反應擴散方程之進行波解的存在性

國 立 交 通 大 學

應用數學系

博 士 論 文

反應擴散方程之進行波解的存在性

Geometric Approach to the Existence of Traveling

Wave Solutions for Reaction Diffusion Systems

研究生：楊其儒

指導教授：林松山教授

反應擴散方程之進行波解的存在性

Geometric Approach to the Existence of Traveling

Wave Solutions for Reaction Diffusion Systems

研究生：楊其儒 Student：Chi-Ru Yang

指導教授：林松山 Advisor：Song-Sun Lin

國 立 交 通 大 學

應 用 數 學 系

博 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

National Chiao Tung University

in partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

in

Applied Mathematics

May 2010

Hsinchu, Taiwan, Republic of China

i

反應擴散方程之進行波解的存在性

研 究 生：楊其儒 指導教授：林松山 教授

國 立 交 通 大 學

應 用 數 學 系

摘要

本論文主要利用幾何觀點，來研究反應擴散方程的進行波解的

存在性。在第一個部份，我們介紹幾何奇異擾動的理論方法，並進一

步用這種方法來發展某一類 FitzHugh-Nagumo type equation 的進行波

解的多樣性。幾何奇異擾動理論，主要利用不同的時間變數(time scales)

把原本的系統分割成慢系統( Slow System )跟快系統( Fast System )，進

而減少系統的維度、簡化問題的難度。理論起源應追溯到 Fenichel 發

展的一系列不變流形( Invariant Manifold )理論。Christopher K.R.T. Jones

和 Kopell 利用這些理論發展出來的 Fenichel coordinate 來追蹤慢流形

(Slow Manifold) 附 近 的 行 為 ， 這 是 有 名 的 交 換 引 理 ( Exchange

Lemma )。然而，伴隨著轉點( Turning Point )的出現， Weishi Liu 和 Van

Vleck 發展出 Fenichel-type coordinate 進而證明轉點出現的情況下，方

程式的解產生更豐富的動態行為。我們利用這兩種交換引理得到進行

波解存在性的充分條件，並且把所有可能存在的進行波波形做一個分

類。

自從 Dunbar 在 1983 年開始利用 Wazewski Theorem 和 LaSalle

Invariance Principle 來證明 Lotka-Volterra model 進行波解的存在，後續

不斷有學者把這套理論應用到其他的掠食者模型: Diffusive Holling

type II, Holling type III。在第二個部分中，我們試著利用同樣的技巧，

做些修改，推廣到更一般的模型中，甚至涵蓋了某種傳染病模型。另

外，特別對於非單調系統，這套理論提供了一種很好的工具，讓我們

對於非單調系統進行波解的結構可以有更進一步的瞭解。

iii

Geometric Approach to the Existence of Traveling

Wave Solutions for Reaction Diffusion Systems

Student：Chi-Ru Yang Advisor：Prof. Song-Sun Lin

Department of Applied Mathematics

National Chiao Tung University

Abstract

This dissertation investigates the existence of traveling wave

solutions in Reaction Diffusion system using geometric approach. The

first part of the dissertation applies the geometric singular perturbation to

establish the diversity of traveling wave solutions of a FitzHugh-Nagumo

type equation with turning point. There coexist different front of traveling

wave and we give a complete classification for all of these fronts of

traveling waves.

The second part of the dissertation applies high-dimensional phase space

analysis together with Wazewski Theorem and LaSalle Invariance

Principle to prove the existence of the traveling wave solution connecting

two equilibria in a general diffusive predator-prey system. The system we

considered contains many known model as examples, such diffusive

Lotka-Volterra model, diffusive Holling type II, Holling type III and even

some SIR model.

