By Albert C. J. Luo
This ebook examines discrete dynamical structures with memory—nonlinear platforms that exist largely in organic organisms and monetary and monetary firms, and time-delay platforms that may be discretized into the memorized, discrete dynamical platforms. It e-book additional discusses balance and bifurcations of time-delay dynamical structures that may be investigated via memorized dynamical structures in addition to bifurcations of memorized nonlinear dynamical structures, discretization tools of time-delay structures, and periodic motions to chaos in nonlinear time-delay systems.
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Extra resources for Memorized Discrete Systems & Time-Delay
0 0 0 0 0 .. ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ .. 0 ÁÁÁ 0 0 0 .. 0 0 0 .. 0 0 0 .. 0 I2Â2 0 0 0 0 3 0 07 7 07 7 ; 07 7 .. 5 0 0 2lÂ2l 0 0 0 0 .. ÁÁÁ 0 ÁÁÁ 0 ÁÁÁ 0 .. . ÁÁÁ 0 ÁÁÁ 0 3 ð1:110Þ 0 07 7 07 .. 7 7 05 0 2lÂ2l where N ¼ diagð02ðiÀ1ÞÂ2ðiÀ1Þ ; Nj 2 lÂ2 l ; 02ðmÀiÀl þ 1ÞÂ2ðmÀiÀl þ 1Þ Þ; 2 2 N ¼ diagð02ðiÀ1ÞÂ2ðiÀ1Þ ; N j2 lÂ2 l ; 02ðmÀiÀl þ 1ÞÂ2ðmÀiÀl þ 1Þ Þ; .. 3 Iterative Solutions with Multiplicity 27 From the previous discussion, the solutions for homogenous discrete systems were presented for distinct and repeated eigenvalues.
12 For a memorized linear discrete system of xk þ 1 ¼ axk þ Rsj¼1 bj xkÀj in Eq. 2) with the initial state xk and memorized state xkÀj ðj ¼ 1; 2; Á Á Á ; sÞ, the corresponding equivalent system is yk þ 1 ¼ Ayk with a matrix AmÂm ðm ¼ nðs þ 1ÞÞ. The matrix A possesses m eigenvalues ki ði ¼ 1; 2; Á Á Á ; mÞ. Set N ¼ f1; 2; Á Á Á ; l; l þ 1; Á Á Á ; ðm þ lÞ=2g; Nb ¼ fj1 ; j2 ; Á Á Á ; jnb g [ £ with jp 2 Nðp ¼ 1; 2; Á Á Á ; nb ; b ¼ 1; 2; Á Á Á ; 7Þ; R4b¼1 nb ¼ l and 2R7b¼5 nb ¼ m À l: [ 7b¼1 Nb ¼ N with Nb \ Nc ¼ £ ðc 6¼ o bÞ:Nb ¼ £ if nb ¼ 0.
5 For Xa Rn and KRl with a 2 Z, consider a vector function f a : Xa Â XaÀ1 Â Á Á Á Â XaÀs Â K ! Xa þ 1 where xkÀj 2 XaÀj (j ¼ 0; 1; Á Á Á ; s) and the resultant memorized vector yk ¼ ðxk ; xkÀ1 ; Á Á Á ; xkÀs Þ 2 Xa : The memorized vector function f a ðyk ; pa Þ f a ðxk ; xkÀ1 ; Á Á Á xkÀs ; pa Þ is called to satisfy the Lipschitz condition if jjf a ðzk ; pa Þ À f a ðyk ; pa Þjj Ljjzk À yk jj with yk ; zk 2 Xa . The constant L is called the Lipschitz constant. 6 For a memorized, discrete system xk þ 1 ¼ f a ðxk ; xkÀ1 ; Á Á Á ; xkÀs ; pa Þ, with yk ¼ ðxk ; xkÀ1 ; Á Á Á ; xkÀs Þ and Fa ¼ ðf a ; xk ; Á Á Á ; xk þ 1Às Þ, the equivalent discrete system is yk þ 1 ¼ Fa ðyk ; pa Þ.