隱函數定理的變形與其在差分系統的混沌結果

國 立 交 通 大 學

應用數學系

碩 士 論 文

隱函數定理的變形與其在差分系統的混沌結果

A version of the implicit function theorem and

its consequences for chaos of difference systems

研 究 生：謝俊鴻

指導老師：李明佳 教授

隱函數定理的變形與其在差分系統的混沌結果

A version of the implicit function theorem and

its consequences for chaos of difference systems

研 究 生: 謝俊鴻 Student：Chun-Hung Hsieh

指導教授: 李明佳 Advisor：Ming-Chia Li

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Mathematics June 2007

Hsinchu, Taiwan, Republic of China

隱函數定理的變形與其在差分系統的混沌結果

學生: 謝俊鴻 指導教授: 李明佳 教授

國立交通大學應用數學系(研究所)碩士班

摘 要

我們指一個差分系統的形式為G

(x

,…,x

,x

,x

,…,

x

)=0，此式G

為一個(μ+1+ν)個變數映至R

的函數，並且每

個變數屬於R

。我們考慮僅和x

變數有關的差分系統稱之為靜

態系統，並且引入在某些性質相當類似於靜態系統的差分系統

稱之為半靜態系統。我們提供隱函數定理的一個變形版本。我

們呈現再加一些條件下，一個靜態系統是混沌的。我們使用這

個隱函數定理的變形去呈現對於正則靜態系統的些微

C

擾動

下，混沌現象的穩定性。

A version of the implicit function theorem and

its consequences for chaos of difference systems

student：Chun-Hung Hsieh Advisor：Ming-Chia Li

Department (Institute) of Applied Mathematics

National Chiao Tung University

ABSTRACT

By a difference system, we mean a system of the form

t−µ, ..., xt−1, xt ,

xt+1 , ..., xt+ν ) = 0

, where each side of this equation is an

column vector

and

_with

_{. We consider a static system as}

a difference system that depends only on

and a quasi-static system as a

difference system that is in a certain sense relatively close to a static system.

We provide a

modified version of the implicit function theorem. We show that

under additional conditions, a static system is chaotic. We use this version of

the implicit function theorem to show the stability of chaos for regular static

system under small

_{perturbations.}

誌 謝

這篇論文的完成，首先要感謝我的指導老師 李明佳教授。

在這兩年來，老師除了在學問上的諄諄教誨令我收穫很多之外，

其對於研究事物的態度更是讓人敬佩，謹此致上我最誠摯的敬意

與謝意。口試期間，也承蒙陳國璋老師、陳怡全老師、莊重老師

費心審閱並提供了寶貴的意見，使得本論文得以更加的完備，永

誌於心。

在這兩年求學的過程中，感謝胡忠澤學長和呂明杰學長在我

遇到問題時，總是幫了很大的忙，在此獻上我最大的感謝之意。

感謝蕭亦廷同學和我一同為了論文奮鬥、互相加油打氣。

除此之外，更要感謝我研究所的全體同學們，謝謝他們讓我

的研究所生活更多采多姿，有了他們的陪伴與支持，讓我擁有這

些美好的回憶。

最後，要感謝我的家人的支持，以及女友雅羚這些日子的用

心陪伴。願與所有關心我的人一起分享這份喜悅，再次地感謝所

有幫助過我及關心過我的人，謝謝大家！

目

錄

中文提要

………

i

英文提要

………

ii

誌謝

………

iii

目錄

………

iv

一、

Introduction ………

1

二、

A version of the implicit function theorem

…………

2

三、

3.1

3.2

四、

4.1

4.2

Chaos

………

Chaos

………

Static system and chaos

………

Stability of chaos

………

Quasi-static system

………

Stability of chaos for regular static system under small

C

perturbations

………

9

9

10

14

14

15

五、

Appendix

………

19

References

………

20

1

Introduction

Consider a di¤erence system

G(xt 1, xt, xt+1) = 0. (1)

De…ne an orbit as a sequence fxtg satisfying Eq. (1) for all t 2 Z. Suppose G reduces

to a static system G , by which we mean that G is a function of xt alone:

G (xt 1, xt, xt+1) = F (xt).