誌 謝

首先感謝我的指導教授 林松山老師，除了在交大生活上的幫忙，也教導我

許多事情。研究生該關心的議題，做學問該擁有的態度，謝謝您叮嚀我。求學期

間，您也經常鼓勵我去參加國內外短期課程與研討會，希望可以培養更寬廣的國

際觀。認識您，我像是多了一副專業的眼鏡，謝謝您帶領我用不一樣的角度去看

待世界以及面對問題。

許正雄學長，感謝你的指導、包容、建議和幫助，與你相處讓我獲益良多

。我知道要把學問做好真的不是一件容易的事情，從你身上我體悟到自己必須不

斷的努力與加倍的認真。 楊智烜學長，有很多問題跟你討論後，我的盲點才真

正的浮現，對問題才能有更深一層的瞭解，謝謝你對我的耐心與指導。 楊定揮

學長，和你一起做研究是很愉快的經驗。在去中央的車上，你的經驗分享以及問

題的討論都是我難忘的回憶。

在交大應用數學系上，老師們給了我很多的幫助與關懷。 秋媛姐，可以聽

到您的問候真的是件很窩心的事情。 賴明治老師，您的關心是我鞭策自己的動

力。 石至文老師，謝謝您常策劃一些很不錯的研討會與演講。另外，我在系上

擔任過許多科目的助教，感謝這期間老師們的照顧，王夏聲、吳慶堂、石志文、

李明佳、吳培元老師，能參與你們的教學過程，也讓我學習到不少。系辦的麗君

姊、盈吟姐、慧珊、雅鈴、宗智與妮臻，由衷的感謝你們無數的幫忙，你們辛苦

了！

博士班期間修過的課並不多，但是聽過的課，仔細數起來卻意外的多: 程守

慶、林松山、蔡東和、文蘭、翁志文、洪盟凱、林長壽、賴明治、許正雄、楊智

烜、林欽誠、楊肅煜、杜一宏、趙一峰、高華隆、劉維士、唐雲、陳國璋、樓元、

班榮超、趙曉強老師以及歷屆動態演討會、京都 ICDEA、義大利 ICTP 短期課程

的所有老師，感謝你們曾經給過我的指導與幫助。

漫長的博士生涯多虧了學長姐及學弟妹的陪伴，謝謝你們豐富了我的求學旅

程：Ban 學長、Amanda 學姐、小拉學長，你們畢業後我就開始懷念起一起去唱歌

的日子、一起煮咖啡的下午。昌源、金龍和 David 郭學長，我還記得週一晚上有

讀書會、週三傍晚則是打羽球的時間。琲琪、狐狸學姐，看妳們把課業和生活中

每件事情都處理得很好，一直讓我很崇拜。我的麻吉忠澤，一路走來還好有你可

以互相幫助。我的室友小明，我永遠會記得我們一起拼資格考的日子。同師門的

文貴、園芳、怡菁、倖綺、慧萍、玉雯、佳玲；同研究室的志弘、明淇、忠逵，

附近研究室的 Andan 學長、貓頭、小培、惠蘭、小巴;一起帶過助教的金龍、忠

澤、小明、康伶、明杰、瑜堯、詩妤;游日月潭的泳伴們、友聲合唱團的同學、

清華電台的 DJ，謝謝你們完整了我多采多姿的交大生涯。

碩班的指導教授 王懷權老師，謝謝你給予我的照顧與關懷，跟您聯絡近況

文、其棟、勝鴻、以前的同學們，偶而碰頭聊起以前的事，總是可以讓人暫時忘

卻眼前的困境，重新再出發。六家國中 203 班的小朋友、長庚大學和新竹教育大

學我帶過的學生、教過的家教學生，陪著你們成長的過程，我自己也更成熟了些。

最後要感謝我親愛的父母、珠珠姨、姐姐、姐夫以及弟弟，雖然在這期間家

中發生很多事情，謝謝你們始終還是一樣的愛我、支持我。另外，特別要感謝兩

個人:一個是從小疼我到大的外婆，另一個是剛通往天國的爺爺。當外婆回憶起

兒時的我，臉上露出的笑容;爺爺因為賣力擔青菜去市場賣，而無法挺直的腰，

這些常常都在夜深人靜時，鼓勵著我應該要更努力向上。

楊其儒 2010\6\30 於交大

Contents

1 Introduction 1 2 Geometric Singular Perturbation(GSP) Theory 4

2.1 Limiting System . . . 4

2.2 Results for Normally Hyperbolic Case . . . 5

2.3 Results for Non-Hyperbolic Case . . . 6

2.3.1 Delay of Stability Loss . . . 6

2.3.2 Exchange Lemma . . . 7

3 Applications to the Existence of Traveling Waves Solutions in Fitz-Hugh Nagumo Type Equations 8 3.1 Formulation of GSP Problems . . . 8

3.1.1 Dynamics for the Limiting Slow System . . . 9

3.1.2 Dynamics for the Limiting Fast System . . . 9

3.1.3 Delay of Stability Loss and Exchange Lemma . . . 14

3.1.4 Admissible Conditions for Singular Orbits . . . 17

3.2 Existence of Traveling Waves . . . 18

3.3 Proof of the Existence Results . . . 20

3.3.1 Melnikov function and Transversality of Manifolds . . . 20

4 Wazewski Theorem and Lasalle Invariance Principle 30 4.1 Wazewski theorem . . . 30

4.2 LaSalle’s Invariance Principle . . . 30

5 Applications to the Existence of Traveling Waves in Predator-Prey Type Reaction Diffusion Systems 31 5.1 Construction of Wazewski set . . . 33

5.1.1 Routh-Hurwitz Criterion . . . 34

5.1.2 The Exit Set W−. . . 35

5.1.3 Construction of Σ0 _{. . . . 36}

5.1.4 Existence of An Invariant Orbit . . . 45

5.2 The Lyapunov Function for the Invariant Orbit . . . 46

5.4.1 Applications to System (1.3) . . . 51

5.4.2 Applications to System (1.4) . . . 51

5.4.3 Applications to System (1.5) . . . 53

5.5 Appendix . . . 56

5.5.1 Appendix I: The Proof of Proposition 5.4 . . . 56

5.5.2 Appendix II: The Proof of Lemma 5.16 . . . 64

6 Reference 69

List of Figures

2 Regions of Ω, Gi and Hi. . . 11

3 Admissible heteroclinic orbits with respect to regions. . . 12

4 Equilibria, turning points, dynamics on the slow manifold and heteroclinic orbit Γ of the fast dynamics which connects E0 and (1, 0, 0) when a(0) = a1(c) or a(0) ≤ a6(0). The red segments M_0,1− on M0,1 are defined by M0− ={(0, 0, w) ∈ M0| λ0−(w, c) < 0} and M₁−=_{{(1, 0, w) ∈ M}1| λ1−(w, c) < 0}. . . 13

5 Part (1) of Proposition 3.2. In this graph, we denote Cδ(c) := (c− δ, c + δ) for some δ > 0 and identify the (w, c)-plane with the vertical axis. . . 16

6 Part (2) of Proposition 3.2. In this graph, we denote Cδ(c) := (c− δ, c + δ) for some δ > 0 and identify the (w, c)-plane with the vertical axis. . . 17

7 The projection of P, Q, R, S in the uw-plane. . . 36

8 Phase plane diagram for Lemma 5.5. . . 38

9 Projection to the uw-plane of the trajectory y(s; y1) in Lemma 5.10. . . 42

10 Construction of the Σ. . . 44

11 Projection of Ω on (u, v)-plane. . . 49

12 The graph of Λ in the case w < w∗ . . . 67

13 The graph of Λ in the case w = w∗ . . . 67

14 The graph of Λ in the case w > w∗ . . . 68

15 The deformation retrace of Λ in the u-z space . . . 68

1

Introduction

This thesis concerns with the existence of traveling wave solutions for reaction diffusion systems by using some geometric methods. In this article, we mainly use the geometric singular perturbation theory and Wazewski theorem to prove the existence of traveling wave solution for FitzHugh-Nagumo type equations and predator-prey type models, respectively.

