Nonlinear Time-discrete Systems: A General Approach by by Dr. sc. nat. M. Göossel (eds.)

By Dr. sc. nat. M. Göossel (eds.)

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Xt), t > O. Explicitly this will be done in the cases 2. F(Xl,X 2) = x I + x 2, G(Xl,X 2) = I + x I + x 2 = x I [] x 2 and 5. F(Xl,X 2) = G(Xl,X 2) = x I + x 2 + XlX 2 = x I v x 2. The other eases are similar. ,x t) is supposed to be in its antivalent normal form t ~--t ct gt(xl .... 'xt) = Co + ci xi + > ij xixj + "'" l~i~t 1~i~j~t • .. °t XlX 2 ... x t. 48 (19) Let F(x 1,x 2) = x I + x 2 ana O(x 1,x 2) = I + x I + x 2 (case 2). -,x t) (20) h(x 1,:x 2) = 1 + BlX 1 + B2x2 = g~(xl,x~2,F(x3,x3),o o .

X i + ~--aij,x i . xj + ... 1~i~t l

X~,x~ ..... ,x t o x~) = h~(x I o x ~ v ... ~ h ~ = h~(~1) ~ ~ ( x ~ = h~(x I) V ~ x~ E X t o x~) = ~ h~Cx t) ~ h~(x~) = ... V h ~ ( x t) V h ~ ( x ~ ) V ... V h ~ ( x ~ ) = = gt(xl,. ,x t) V gt(x~,-- . ,x't ), with h it • H o v , i = I .... ,t. b. Let ~ be (o,v)-superponable. Then for every t > 0 we define h (x) = g÷(e ..... e, x, e ... e) i -- I, .... t. (59) i - I Then we have ht(x o x') = g t ( ~ , x o x', e ..... e) = gt(e o e .... ,e o e) ..... o) . i- I ..... i- = ), I i. e. ,t and ht(x I) v ...

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