In [1], we have the fact that if F (xt) = 0 has multiple solutions at which the

Jacobian matrix DF is nonsingular, then for G is in a certain sense relatively close to G , G displays chaotic dynamics.

The result will be based on simultaneous control of appropriate perturbation of static di¤erence system. We provide a modi…ed version of the implicit function the-orem which inspired by the concept of Li and Malkin in [2].

In this paper, section 2 presents a modi…ed version of the implicit function theorem and proof. Section 3 gives the de…nition of chaos and show that under additional con-ditions, a static system is chaotic. Section 4 de…nes quasi-static systems, establishes their properties, and presents stability of quasi-staticness and chaos.

2

A version of the implicit function theorem

Let x 2 Rn_{, x = (x}

1, x2,..., xn), k k₂ denotes the Euclidean norm, i.e. kxk₂ =

(x2

1 + x22+...+x2n)

1 2, k k

1 denotes the sup norm, i.e. kxk1 = max_{1 i n}jxij. In fact, for a

m n matrix B, kBk1= max_2Rn kB k1 k k₁ = max1 i m n X j=1 jBijj . Let m, n 2 N, for H Rm Rn_{, and C}1_(H

, Rn_{) denote the set of C}1 _functions

F : Dom(F )_{! R}n _{such that Dom(F ) H}

. Let E = fF jH: F 2 C1(H, Rn)g. For F1, F2 2 E, de…ne (F1, F2) max_{f sup} (y, z)2HkF 1(y, z) F2(y, z)k2, sup (y, z)2HkDF 1(y, z) DF2(y, z)k₁g,

then (E, ) is a metric space. We will use the notation U (x, r) and U [x, r] for the open and closed ball, respectively, of radius r centred at the point x 2 X, where X is a metric space. Let

N (F _{, H) = fF 2 E j (F , F ) < , Dom(F )} Dom(F )_g.

Theorem 1 _{Let m, n 2 N, Y R}m, Z _Rn, H _Rm _Rn_{, and F 2 E. Suppose:} (a1) Y , Z, and H are compact, and Y Z H.

(a2) There is a unique function f : Y ! Z such that for all y 2 Y , F (y, f (y)) = 0. (a3) f (Y ) _{the interior of Z and for any y 2 Y , D}2F (y, f (y)) is nonsingular.

Then

(c1) there exists a > 0 for any F _{2 N (F , H), there is a unique function} fF _{: Y}

! Z such that for any y 2 Y , F (y, fF_{(y)) = 0, and the unique}

(c2) sup

F 2N (F ,H)

(max_fsup

y2Y

fF(y) f (y) , sup

y2Y

DfF(y) Df (y) _{g) ! 0 as} # 0.

Proof of Theorem 1. Since f is continuous and Y is compact, we have f (Y ) is compact. From (a3), for any y 2 Y there exists a "y > 0such that U (f (y), "y) Z.

Because f (Y ) is compact, there exist "y1, "y2,..."yn; y1, y2,...yn such that f (Y )

[n

i=1U (f (yi), "yi). Let 0 = min_{1 i n}"yi. Denote V1 = U (F , 1), W 0 = [y2YU (f (y),

0). Then W 0 Z. For any y 2 Y , we de…ne a function gy : V1 U (f (y), 0)! R

n

by

gy(F, z) = z (D2F (y, f (y))) 1F (y, z)

then gy(F , f (y)) = f (y) and D2gy(F , f (y)) = 0. We denote Ty = D2F (y,

f (y))_{. By assumption (a3) and the map T : A ! A} 1 is continuous, there exists a constant M > 0 such that for any y 2 Y ,

(D2F (y, f (y))) 1 < M.

Since D2F is continuous on the compact set Y f (Y ), there exists a 1, 0 < 1 < minf_4M1 , 1, 0g such that for any y 2 Y ,

kD2F (y, z) D2F (y, f (y))k

kD2F (y, z) D2F (y, z)k + kD2F (y, z) D2F (y, f (y))k

< _{kDF (y, f (y))} DF (y_{, f (y))k +} 1 4M < (F, F ) + 1 4M < 1 2M 1 2 T 1 y

provided F 2 U[F , 1], z 2 U[f (y), 1].