In the Section 2, we recall some known results of the geometric singular perturbation theory. Then, in the Section 3, we apply the geometric singular perturbation theory to investigate the existence of traveling wave solutions of FitzHugh-Nagumo type equations. The equations can be described as the following:      ut(x, t) = uxx(x, t) + f (u(x, t), w(x, t)), wt(x, t) = εg(u(x, t), w(x, t)), (1.1) where ε > 0, f (u, w) = u(u_{− a(w))(1 − u) for some smooth function a(w) and g(u, w) = u − w. When} f (u, w) = u(1_{− u)(u − α) − w and g(u, w) = u − γw with constants α and γ, (1.1) is the prototype} FitzHugh-Nagumo equation, which is a simplification of the Hodgkin-Huxley equation describing the propagation of action potentials in the nerve axon of the squid, cf. [17]. The dynamics of such specific equations, especially the traveling wave solutions, have been widely studied in the past, see [11, 18, 22, 26, 29, 40] and the references therein.

Recently, Liu and Van Vleck [33] consider the co-existence of different traveling wave fronts of (1.1) by allowing a(w) to cross 0 and 1, then the profile equations of (1.1) can be reduced as a singularly perturbed system with turning points. Those special turning points exhibit the so-called delay of stability loss. Applying the geometric singular perturbation (GSP) theory and the exchange lemma for turning points (cf. [32]), Liu and Van Vleck show the existence of various types of traveling wave solutions which posses a special set of turning points. The slow manifold M for such singularly per-turbed system consists of three parts, by M = M0∪ Ma∪ M1. They studied traveling wave solutions

whose slow orbits lie only on the portions M0 and M1 of the slow manifold.

Motivated by the work of [33], in this article we generalize their results to the cases of traveling waves of (1.1) which involves all the portions M0, Ma, M1 of the slow manifold. The main difficulties

in applying the GSP theory to our problem is to investigate the transversality of invariant manifolds by computing the Melnikov functions. In [33], the slow orbits lie only on the portions M0 and

M1, then the Melnikov functions does not vanish obviously. However, due to the consideration of

Melnikov functions (first and second order) in terms of Beta or Gamma functions. Thus we can apply the exchange lemma to track the evolution of invariant manifolds as they pass the vicinity of the slow manifold. Under the consideration of Ma, there are more complicated and richer dynamics of traveling

wave solutions than those of [33]. Moreover, we give a complete classification of all different fronts of traveling waves.

In Section 4 and Section 5, we use the high dimensional phase space analysis together with LaSalle invariance principle and Wazewski theorem to investigate the existence of traveling wave solutions of the following diffusive predator-pray system

   ut= d1uxx− h(u)[g(w) − p(u)], wt= d2wxx− `(w)q(u), (1.2) where the functions p, g, h, ` and q satisfy

(A1) p0(u) < 0 for u > 0 and p(K) = 0 for some u = K. (A2) q0(u) < 0 for u > 0 and q(u∗) = 0 for some u∗∈ (0, K).

(A3) g0(w) > 0, `0(w) > 0, `00(w)_{≤ 0 for w ∈ R and g(0) = `(0) = 0.} (A4) h0_{(u) > 0 for u∈ R and h(0) = 0.}

Equation (1.2) is a general form of predator-prey system containing many known models as examples. In the following, we illustrate some of these examples of which the existence traveling waves has been studied in the past. In 1983, Dunbar [8, 9] considered the existence of traveling wave solutions for the following reaction-diffusion system based on the Lotka-Volterra differential equation model of a predator and prey interaction:

     ut= d1uxx+ Au(1− u K)− Buw, wt= d2wxx− Cw + Duw, (1.3) where d1, d1, A, B, C, D, K are positive constants. The functions u(x, t) and w(x, t) represent the

species densities of the prey and predator, respectively. d1 and d2 are the diffusion coefficients; A is

the intrinsic rate of increasing for the prey species; C is the death rate for the predator in the absence of prey; K is the carrying capacity of the environment; B and D are the interaction rates of the two species. By using the Wazewski theorem (an extension of shooting argument in higher dimension) together with a Liapunov function and LaSalle’s invariance principle, he proved the existence of traveling wave solutions which are equivalent to heteroclinic orbits in 4-dimensional phase space.

Later, Dunbar [10] further considered the existence of traveling wave solutions for system (1.3) but with Holling type II functional response H2(u) = _1+Euu , i.e.,

     ut= d1uxx+ Au(1− u K)− BH2(u)w, wt= d2wxx− Cw + DH2(u)w. (1.4) with E > 0. Different to system (1.3), system (1.4) includes the effects of predator satiation. E is the parameter which measures the satiation effect : the consumption of prey by a unit number of predators cannot continue to grow linearly with the number of prey available but must “saturate” at the value 1/E. Assume d1= 0, Dunbar used the method similar to that in [8, 9] and invariant manifold

theory to demonstrate the existence of periodic traveling wave train and traveling front solutions for system (1.4). The case for d16= 0 was then considered by Huang, Lu and Ruan [24]. Using the same

shooting argument and the Hopf bifurcation theorem, they established the existence of the traveling wave solutions connecting two rest states as well as the existence of small amplitude traveling wave train solutions.

