III. 3 [4], a motion π ( t, h ) lies in a compact set if h is a bounded and uniformly continuous function. Assuming this, by means of Theorem 2.2, we obtain that an unpredictable function h determines chaos if it is bounded and uniformly continuous. On the basis of this result, one can say that if a differential equation admits an unpredictable solution which is uniformly continuous and bounded, then chaos is present in the set of solutions. In the next section, we will prove the existence of an unpredictable solution whose quasi-minimal set is a chaotic attractor. Consider the following quasi-linear system, where the m × m constant matrix A has eigenvalues all with negative real parts, the function f : R m → R m is continuous and g : R → R m is a uniformly continuous and bounded function. Since the eigenvalues of the matrix A have negative real parts, there exist positive numbers K and ω such that ‖ e $^{At}$‖ ≤ Ke − $^{ωt}$, t ≥ 0 [13]. The following conditions are required. The main result of the present study is mentioned in the next theorem. Theorem 4.1 Suppose that the conditions ( C 1) − ( C 3) are valid. If the function g ( t ) is unpredictable, then system (4.2) possesses a unique uniformly exponentially stable unpredictable solution, which is uni- formly continuous and bounded on R . 4 (C3) KL$_{f}$ − ω < 0 . (C2) There exists a positive number L$_{f}$ such that ‖ f ( x$_{1}$ ) − f ( x$_{2}$ ) ‖ ≤ L$_{f}$ ‖ x$_{1}$ − x$_{2}$ ‖ for all x$_{1}$, x$_{2}$ ∈ R $^{m}$; (C1) There exists a positive number M$_{f}$ such that sup x ∈ R m ‖ f ( x ) ‖ ≤ M$_{f}$ ; x ′ = Ax + f ( x ) + g ( t ) , (4.2) 4 Unpredictable Solutions of Quasi-linear Systems d ( h$_{1}$, h$_{2}$ ) = ∞ ∑ k =1 2 − $^{k}$ρ$_{k}$ ( h$_{1}$, h$_{2}$ ) , (3.1)