And therefore

k(D2gy(F, z)k = I Ty 1D2F (y, z) = Ty 1Ty Ty 1D2F (y, z)

T_y 1 _kTy D2F (y, z)k

1 2.

By Mean Value Theorem applied to gy(F, ), For any F 2 U[F , 1], y 2 Y , and

any two points z1, z2 2 U[f (y), 1],

kgy(F, z1) gy(F, z2)k

1

2kz1 z2k .

Now, we choose a , 0 < < min_f2M1 1, 1g such that for any y 2 Y , and

F _{2 U[F , ],}

kF (y, f (y))k = kF (y, f (y)) F (y_{, f (y))k} (F, F ) < 1 2M 1 1 2 T 1 y 1, and therefore

kgy(F, f (y)) f (y)k = Ty 1 F (y, f (y))

T_y 1 _{kF (y, f (y))k <} 1 2 1. Thus for any y 2 Y , F 2 U[F , ] and z 2 U[f (y), 1]one has

kgy(F, z) f (y)k kgy(F, z) gy(F, f (y))k + kgy(F, f (y)) f (y)k

< 1

2kz f (y)k + 1

2 1 1.

This implies that for any y 2 Y and any (…xed) F 2 U[F , ], the map z ! gy(F,

z)is a contraction of the complete metric space U [f (y), 1]into itself. Hence by the

contraction mapping principle, there exists a unique …xed point, say _y(F ), and so gy(F, y(F )) = y(F ) or, equivalently, F (y, y(F )) = 0.

Given a F 2 U[F , ], for any y 2 Y there exists a unique y(F ) such that F (y, y(F )) = 0. We de…ne the function fF : Y ! Z by fF(y) = y(F ). Therefore, for

any F 2 N (F , H), there is a unique function fF : Y _{! Z such that for all y 2 Y ,} F (y, fF(y)) = 0. It remains to show that fF is C1.

For all F 2 N (F , H),

D2F (y, fF(y)) D2F (y, f (y)) <

1 2M <

1

M < D2F (y, f (y))

Hence D2F (y, fF(y)) is nonsingular for all y 2 Y . (see [3], p. 209)

Therefore for all F 2 N (F , H), there is a unique function fF _{: Y}

! Z such that for all y 2 Y , F (y, fF_{(y)) = 0, and D}

2F (y, fF(y)) is nonsingular. By implicit

function theorem (see [4], p. 374), the function fF _{is unique and C}1_{. We complete}

the proof of (c1).

Next, we prove the (c2). Let y 2 Y and F 2 N (F , H). Then

y(F ) y(F ) = gy(F, y(F )) gy(F , y(F )) gy(F, y(F )) gy(F, y(F )) + gy(F, y(F )) gy(F , y(F )) 1 2 y(F ) y(F ) + gy(F, y(F )) gy(F , y(F )) Thus y(F ) y(F ) 2 gy(F, y(F )) gy(F , y(F )) = 2 T_y 1(F (y, y(F ) F (y, y(F )) 2M F (y, y(F ) F (y, y(F ) 2M sup (y, z)2HkF (y, z) F (y_{, z)k} 2M _{! 0 as ! 0.} That is, sup F 2N (F ,H) sup y2Y fF(y) f (y) _{! 0 as ! 0.} (2) Now, we remain to prove

sup

F 2N (F ,H)

sup

y2Y

DfF(y) Df (y) _{! 0 as ! 0.}

that kD1F (y, z)k N1 for all (y, z) 2 Y Z. Let y 2 Y and F 2 N (F , H). Then kD1F (y, z)k kD1F (y, z)k kD1F (y, z) D1F (y, z)k kDF (y, z) DF (y_{, z)k} sup (y, z)2HkDF (y, z) DF (y_{, z)k} < 1. Thus kD1F (y, z)k 1 +kD1F (y, z)k 1 + N1 (3)