Later on, Li and Wu [31] also consider the system (1.4) but with Holling type-III functional response H3(u) = u 2 1+Eu2, i.e.,      ut= d1uxx+ Au(1− u K)− BH3(u)w, wt= d2wxx− Cw + DH3(u)w. (1.5) By using the similar methods of [8, 9], they establish the existence of traveling wave solutions of (1.5) for the case d1= 0. Recently, we generalize the results of [31] to the case d16= 0, see [41].

For other examples, Owen and Lewis [36] consider the the following general system      ut= εα0uxx+ α1uf1(u)− α2wf2(u), wt= α0wxx+ α3wf2(u)− α4w, (1.6) where all αi and  ≈ 0 are positive constants. They study the mechanism for which predation can

slow, stall and reverse a spatial invasion of prey, i.e., traveling wave by demonstrating the numerical results for specific f_i0s described below. The f1is given by f1(u) = (1− u) or f2(u) = k(1− u)(u − a)

with suitable choice of constants k and a while f2 is given by Holling type I (f2(u) = u), type II, or

type III. However there was no theoretical proof for those numerical results.

Note that (A1)_{∼(A4) hold for the system (1.3)∼ (1.6) provided corresponding parameters lying a} suitable region. For example, p(u) = A(1_{− u/K), g(w) = Bw, h(u) = u, `(w) = w and q(u) = C − Du}

Extending the ideas of [8, 9], we apply the Wazewski theorem (see Lemma 4.1) together with LaSalle’s invariance principle (see [21]) to prove Theorem 5.1. Note that although we apply the techniques similar to those of [8, 9], there are some differences. First, the model that we consider is more general, and our results contain all the results of the above mentioned examples and some other models, e.g., some typical S.I.R models, such as Kendrick Kermack model, for details see p.6 in [42]. Second, due to the general setting of system (1.2), the construction of Wazewski set is more complicated than those of [8, 9], and it’s more difficult to find an invariant orbit of system (5.29) in the Wazewski set. Third, we construct the Liapunov function for system (1.2) is general to prove the existence results.

2

Geometric Singular Perturbation(GSP) Theory

In this section, we recall geometric singular perturbation theory. The well-developed theorem was retraced to the invariant manifold theorems developed by Fenichel, see [13, 14]. The GSP theory is a powerful tool reducing a high dimensional system into two lower dimensional subsystems. Here, we introduce the main results. For more complete matter, we refer the reader to the book of Christopher K.R.T. Jones, [27].

2.1

Limiting System

Consider the singular perturbed system in slow form with slow time τ        εdx(τ ) dτ = F (x, y; ε), dy(τ ) dτ = G(x, y; ε), (2.7) where (x, y)_{∈ R}m

× Rn_{, ε > 0 small. In terms of the fast time t = τ /ε, we have the fast system}

       dx(t) dt = F (x, y; ε), dy(t) dt = εG(x, y; ε), (2.8) For ε _{6= 0, those two systems are equivalent. Alternatively, for ε = 0 we have the following limiting} slow system      0 = F (x, y; 0), dy(τ ) dτ = G(x, y; 0). (2.9)

and limiting fast system        dx(t) dt = F (x, y; 0), dy(t) dt = 0, (2.10) from systems (2.7) and (2.8). The limiting slow and fast systems are totally different but they are complimentary. The dynamics of original system can be viewed as the combination and interaction of dynamics of the systems (2.9) and (2.10).

We call the set Z0 :={(x, y) : F (x, y; 0) = 0} the slow manifold. For sufficient good function F ,

we assume Z0 ={(x, y) : x = H0(y) for y∈ D ⊂ Rn}. It is obvious that the slow manifold Z0 is a

set of equilibria of the limiting fast system (2.10). The limiting slow system (??) provides limiting dynamics for y-variables on Z0 and The limiting fast system (2.10) provides limiting dynamics for

x-variables with parameters y.

Linearize system (2.10) at (x, y) = (H0(y), y)∈ Z0, then we yield the corresponding variational

matrix   Fx(H0(y), y; 0) Fy(H0(y), y; 0) 0n×m 0n×n  .

There are n zero eigenvalues corresponding the eigenspace T(H0(y),y)Z0 and the other m eigenvalues

come from Fx(H0(y), y; 0).

Definition 2.1

(1) The set Z0 is normally hyperbolic if all eigenvalues of Fx(H0(y), y; 0) have nonzero real part;

(2) if Fx(H0(y∗), y∗; 0) have zero real part eigenvalue for some y∗∈ D, Z0is not normally hyperbolic

and (H0(y∗), y∗) is called a turning point.

2.2

Results for Normally Hyperbolic Case

Assume Z0is normally hyperbolic and Fx(H0(y), y; 0) has ` eigenvalues, denoted by αj(y), j = 1, .., `,

with positive real parts and k := (m_{− `) eigenvalues, denoted by β}j(y), j = 1, .., k, with negative real

parts.

