By Nomizu K., Sasaki T.

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**Example text**

Xn are given functions, real and uniform depending on x1 , . . , xn , analytical with respect of these variables, and where t is the independent variable. Variables x1 , . . ” This is exactly the same deﬁnition as previously proposed by Henri Poincar´e (1886, p. 168) in one of his famous memoirs entitled: Sur les courbes d´eﬁnies par une ´equation diﬀ´erentielle. “. . , dxn = Xn dt where Xi are real polynomials. ” — H. Poincar´e — The notion of dynamical system is the mathematical description of the dynamics of a given physical, mechanical, electronic, biological, ecological, economical system from the point of view of a deterministic process which is expressed in terms of state variables, making it possible to deﬁne the instantaneous state of the system, and equations of evolution of these variables between an initial and ﬁnal instant.

2004, p. 311) there is no universally accepted deﬁnition of an attractor. Moreover, fractal dimension of the attractor implies that the phase space dimension might be greater than (or equal to) three. Sensitive dependence on initial conditions is the hallmark of chaotic system. So, strange attractor are often called chaotic attractor. 6. 12), V a volume of the phase space at time t = 0 and V (t) = Φt (V ) the image of V by Φt , then: dV (t) dt n ∇· = t=0 V Proof. 13) Cf. Dang-Vu (2000, p. 23) ; Hubbard and West (1991).

Another kind of attractor is the limit cycle and although there is no analogue for the Poincar´e-Bendixson’s theorem for proving their existence high-dimensional dynamical systems may have limit cycles. 5. , fn (X)]t ∈ E ⊂ Rn deﬁnes in E a vector ﬁeld whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem with the ﬂow Φt . A set Λ is called an attractor if: • Λ is compact and invariant. • There is an open set U containing Λ such that for each X ∈ Λ, Φt (X) ∈ U for all t ≥ 0 and ∩t≥0 Φt (U ) = Λ.