低維度擾動到高維度系統的混沌動態

Abstract

延續稍早前的論文 [1, 2, 3], 我們打算進一步研究低維度局部系統擾動到高維度系統的混沌問題， 分別以下列不同的假設情況，做出結論的推廣： (SP1) 當局部函數可化成高階差分方程時； (SP2) 當局部函數具有穩定和不穩定方向的雙曲不變集時； (SP3) 當局部函數具有二次錐條件的非雙曲不變集時； (SP4) 當局部函數具有偽軌性質時； (SP5) 當局部函數具有snap-back repeller拓撲屬性時； (SP6) 當低維度系統擾動到無限維系統時； (SP7) 應用上述所得結論到多維度網格系統、反應擴散系統離散模型、及實質的經濟模型等。 [1] M.-C. Li and M. Malkin, 2006, Topological horseshoes for perturbations of singular difference equations, Nonlinearity, 19, 795-811. [2] J. Juang, M.-C. Li, and M. Malkin, 2008, Chaotic difference equations in two variables and their multidimensional perturbations, Nonlinearity, 21, 1019-1040. [3] M.-C. Li, M.-J. Lyu and P. Zgliczynski, 2008, Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps, Nonlinearity, 21, 2555-2567.

Continuing our earlier works [1,2,3], we plan to study multidimensional perturbations from a low-dimensional local map to a high-dimensional map and generalize results in the following subprojects: (SP1) when the local difference equation is high-order; (SP2) when the local map has hyperbolic invariant set with both stable and unstable directions; (SP3) when the local map has non-hyperbolic invariant set with quadratic cone condition; (SP4) when the local map has shadowing property; (SP5) when the local map has topological property of snap-back repeller; (SP6) when the low-dimensional system is perturbed to infinite-dimensional ones; (SP7) when the above results are applied to high-dimensional lattice systems, numerical models of reaction-diffusion systems, and practical economic models, etc. [1] M.-C. Li and M. Malkin, 2006, Topological horseshoes for perturbations of singular difference equations, Nonlinearity, 19, 795-811. [2] J. Juang, M.-C. Li, and M. Malkin, 2008, Chaotic difference equations in two variables and their multidimensional perturbations, Nonlinearity, 21, 1019-1040. [3] M.-C. Li, M.-J. Lyu and P. Zgliczynski, 2008, Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps, Nonlinearity, 21, 2555-2567.