The normally hyperbolic theory implies the persistence of the slow manifold Z0, invariant manifold

Ws,u_(Z

0) and invariant foliations. As a consequence, the dynamics in the vicinity of the slow manifold

Using normally hyperbolic theory, we know that there exists a Fenichel coordinates (u, v, y) in a neighborhood of Z0 such that

               du(t)

dt = A(u, v, y; ε)u + F1(u, v, y; ε), dv(t)

dt = B(u, v, y; ε)v + F2(u, v, y; ε), dy(t)

dt = εG(y; ε) + εh(u, v, y; ε)u⊗ v,

(2.11)

,where σ(A(0, 0, y; ε)) =_{αj(y)}, σ(B(0, 0, y; ε)) = {βj(y)} and Fj’s are higher order terms in (u, v)∈

R`× Rk. We have Zε ={u = v = 0}, Ws(Zε) ={v = 0}, Wu(Zε) ={u = 0}, Ws(y) = {(u, 0, y) :

|u| < δ} and Wu_{(y) =}

{(0, v, y) : |v| < δ}. Fenichel coordinates was used to track the solution near the slow manifold. With such powerful tool, Jones and Kopell ,[29, 30], developed the exchange lemma in hyperbolic case: Let Mε_{be a C}1 _{invariant manifold of dimension (k + σ) that is smooth in ε ,where}

1_{≤ σ ≤ n. Here, we make two assumptions:}

(H1) M0_{∩ W0}s(Z0) transversally on ∂N , where N is the neighborhood where Fenichel coordinate

works.

(H2) dim ω(N ) = σ_{− 1 and G(y; 0) /∈ T}yω(N ) for y∈ ω(N).

Let N = M0

∩ Ws_(Z

0). Then dimN = (k + σ) + (n + l)− (m + n) = σ ≥ 1.

For any τ0 > 0, let 0 < δ < τ0. For any q ∈ W0u(ω(N )· (τ0 − δ, τ0 + δ)), let y1 = α(q) ∈ Z0.

∃!(y0, τ∗) ∈ ω(N) · (τ0− δ, τ0+ δ) such that y1 = y0· τ∗. Furthermore, ∃!p ∈ N ∩ ∂N such that

ω(p) = y0.

Theorem 2.2 For ε > 0 small, after time τ∗, a portion of Mε _{close to p is mapped into an O(ε)}

neighborhood of q; Near q, Mεis O(ε)-close to W₀u(ω(N )_{· (τ}0− δ, τ0+ δ)).

2.3

Results for Non-Hyperbolic Case

With the presence of initial turning points, slow manifold fails to be normally hyperbolic in the vicinity of turning points. Liu [32] used the center manifold theory to achieve a Fenichel-type coordinate system.

2.3.1 Delay of Stability Loss

Before we start to state the exchange lemma with turning point, there is a local property of turning points (so-called delay of stability loss) which was studied by Pontryagin. Let us recall with the

simplest situation. Consider the singularly perturbed system    ε ˙x = xf (x, y; ε), ˙ y = g(x, y, ε). (2.12) where f and g are smooth and g(0, y; 0) > 0, f (0, 0; 0) = 0 and f (0, y; 0)y > 0 for all y_{6= 0.}

We have slow manifold S0 ={x = 0} is a branch of {x : f(x, y, 0) = 0}, S− ={(0, y); y < 0} is

stable and S+={(0, y); y > 0} is unstable. For a fixed large number K >> 1, define P0: (−K, 0) →

(0,_{∞) by} P0(y0) = y1,where Z y1 y0 f (0, y; 0) g(0, y; 0) = 0

Fix δ > 0. For any (δ, y0) with y0< 0. Let (x(τ, ε), y(τ, ε)) be the solution with initial condition (δ,

y0). Let τε> 0 be the time so that x(τε, ε) = δ. Define Pε(y0) = y(τε; ε). Pointryagin demonstrated

the following result.

Lemma 2.3 Pε→ P0 smoothly as ε→ 0

Assume Z0is not normally hyperbolic and Fx(H0(y), y; 0) has ` eigenvalues, denoted by αj(y), j =

1, .., `, with positive real parts and k := (m<sub>− ` − 1) eigenvalues, denoted by β</sub>j(y), j = 1, .., k, with

negative real parts. The rest eigenvalue λ(y) of Fx(H0(y), y; 0) have the same sign with y1. Let

Z+ = x = 0, y1> 0, T = x = 0, y1= 0 and Z− = x = 0, y1< 0. Assume there exists α0< 0 < β0such

that Reαj(y) < α0< λ0(y) < β0< βj(y) for all j. Assume G(0, y; 0) > 0. Define P0: Z− → Z+ by

P0(y0) = y1, where y0· τde= y1 and

Z τde

0

λ0(y0· τ)dτ = 0

2.3.2 Exchange Lemma

Let Mε _{be a C}1 _{invariant manifold of dimension (k + σ) that is smooth in ε where 1}

≤ σ ≤ n. Here, we make two assumptions:

(H3) M0

∩ Wcs_(Z

0)\ Ws(Z0) transversally on ∂N , where N is the neighborhood where Fenichel

coordinate works.

(H4) dim ω(N ) = σ_{− 1 and G(0, y; 0) /∈ T}yω(N ) for y∈ ω(N).

Let N = M0

∪Wcs_(Z

0). Then dimN = (k + σ) + (n + l + 1)−(m+n) = σ ≥ 1. For any τ0> 0 > 0, let

0 < δ < τ0. Consider ω(N )·(τ0−δ, τ0+ δ). For any q∈ W0u(ω(N )·(τ0−δ, τ0+ δ)), let y1= α(q)∈ Z0.

