Memorized Discrete Systems and Time-delay by Albert C. J. Luo

By Albert C. J. Luo

This booklet examines discrete dynamical platforms with memory—nonlinear structures that exist commonly in organic organisms and fiscal and monetary enterprises, 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 platforms that may be investigated via memorized dynamical platforms in addition to bifurcations of memorized nonlinear dynamical platforms, discretization tools of time-delay platforms, and periodic motions to chaos in nonlinear time-delay systems.

The ebook is helping readers locate analytical ideas of MDS, switch conventional perturbation research in time-delay platforms, notice movement complexity and singularity in MDS; and be certain balance, bifurcation, and chaos in any time-delay system.

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Extra resources for Memorized Discrete Systems and Time-delay

Example text

On the invariant eigenvector vk ¼ vi , consider ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ yk ¼ ck vi and yk þ 1 ¼ ck þ 1 vi ¼ ki ck vi , thus, ck þ 1 ¼ ki ck . 5 For a memorized linear discrete system of xk þ 1 ¼ anÂn xk þ bnÂn xkÀs 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. Consider a pair of complex eigenvalue ai Æ ibi of matrix Aði 2 N ¼ f1; 2; Á Á Á ; pffiffiffiffiffiffiffi ng; i ¼ À1Þ and there is a corresponding eigenvector ui Æ ivi .

Q ¼ ðq1 ; q2 ; Á Á Á ; qr ÞT and j7 ¼ ðj71 ; j72 ; Á Á Á ; j7r ÞT with j7j 2 f£; m7j gðj ¼ 1; 2; Á Á Á ; rÞ. m7 ¼ ðm71 ; m72 ; Á Á Á ; m7r ÞT : I. Nondegenerate cases o m o (i) The origin is a ð½nm 1 ; n1 Š : ½n2 ; n2 Š : £ : £jn5 : n6 : £Þ hyperbolic point for the memorized linear discrete system. o (ii) The origin is a ð½nm 1 ; n1 Š : ½£; £Š : £ : £jn5 : £ : £Þ-sink for the memorized linear discrete system. o (iii) The origin is a ð½£; £Š : ½nm 2 ; n2 Š : £ : £j£ : n6 : £Þ-source for the memorized linear discrete system.

42 1 Memorized Linear Discrete Systems o m o (x) An ð½nm 1 ; n1 Š : ½n2 ; n2 Š : ½n3 þ k3 ; j3 Š : ½n4 þ k4 ; j4 Šjn5 : n6 : ½n7 þ k7 ; q; j7 ŠÞ state of the origin for the memorized linear discrete system is a o m o boundary of its ð½nm 1 þ k3 ; n1 þ k4 Š : ½n2 ; n2 Š : ½n3 ; j3 Š : ½n4 ; j4 Šjn5 þ m o o k7 : n6 : ½n7 ; q; j7 ŠÞ state and ð½n1 ; n1 Š : ½nm 2 þ k3 ; n2 þ k4 Š : ½n3 ; j3 Š : ½n4 ; j4 Šjn5 : n6 þ k7 : ½n7 ; q; j7 ŠÞ state. 14 For a memorized linear discrete system of xk þ 1 ¼ anÂn xk þ bnÂn xkÀs in Eq.

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