for any y 2 Y and F 2 N (F , H). Similarly, there exists a N2 > 0 such that

kD2F (y, z)k N2 for all (y, z) 2 Y Z and

kD2F (y, z)k 1 +kD2F (y, z)k 1 + N2

for any y 2 Y and F 2 N (F , H). Let y 2 Y and F 2 N (F , H). Then

[D2F (y, fF(y))] 1 [D2F (y, f (y))] 1

[D2F (y, fF(y))] 1 [D2F (y, f (y))] 1

[D2F (y, fF(y))] 1 D2F (y, fF(y)) D2F (y, f (y)) [D2F (y, f (y))] 1

[D2F (y, fF(y))] 1

1 2M M Thus

[D2F (y, fF(y))] 1 2 [D2F (y, f (y))] 1 2M

for any y 2 Y and F 2 N (F , H).

0 < 2 < such that for any y 2 Y

[D2F (y, fF(y))] 1 [D2F (y, f (y))] 1

[D2F (y, fF(y))] 1 D2F (y, fF(y)) D2F (y, f (y)) [D2F (y, f (y))] 1

2M D2F (y, fF(y)) D2F (y, f (y)) M

2M2( D2F (y, fF(y)) D2F (y, fF(y)) + D2F (y, fF(y)) D2F (y, f (y)) )

2M2( sup (y, z)2HkDF (y, z) DF (y_{, z)k + )} 2M2( 2 + ) provided F 2 N 2(F , H), f F_(y) 2 U[f (y), 2]. That is, sup F 2N 2(F ,H) sup y2Y

[D2F (y, fF(y))] 1 [D2F (y, f (y))] 1 ! 0 as ! 0. (4)

Similarly, since D1F is continuous on the compact set Y Z, for this 2 > 0

there exists a 0 < 3 < 2 such that for any y 2 Y

D1F (y, fF(y)) D1F (y, f (y))

D1F (y, fF(y)) D1F (y, fF(y)) + D1F (y, fF(y)) D1F (y, f (y))

sup (y, z)2HkDF (y, z) DF (y_{, z)k +} 2 3+ 2 provided F 2 N 3(F , H), f F_(y) 2 U[f (y), 3]. That is, sup F 2N3(F ,H) sup y2Y

D1F (y, fF(y)) D1F (y, f (y)) ! 0 as 2 ! 0. (5)

Let y 2 Y and F 2 N 3(F , H).

By Eqs. (3), (4) and (5), we have DfF(y) Df (y)

[D2F (y, fF(y))] 1D1F (y, fF(y)) [D2F (y, f (y))] 1D1F (y, f (y))

D1F (y, fF(y))([D2F (y, fF(y))] 1 [D2F (y, f (y))] 1) +

[D2F (y, f (y))] 1(D1F (y, fF(y)) D1F (y, f (y)))

D1F (y, fF(y)) [D2F (y, fF(y))] 1 [D2F (y, f (y))] 1) +

[D2F (y, f (y))] 1 D1F (y, fF(y)) D1F (y, f (y))

(1 + N1)[2M2( 2+ )] + M ( 3+ 2)

for any y 2 Y and F 2 N 3(F , H).

That is, sup F 2N3(F ,H) sup y2Y DfF(y) Df (y) _{! 0 as ! 0.} (6) By Eqs. (2) and (6), we have

sup

F 2N 3(F ,H)

(max_fsup

y2Y

fF(y) f (y) ; sup

y2Y

DfF(y) Df (y) _{g) ! 0 as} 3 # 0:

3

Chaos

3.1. Chaos

For any function G , we denote its domain by Dom(G ). We consider di¤erence systems of the form

G (xt ,:::, xt 1, xt, xt+1,:::, xt+ ) = 0, (7)

where each side of Eq. (7) is an N 1 column vector and G : Dom(G ) (RN₎ +1+

! RN _{with N , ,}

2 N. By a di¤erence system, we always mean a system of the form in Eq. (7), which we denote simply by G .

We de…ne an orbit of G as a bi-in…nite sequence fxtg1_{t= 1} such that for all t 2 Z,

G (xt ,..., xt 1, xt, xt+1,..., xt+ ) = 0.