∃!(y0_{, τ}∗₎

∈ ω(N) · (τ0− δ, τ0+ δ) such that y1 = y0· τ∗. Furthermore, ∃!p ∈ N ∩ ∂N such that

(i) If y1_{< P}

0(y0), then a portion of Mεclose to p is mapped into an O(ε) neighborhood of q. Near

q, Mε is C1-close to W₀u(ω(N )_{· (τ}0)− δ, τ0) + δ)p)

(ii) If y1_{= P}

0(y0), then a portion of Mε will be C1-closed to Wcu(P0(ω(N ))).

(iii) If y1_{> P}

0(y0), then there is no portion of Mεthat comes near q.

3

Applications to the Existence of Traveling Waves Solutions

in Fitz-Hugh Nagumo Type Equations

3.1

Formulation of GSP Problems

In this section, we consider the traveling wave solutions of system (1.1) by assuming u(x, t) = u(x + ct) = u(ξ) and w(x, t) = w(x + ct) = w(ξ) for some real constant c > 0, which is the speed of traveling waves. Under such assumptions, the profile equations of (1.1) yield to

     cu0(ξ) = u00(ξ) + f (u(ξ), w(ξ)), cw0(ξ) = εg(u(ξ), w(ξ)). (3.13) Introducing v = u0, then (3.13) can be rewritten as

             u0(ξ) = v(ξ), v0(ξ) = cv(ξ)_{− f(u(ξ), w(ξ)),} cw0(ξ) = εg(u(ξ), w(ξ)). (3.14)

In terms of the slow variable η := εξ, we have              ε ˙u(η) = v(η), ε ˙v(η) = cv(η)_{− f(u(η), w(η)),} ˙ w(η) = c−1g(u(η), w(η)), (3.15)

here “_{· ” means dη}d . Systems (3.14) and (3.15) are equivalent which give the standard singularly perturbed system in fast and slow scales respectively. Assume set E := _{{w | w = a(w)} is a} non-empty set, then system (3.14) or (3.15) has equilibria: (0, 0, 0), (1, 0, 1) and (a(w0), 0, w0) with w0∈ E.

We are interest in traveling waves solutions related to such equilibria.

The main application of geometric singular perturbation theory to the problem is to lift limiting singular orbits to traveling wave solutions. The following we examine the limiting slow and fast dynamics of (3.15) and (3.14) respectively.

3.1.1 Dynamics for the Limiting Slow System The limiting slow dynamics is governed by

0 = v, 0 = cv_{− f(u, w),} w = c˙ −1g(u, w). (3.16) Thus the slow manifold M consists of three parts by M := M0∩ Ma∩ M1, where

M0:={u = v = 0}, Ma :={u = a(w), v = 0}, M1:={u = 1, v = 0}.

It is easy to see that M0and M1are invariant with respect to the flow (3.14) for all ε, and equilibrium

(0, 0, 0) or (1, 0, 1) attracts all solutions of (3.16) on M0 or M1 respectively. If we allow a(w) crossing

0 and 1, then there exists a special type of turning points on M0 and M1. We will see that the

invariance of M0and M1 plays a crucial role when we consider the limiting slow orbits pass through

the turning points.

3.1.2 Dynamics for the Limiting Fast System The limiting fast dynamics is governed by

u0= v, v0= cv_{− f(u, w), w}0= 0, (3.17) According to (3.16), the slow manifold M consists of equilibria of (3.17). From above equations, we know that each plane_{{w = const} is invariant, and there exist three equilibria of system (3.17):}

E0:= (0, 0, w)∈ M0, Ea(w) := (a(w), 0, w)∈ Ma and E1:= (1, 0, w)∈ M1.

Let λ±₀(w, c), λ±_a(w, c), λ±₁(w, c), be the linearized eigenvalues of system (3.17) with respect to E0, Ea

and E1 respectively. Then we have

λ±₀(w, c) =c±pc 2_{+ 4a(w)} 2 , (3.18) λ±_a(w, c) =c±pc 2_{+ 4a(w)(a(w)− 1)} 2 , (3.19) λ±₁(w, c) =c±pc 2_{+ 4(1− a(w))} 2 . (3.20) If c_{≥ 1, then λ}±a(w, c) are real. If c < 1 then the sign of above real eigenvalues with respect to the

CPX CPX CPX 0 < λ−0 < λ+0 λ−0 < 0 < λ+0 λ− a < 0 < λ+a 0 < λ−a < λ+a a(w) 0 < λ− a < λ+a λ−a < 0 < λ+a λ−₁ < 0 < λ+ 1 0 < λ−1 < λ+1 −c 2 4 0 1−√1− c2 2 1 +√1− c2 2 1 1 + c2 4 2

Figure 1: Sign of linearized eigenvalues with respect to the range of a(w), where CPX means that the eigenvalues are conjugate complex numbers in the range of a(w).

Therefore, all the linearized eigenvalues are real in the region Ω defined by Ω := _{{(w, c) ∈ [0, 1] × R}+: a(w)_{∈ [−}c 2 4, 1 + c2 4] for c > 1; or a(w)_{∈ [−}c 2 4,∞) \ ( 1₋√1_{− c}2 2 , 1 +√1_{− c}2 2 ) for c < 1}.

