混沌嵌入與全局隱函數定理

Abstract

在這個計畫中，我們有三個目標。首先，對於擾動方程我們將建立全局隱函數定理，這個版本有別於 [4,6]裡的定理。我們的證明將使用contraction mapping principal 和其他成分。 其次，我們將研究下列高階差分方程解的複雜性 Φ(x_{i-m}，...，x_{i-1}，x_{i}，x_{i+1}，...，x_{i+n})= 0，i∈Z， 其中Φ是一個從(R^ ℓ)^ {m+n+1}到R^ ℓ 的C¹ 函數。我們的主要結果將提供一個充分條件使得任何Φ函數的C¹ 小擾動都具有符號混沌嵌入。當Φ取決於少個變數時，譬如有名的logistic 和 Henon函數族，我們的充分條件能夠很容易地被驗證。我們的證明將使用第一階段完成的全局隱函數定理及Brouwer固定點定理。我們的新方法不需要依賴雙曲性，將推廣 [1,2,5,7] 的結果。 最後，跟隨著[3,8]，我們將研究任意有限圖耦合映射網絡。使用耦合映射網絡的覆蓋關係及非擾動網絡的拓撲熵，我們將提供擾動耦合映射網絡拓撲熵的下界。我們不假設局部動態的雙曲性，也不假設耦合的線性化，所得結論的應用相當廣。

In this project, we have three goals. First, we establish a global version of the implicit function theorem for perturbed equations, which is different from the ones in [4, 6]. Our proof will be based on the contraction mapping principle and other ingredients. Second, we study complexity of solutions of a high-dimensional difference equation of the form #8;(xi−m, . . . , xi−1, xi, xi+1, . . . , xi+n) = 0, i 2 Z, where #8; is a C1 function from (R`)m+n+1 to R`. Our main result will provide a sufficient condition for any sufficiently small C1 perturbation of #8; to have symbolic embedding, that is, to possess a closed set of solutions #3; that is invariant under the shift map, such that the restriction of the shift map to #3; is topologically conjugate to a subshift of finite type. The sufficient condition can be easily verified when #8; depends on few variables, including the logistic and H´enon families. The proof of the main result is based on the Brouwer fixed point theorem and the global implicit function theorem we established first. We do not assume hyperbolicity and such a novel approach will extend results in [1, 2, 5, 7]. Third, following [3, 8], we study coupled map networks over arbitrary finite graphs. An estimate from below for a topological entropy of a perturbed coupled map network via a topological entropy of an unperturbed network by making use of the covering relations for coupled map networks is obtained. The result will be quite general, particularly no assumptions on hyperbolicity of a local dynamics or linearity of coupling are made. References [1] J. Juang, M.-C. Li, and M. Malkin, Chaotic difference equations in two variables and their multidimensional perturbations, Nonlinearity 21 (2008), 1019-1040. [2] T. Kamihigashi, Chaotic dynamics in quasi-static systems: theory and applications, J. Mathematical Economics 31 (1999), 183-214. [3] Koiller, J. and Young, L.-S., Coupled map networks, Nonlinearity 23 (2010), 1121-1141. [4] S. G. Krantz and H. R. Parks, The Implicit Function Theorem: History, Theory, and Applications, Birkh¨auser, Boston, 2002. [5] M.-C. Li and M. Malkin, Topological horseshoes for perturbations of singular difference equations, Nonlinearity 19 (2006), 795-811. [6] I. W. Sandberg, Global implicit function theorems, IEEE Transactions on Circuits and Systems, Cas-28 (1981), 145-149. 1 [7] D. Sterling, H. R. Dullin, and J. D. Meiss, Homoclinic bifurcations for the Henon map, Phys. D 134 (1999), 153-184. [8] Young, L.-S., Chaotic phenomena in three settings: large, noisy and out of equilibrium, Nonlinearity 21 (2008), T245-T252.