Let k : k be the sup norm whenever its argument is a vector or a sequence. Let y = _fytg1t=l, l 1, be any sequence. If there exists a n 2 N for all t l such

that yt+n = yt, then y is called periodic. If n 2 N is the smallest such number, then

y _{is called n-periodic. Suppose y is a sequence in R}m_{, m 2 N. We say y is called} asymptotically periodic if there is a periodic sequence fytg such that kyt ytk ! 0 as

t_{! 1. If y is not asymptotically periodic, then y is called asymptotically nonperiodic.}

De…nition 2 We say that a di¤erence system G is chaotic if (T1) and (T2) below hold:

(T1) There exists a m 2 N, for all n m, G has an n-periodic orbit.

for all x, y 2 (x_{6= y)}

lim sup

t!1 kx

t ytk > 0, (8)

for all n 2 N, lim inf

t!1 k(xt n,..., xt+n) (yt n,..., yt+n)k = 0. (9)

Condition (T2) means that any two orbit in never converge to each other but they become arbitrarily close in…nitely often.

3.2. Static system and chaos

Let G : Dom(G ) _(RN) +1+ _{! R}N with N , , _{2 N be a di¤erence system.} We denote

Dom(G )0 =fx0 2 RN j (x ,..., x 1, x0, x1,..., x ) 2 Dom(G )

where x ,..., x 1, x1,..., x 2 RNg.

We say that G is static or a static system if there is a function Gs_{: Dom(G )} 0

RN

! RN _{such that G}s_(x

0) = G (x ,..., x 1, x0, x1,..., x ) for all (x ,..., x 1, x0,

x1,..., x ) 2 Dom(G ). If G is static, we de…ned a static point of G as a point

2 Dom(G )0 such that Gs( ) = 0.

Let K1,..., KM Dom(G )0. We de…ned a pattern as a vector of + 1 + v

natural numbers; a sequence of natural number is called a symbolic sequence. We say that a pattern p = (p ,..., pv) is a feasible pattern (w.r.t. G and K1,..., KM) if

Kp ... Kpv Dom(G ). Let P (G , K1,..., KM)be the set of pattern feasible w.r.t.

G and K1,..., KM. We say that a symbolic sequence fstg1t=l, 1 l 1, is

feasible (w.r.t. G and K1,..., KM) if for all t = l + ,..., v, (st ,..., st+v)2 P (G ,

Let p, q 2 P (G, K1,..., KM). We say that q is reachable from p if one of the

following three cases holds: (i) there exists a n 2 N, there is a symbolic sequence fstgnt=1 such that fp ,..., p , s1,..., sn, q ,..., q g is feasible; (ii) fp ,..., p , q ,...,

q _{g is feasible; (iii) there exists a m 2 f1,...,} + _{g, fp ,..., p , q} m+1,..., q g is

feasible and for all i = + m,..., pi = qi m.

Theorem 3 Let G be a static system with static points 1,..., M 2 Dom(G )0 and

there are p, q 2 P (G , 1,..., M) with p 6= q such that p =... = p and p, q are

reachable from each other. Then G is chaotic.

Proof. Without loss of generality, assume p =... = p = 1. If q is reachable from p_{with case (ii) or (iii) holding, then case (i) also holds for any n 2 N, if we let S}t= 1

for all t = 1,..., n. Hence in any case, there is a symbolic sequence S _fsigni=1 such

that fp ,..., p , s1,..., sn, q ,..., q g is feasible. De…ne T ftigmi=1, similarly. Let

v2 =_{fp ,..., p , s}1,..., sn,q ,..., q , t1,..., tm, p ,..., p g.