Now we consider the dynamics of (3.17). On each plane _{{w = const}, the limiting system is} that for a prototype FitzHugh-Nagumo equation with specific cubic nonlinearity. The existence of heteroclinic orbits on the plane is well understood, cf. [4]. To classify all the possible heteroclinic orbits of (3.17), we first introduce the following notations:

a1(c) := max{0, 1₋√2c 2 }, a2(c) := min{1, 1 +√2c 2 }, a3(c) := 2 + √ 2c; a4(c) :=−1 − √ 2c, a5(c) := max{1, 2 − √ 2c_{}, a}6(c) := min{0, −1 + √ 2c_}; Hi(c) :={w ∈ (0, 1) : a(w) = ai(c), a0(w)6= 0}, i = 1, · · · , 6, G1(c) :={w ∈ (0, 1) : a(w) ≤ a6(c)}, G2(c) :={w ∈ (0, 1) : a(w) ≥ a5(c)}, G3(c) :={w ∈ (0, 1) : 0 > a(w) > a4(c)}, G4(c) :={w ∈ (0, 1) : 1 < a(w) < a3(c)}, G5(c) :={w ∈ (0, 1) : 0 < a(w) < a2(c)}, G6(c) :={w ∈ (0, 1) : 1 > a(w) > a1(c)}.

Furthermore, for any fixed w_{∈ [0, 1] we denote r → s to be the heteroclinic orbit connecting (r, 0, w)} to (s, 0, w), where r _{6= s and r, s ∈ {0, a(w), 1}. According to the results of [4] and phase plane} analysis, various types of heteroclinic orbits with respect to different regions of the parameters can be classified in Table 1.

Type of Orbit Admissible Parameter Condition Region 0_{→ 1} a(w) = a1(c) or a(w)≤ a6(w) w∈ H1∪ G1

1_{→ 0} a(w) = a2(c) or a(w)≥ a5(w) w∈ H2∪ G2

0_{→ a(w)} a(w) = a3(c) or a4(c) < a(w) < 0 w∈ H3∪ G3

1_{→ a(w)} a(w) = a4(c) or 1 < a(w) < a3(c) w∈ H4∪ G4

a(w)_{→ 0} a(w) = a5(c) or 0 < a(w) < a2(c) w∈ H5∪ G5

a(w)_{→ 1} a(w) = a6(c) or a1(c) < a(w) < 1 w∈ H6∪ G6

Table 1: Classification of admissible heteroclinic orbits.

Note that the linearized eigenvalues are real in the region Ω. Throughout this work, we redefine sets Hi and Gi in Table 1 by Hi∩ Ω and Gi∩ Ω, with a slight abusing the notation, we keep the same

notations. The regions of Ω, Hi and Gi are illustrated in Figure 2.

c a(w) 1 a1(c) a2(c) H1 H2 a6(c) a5(c) a4(c) a3(c) G5 G6 H1∩ H6 H2∩ H5 G4 G3 H6 H5 G2 G1 H3 H4 G1∩ G3 G2∩ G4 G5∩ G6 1 2 Ω Ω Ω a(w) = 1 +c 2 4 a(w) = −c 2 4 a(w) =1 + √ 1 − c2 2 a(w) =1 − √ 1 − c2 2

Remark 3.1

(1) As shown in [4], if w = w0 ∈ Hi(c), i = 3, 4, 5, 6, then the exact formulas for the heteroclinic

orbits (u(t; w0), v(t; w0)) of (3.17) can be expressed as follows:

u(t; w0) =                     

a(w0)− a(w0)(1 + ea(w0)t/ √

2₎−1_{, if w}

0∈ H3(c);

a(w0) + (1− a(w0))(1 + e(1−a(w0))t/ √ 2₎−1_{, if w} 0∈ H4(c); a(w0)(1 + ea(w0)t/ √ 2₎−1_{, if w} 0∈ H5(c); 1_{− (1 − a(w}0))(1 + e(1−a(w0))t/ √ 2₎−1_{, if w} 0∈ H6(c).

Base on the above formulas, the Melnikov functions (first and second order) for invariant man-ifolds of connecting orbits can be derived explicitly in terms of Beta or Gamma functions, for details see Section 4.

(2) In [33], they examed traveling waves whose slow orbits lie only on the portions M0 and M1 of

the slow manifold, thus only regions H1, H2, G1, G2 are considered (see dash paths of Figure 3).

To generalize their work to traveling waves whose slow orbits lie on all portions of M , we need to consider some additional regions than those of [33] (see the non-dash path of Figure 3).

H4 H3 G5 G6 H1∩ G1 H2∩ G2 G3 G4 H5 H6 a a a 0 1

Figure 3: Admissible heteroclinic orbits with respect to regions.

Next, we investigate the normal hyperbolicity of the slow manifolds. The normal hyperbolicity of the slow manifold of M0 or M1 is determined by the eigenvalues λ±0(w, c) or λ

±

1(w, c), respectively.

If (0, 0, w) _{∈ M}0 at which a(w) = 0, then λ−0(w, c) = 0 and the slow manifold M0 loses normal

hyperbolicity at this point. Similarly, the slow manifold M0 loses normal hyperbolicity at points

(1, 0, w)_{∈ M}1 satisfying a(w) = 1. All such points are called turning points. Since M0 and M1 are

invariant, the existence of turning points on them can cause the phenomena of delay of stability loss, see [32]. To describe the results for delay of stability loss, exchange lemma with turning points and our main theorems, in this article we assume the curve u = a(w) crossing u = 0 and u = 1 on the plane v = 0, and satisfying the following assumption:

(H) there exist (increasing) ordered sets_{Ti 0} p i=1,{T j 1} q j=1⊆ [0, 1] such that a(T₀i) = 0, a(T₁j) = 1, a0(T₀i)_{6= 0, a}0(T₁j)_{6= 0,} for all 1_{≤ i ≤ p and 1 ≤ j ≤ q.}

By (H), the sets of points_{{(0, 0, T}i 0)} p i=1 and {(1, 0, T j 1)} q

j=1are turning points on the slow manifold

M0 and M1 respectively. For the position of equilibria and turning points, dynamics on the slow

manifold and heteroclinic orbit for fast dynamics, see Figure 4.

