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
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 : ½n2 ; n2 : £ : £jn5 : n6 : £Þ hyperbolic point for the memorized linear discrete system. o (ii) The origin is a ð½nm 1 ; n1 : ½£; £ : £ : £jn5 : £ : £Þ-sink for the memorized linear discrete system. o (iii) The origin is a ð½£; £ : ½nm 2 ; n2 : £ : £j£ : n6 : £Þ-source for the memorized linear discrete system.
42 1 Memorized Linear Discrete Systems o m o (x) An ð½nm 1 ; n1 : ½n2 ; n2 : ½n3 þ k3 ; j3 : ½n4 þ k4 ; j4 jn5 : n6 : ½n7 þ k7 ; q; j7 Þ 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 : ½n2 ; n2 : ½n3 ; j3 : ½n4 ; j4 jn5 þ m o o k7 : n6 : ½n7 ; q; j7 Þ state and ð½n1 ; n1 : ½nm 2 þ k3 ; n2 þ k4 : ½n3 ; j3 : ½n4 ; j4 jn5 : n6 þ k7 : ½n7 ; q; j7 Þ state. 14 For a memorized linear discrete system of xk þ 1 ¼ anÂn xk þ bnÂn xkÀs in Eq.