Let m = 3( + 1 + ) + n + m and v1 ₌

f1, 1,..., 1g with m 1‘s; v1_{and v}2 _have

the same dimension. For each bi-in…nite sequence of 1 and 2 (i.e.., i 2 f1, 2g for

all i 2 Z), let s( ) be the symbolic sequence such that for all i 2 Z

s( )im,..., (i+1)m 1= v i. (10)

Note that the mapping _{! s( ) is one-to-one and that s( ) is always feasible.} We …rst verify (T2). For r 2 R, let [r] denote the largest integer less than or equal to r . For w 2 (0, 1), de…ne a bi-in…nite symbolic sequence w _{as follows. For i 0,}

let w

i = 1. For i 1, de…ne wi as follows: w 1;10=f1,..., 1_{| {z }} [10w]1`s , 2,..., 2 | {z } (10 [10w])2`s g

w 11;110 =f1,..., 1_{| {z }} [100w]1`s , 2,..., 2 | {z } (100 [100w])2`s g w 111;1110=f 1,..., 1_{| {z }} [1000w]1`s , 2,..., 2 | {z } (1000 [1000w])2`s g

and so on. More precisely, letting Tn = 1 + 10+... +10n for n 2 N, we have for all

n_{2 N}

w

i = 1, for all i = Tn,..., Tn+ [10nw] 1, (11) w

i = 2, for all i = Tn+ [10nw],..., Tn+1 1. (12)

Note that for any w, w0 _{2 (0, 1), w 6= w}0. [10n_w]

6= [10n_w0_{] for n large enough. Thus} w

i 6= w0

i for in…nitely many i

0

s. (13) Therefore, for any w 2 (0, 1) there is an orbit xw

such that for all t 2 Z, xw

t =

s( w₎

t. Let = fx

w

j w 2 (0, 1)g; we show that satis…es (T2). Clearly is an uncountable set. Let w 2 (0, 1). Since [10nw] _{" 1 as n " 1,} w is asymptotically nonperiodic; thus xw _{is asymptotically nonperiodic. It remains to show Eqs. (8) and}

(9). Let w, w0 _{2 (0, 1) with w 6= w}0_{. Let w = minfw, w}0_{g. Let m 2 Znf0g. For} n_{2 N, let n}= Tn+ [10

n_w

2 ]. By Eqs. (11) and (12), we have

(xw n m,..., x w n+m) (x w0 n m,..., x w0 n+m) = 0 as n ! 1. That is lim inf t!1 (x w n m,..., x w n+m) (x w0 n m,..., x w0 n+m) = 0:

By Eq. (13), we also have

lim sup

t!1

xw_t xw_t0 > 0

Since w, w0, and m were arbitrary, we have veri…ed Eqs. (8) and (9) and thus (T2).

Now to verify (T1), let =_{f...,1, 2, 1, 2,...g with} 0 = 2. Clearly s( ) is feasible

for all t expect that if t = im for some i 2 Z, smt,..., t+m 1 = v2. For n > m, let sn be

the symbolic sequence such that snt = 1 for all t expect that if t = in for some i 2 Z,

sm

t,..., t+m 1 = v2. Clearly for all n m, sn is feasible and n-periodic. That is for all

4

Stability of chaos

4.1. Quasi-static system

We introduce the concept of quasi-static system. Quasi-static systems are di¤er-ence systems that are in a certain sense relatively close to static systems.

De…nition 4 We say that G is quasi-static (w.r.t. K1,..., KM) if (K1) and (K2)

below hold:

(K1) K1,..., KM are disjoint, compact, and convex.

(K2) For all p 2 P (G , K1,..., KM), ( p ,..., p 1)2 Kp ... Kp 1, and ( p1,...,

p ) 2 Kp1 ... Kpv there is a unique gp( p ,..., p 1, p1,..., p ) 2 Kp0

such that G ( p ,..., p 1, , p1,..., p ) = 0.

Lemma 5 G is quasi-static w.r.t. K1,..., KM Dom(G )0 then

(c1) for each bi-in…nite feasible symbolic sequence fstg1_{t= 1}, G has an orbit fxtg1_{t= 1}

such that for all t 2 Z, xt 2 Kst.

(c2) For each n-periodic bi-in…nite feasible symbolic sequence fstg1_{t= 1}, G has an

n_{-periodic asymptotic orbit fx}tg1_{t= 1} such that for all t 2 Z, xt 2 Kst.

Remark 6 _{For all p 2 P (G , K}1,..., KM), gp : Dp ! Kp0 is continuous, where

Dp = (Kp ... Kp 1) (Kp1 ... Kpv).

