By Milos Marek, Igor Schreiber

Surveying either theoretical and experimental points of chaotic habit, this booklet provides chaos as a version for plenty of doubtless random techniques in nature. simple notions from the speculation of dynamical structures, bifurcation idea and the homes of chaotic suggestions are then defined and illustrated by means of examples. A evaluate of numerical equipment used either in reports of mathematical types and within the interpretation of experimental info can also be supplied. moreover, an intensive survey of experimental commentary of chaotic habit and strategies of its research are used to emphasize common beneficial properties of the phenomenon.

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**Example text**

B0 0 ... 0 A A2 B ... 0 .. , B6 := .. . . . α−2 A B ... B Aα 33 C CA CA2 .. 49) In Chap. 6 we will use this 1D model to characterize so-called pass proﬁle controllability in terms of rank tests on matrices with constant entries. 9. 4 2D Transfer-Function and Related Representations In 1D linear systems theory, the transfer-function (or transfer-function matrix) and similar representations, such as matrix fraction and matrix factorization descriptions, play a central role.

J1 . . 0 0 0 · · · J2 0 0 0 0 J3 J4 .. 0 ··· ··· ··· .. .. 43) (which is structurally similar to the Fornasini-Marchesini state-space model) can be replaced by a dynamically equivalent 1D time-varying linear systems state-space model. 44) expand as q˜ increases. This fact alone has greatly reduced the value of the 1D equivalent model in 2D linear systems theory. e. the matrices and vectors involved are of constant dimensions and have constant entries. 10. Then the equivalent 1D model can be obtained (see, for example, [67]) by a number of routes.

3. Suppose that the linear repetitive process S is asymptotically stable and let {bk }k≥1 be a disturbance sequence that converges strongly to a disturbance b∞ . Then the strong limit y∞ := lim yk k→∞ is termed the limit proﬁle corresponding to {bk }k≥1 . 2. Suppose that the linear repetitive process S is asymptotically stable and let {bk }k≥1 be a disturbance sequence that converges strongly to a disturbance b∞ . 9) Proof. 9). 9) in the form (I−Lα )y∞ = b∞ and noting, by asymptotic stability, that r(Lα ) < 1 and hence (I − Lα ) has a bounded inverse in Eα .