Remark 7 (Brouwer …xed point theorem) Suppose that M is a nonempty, con-vex, compact subset of Rn_{, n 1 and that f : M}

! M is a continuous mapping then f has a …xed point.(see [5], p. 51)

Proof of Lemma. _{For any sequence fy}tg, let

y_t = (yt ,..., yt 1), y+t = (yt+1,..., yt+ )

We …rst prove (c2). The proof of (c1) is similar to that of (c2), and is thus omitted. Let fstg1t=l, 1 l , be a feasible sequence For t l, let St = Kst.Suppose

l = _{1 and fs}tg is n-periodic. Given x1;n (x1, :::, xn) 2 Ks1 Ks2 ::: Ksn.

Let x be the n-periodic sequence such that x1,..., xn are as given. De…ne T1,n :

Ks1 Ks2 ... Ksn ! Ks1 Ks2 ... Ksn by T1,n(x1,n) = gst ;t+ (xt , x +

t ). Since

T1,n is continuous and S1,nis compact and convex, T1,nhas a …xed point x1,n = (x1,...,

x_n) (by the Brouwer …xed point theorem) Clearly, the associated n-periodic orbit x is an orbit of G such that for all t 2 Z, xt 2 Kst.

Note that if G is a static system with static points 1,..., M, then G is a

quasi-static w.r.t.f 1g,..., f Mg, and the conclusions (c1)-(c2) trivially hold with xt 2 f 1,..., Mg for all t. The lemma says that they continue to hold for a quasi-static system

with appropriate compact convex sets replacing static points.

4.2. Stability of chaos for regular static system under small C1 pertur-bations

Let G is static; we say that G is regular if G is C1, if G has only a …nite number of static points 1,..., M 2 Dom(G)0, and DGs( i) is nonsingular for all

i = 1,..., M .

Let G be a C1 _{static system with static points}

1,..., M 2 Dom(G )0. Denote

and

J (G ,N"( 1),..., N ( M)) =fN"( i ) ... N ( i ) Dom(G )j

for all (i _{,..., i ) 2 P (G , N}"( 1); :::; N"( M))g.

Theorem 8 Let G be a regular static system with static points ₁,..., _{M 2 K}0 and

Dom(G ) is open. Let K Dom(G ) be a compact set such that J (G , ₁,..., _M) K (the interior of K). Then there exist " and " > 0 such that for all G2 N "(G , K)

(i) we have G is quasi-static (w.r.t. N ( ₁), N ( ₂),..., N ( _M)) and

(ii) for G , if there are p, q 2 P (G , 1,..., M) with p 6= q such that p =...

= p and p, q are reachable from each other. Then G is chaotic.

Proof. (i) Let G be a C1static system with static points 1,..., M 2 Dom(G )0, and

DGs( _i)is nonsingular for all i = 1,..., M , and Dom(G ) is open. Let K Dom(G ) be a compact set such that J (G , ₁,..., _M) K (the interior of K). Then there is " > 0 such that J (G , ₁,..., _M) J (G , N"( 1),..., N"( M)) K Dom(G ), and

P (G , N"( 1),..., N"( M)) = P (G , 1,M) P. For this " > 0, G is a quasi-static

system w.r.t. N"( 1),..., N"( M). Let Ki = N"( i) for i = 1,..., M . For p 2 P ,

let gp : Dp ! Kp0 be de…ned as in (K2). Note that for all p 2 P and for all

2 Dp, we have gp( ) = p0 2 Kp0 and Dgp( ) = 0. Hence by theorem 1, there

is " > 0 such that for all G 2 N "(G , K) for all p 2 P , there is a unique function

gG

p : Dp ! Kp0 such that for all 2 Dp, G( , g

G p( ), +_{) = 0, g}G p is C1, and max 2Dp DgG p( ) =max 2Dp Dg G

p( ) Dgp( ) < 1. Therefore for all G 2 N (G , K),

G satis…es (K1) and (K2). So, we have G is quasi-static (w.r.t. N ( ₁), N ( ₂),..., N ( _M)_{). From Lemma 5, note that if fs}tg1_{t= 1} is the orbit of G , then there exists a

correspond orbit preserve the period of fstg1_{t= 1}. That is if fstg1_{t= 1} is n periodic

then fxtg1t= 1 is n periodic.

(ii) For G , if there are p, q 2 P (G , 1,..., M) with p 6= q such that p =...

= p and p, q are reachable from each other, then G satis…es (T1) and (T2) and is chaotic. (From Theorem 3). In fact, for all G 2 N"(G , K) G also satis…es (T1) and

there exists an uncountable set G _{of asymptotically nonperiodic orbit such that for}

all xG, yG₂ G (xG _{6= y}G)we have lim sup

t!1

xG_t y_tG > 0.

We remain Eq.(3) to be check. Now we give two remarks as following, and post-pone the proof of Remark 9 to the appendix.

Remark 9 Let G be a C0 _{system. Let H Dom(G )}

0 is compact. Suppose (a) there

is a unique orbit x such that x_{t 2 H for all t 2 Z. Then x is a constant sequence} and for any " > 0 there is n 2 N such that for all t 2 Z and for any orbit x, if xi 2 H

for all i = t n_{,..., t + n, we have kx}t xtk < ".

Remark 10 If G is quasi-static (w.r.t. N ( ₁), N ( ₂),..., N ( _M)) and there are p, q _{2 P (G , N ( 1}),..., N ( _M)) with p _{6= q such that p} =... = p and p, q are reachable from each other and gG

p as given by (K2) is C1 and max 2Dp Dg

G

p( ) < 1.

Then there is a constant sequence f..., , , _{,...g is the unique orbit fx}tg such

that xt 2 N ( p0) for all t2 Z.

By the proof of theorem 3 and Lemma 5 (c1), for any w 2 (0, 1) there is an orbit xw _{such that for all t 2 Z, x}w_{t 2 N}"( s( w₎

t). Let =fx

w

j w 2 (0, 1)g; we show that satis…es (T2). Clearly _{is an uncountable set. Let w 2 (0, 1). Since [10}n_w]

n _{" 1,} w is asymptotically nonperiodic; thus xw is asymptotically nonperiodic. It remains to show Eqs. (8) and (9). Let w, w0 _{2 (0, 1) with w 6= w}0. Let w = min_fw, w0_{g. Let m 2 Znf0g. For n 2 N, let n} = Tn+ [10

n_w

2 ]. By Eqs. (11) and (12) and

Remark 9 and 10, we have (xw n m,..., x w n+m) (x w0 n m,..., x w0 n+m) (x w n m,..., x w n+m) ( ,..., ) + (xw0 n m,..., x w0 n+m) ( ,..., ) ! 0 as n ! 1. That is lim inf t!1 (x w n m,..., x w n+m) (x w0 n m,..., x w0 n+m) = 0:

By Eq. (13), we also have

lim sup

t!1

xw_t xw_t0 > 0

Since w, w0, and m were arbitrary, we have veri…ed Eqs. (8) and (9) and thus (T2). Therefore G satis…es (T1) and (T2) and thus chaotic.

5

Appendix

Here we give the proof of Remark 9.

Proof of Remark 9. Let G be a C0 _{system. Let H Dom(G )}

0 is compact.

Assume (a) above. Since x = fxtg is the unique orbit in H and since fxt+1g is

clearly an orbit, we have xt = xt+1 for all t 2 Z, i.e., x is a constant sequence. Let

= x_{t. Let " > 0. Suppose there is no n 2 N such that for all t 2 Z and for any} orbit x, if xi 2 H for all i = t n,..., t + n, we have kxt k < ". This means that

for all n 2 N there is an orbit yn _{such that for some T}

n 2 Z, yTnn " and

yn

i 2 H for all i = t n,..., t + n. For n 2 N, de…ne xn=fxntg by xnt = ynTn. Note that

for all n 2 N, xn

is an orbit and kxn

0 k ". Taking a subsequence if necessary, we

may assume xn

t ! xt 2 H as n " 1 for all t 2 Z. Then we have kx0 k " and

thus fxtg 6= x . But since G is C0, it follows that fxtg is an orbit, which contradicts

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