¿From Table 1 and the hyperbolicity of slow manifold for the limiting system (2.4), we plan to construct singular orbits (unions of slow and fast orbits) as candidates for limits of traveling wave solutions. Then we can obtain the existence of traveling wave solutions of (3.13) by applying the geometric singular perturbation theorem to lift singular orbits to the true orbits.

u

v

w

M

M

E

E

E

M

T

T

T

T

Γ

M

M

M

M

Figure 4: Equilibria, turning points, dynamics on the slow manifold and heteroclinic orbit Γ of the fast dynamics which connects E0and (1, 0, 0) when a(0) = a1(c) or a(0)≤ a6(0). The red segments M0,1− on

M0,1 are defined by M0−={(0, 0, w) ∈ M0| λ0−(w, c) < 0} and M1−={(1, 0, w) ∈ M1| λ1−(w, c) < 0}.

3.1.3 Delay of Stability Loss and Exchange Lemma

In this section, we recall and reformulate the results in [32, 33] about the delay of stability loss and exchange lemma for turning points. For any fixed c > 0, let’s denote

M₀−:=_{{(0, 0, w) ∈ M}0|λ0−(w, c) < 0} and M1−:={(1, 0, w) ∈ M1|λ1−(w, c) < 0}.

If the above sets are non-empty, then we define two maps P0 and P1on such sets as following.

(P0) Let P0: M0− → M0 be defined by P0(0, 0, w) =    (0, 0, w), if w exists, (0, 0, 0), otherwise, where w _{∈ (0, w) is} the first value such that

Z w w λ0 −(η, c) g(0, η) dη = 0. (P1) Let P1: M1− → M1 be defined by P1(1, 0, w) =    (1, 0, w), if w exists, (1, 0, 1), otherwise, where w _{∈ (w, 1) is} the first value such that

Z w

w

λ1 −(η, c)

g(1, η) dη = 0.

Base on the above two maps, Liu and Van Vleck [33] reformulated the exchange lemma on M0

and M1 for system (3.14) with an extra equation c0= 0, that is

u0(ξ) = v, v0(ξ) = cv_{− f(u, w), w}0(ξ) = εc−1g(u, w), c0= 0. (3.21) To promise the existence of unstable manifold W0u(K) and center manifold W0c(K) for any set K ⊂

M0∪ M1, we restrict c belonging to the following set

S :=_{{c > 0 : a(w) ∈ [−}c 2 4, 1 + c2 4] for all w∈ [0, 1])}. Denote M₁δ(w) := _{{(1, 0, w) ∈ M}1: w∈ (w − δ, w + δ)}, M0δ(w) := {(0, 0, w) ∈ M0: w∈ (w − δ, w + δ)}, M_aδ(w) := _{{(a(w), 0, w) ∈ M}a: w∈ (w − δ, w + δ)},

for any small δ > 0 and any w _{∈ [0, 1]. The exchange lemma for M}1 with turning points is stated as

follows.

Proposition 3.2 (Exchange Lemma with Turning point, cf. [32, 33]) Let Mε _{be a two-dimensional}

invariant manifold of system (3.21) which is smooth in ε. For ε = 0, suppose that M0 intersects W0c(M1× (c1, c2)) transversally. Let N be the intersection. Then dim N =1. Suppose that ω(N ) =

(1) If w2< P1(w1), then for ε > 0 small, a portion of Mεwill approach (1, 0, w1, c∗), follow the slow

orbit from (1, 0, w1, c∗) to (1, 0, w2, c∗), leave the vicinity of M1× (c1, c2), and upon leaving, it

is C1 O(ε)-close to the unstable manifold Wu(M1δ(w2)× {c∗}) for some δ > 0 independent of ε

(see Figure 5).

(2) If w2 = P1(w1) * {T11, T12, ..., T q

1}, then for ε > 0 small, a portion of Mε will approach

(1, 0, w1, c∗), follow the slow orbit from (1, 0, w1, c∗) to (1, 0, w2, c∗), leave the vicinity of M1×

(c1, c2), and upon leaving, it is C1 O(ε)-close to the center-unstable manifold Wcu(1, 0, w2, c∗)

(see Figure 6).

(3) If w2> P1(w1), then for ε > 0 small, there is no portion of Mεthat approaches (1, 0, w1, c∗),

fol-lows the slow orbit from (1, 0, w1, c∗), leave the vicinity of M1in a neighborhood of (1, 0, w2, c∗).

For singular orbits passing no turning point, we use the following exchange lemma without turning points.

Proposition 3.3 (Exchange lemma without Turning point, cf. [27, 30, 38]) Let Mε _{be a}

two-dimensional invariant manifold of system (3.21) which is smooth in ε. For ε = 0, suppose that M0_{intersects W}c

0(Ma× {c∗}) transversally. Let N be the intersection. Then dim N=1. Suppose that

ω(N ) = _{(a(w1), 0, w1, c∗)}. Let w2 be any number such that a(w) 6= 0 or 1, for all w between w1

and w2. then for ε > 0 small, a portion of Mεwill approach (a(w1), 0, w1, c∗), follow the slow orbit

from (a(w1), 0, w1, c∗) to (a(w2), 0, w2, c∗), leave the vicinity of Ma× {c∗}, and upon leaving, it is C1

O(ε)-close to the unstable manifold Wu_(Mδ

u v (w, c) Wu_(M 0× Cδ(c∗)) Wu_(M 1× Cδ(c∗)) E0 M0× Cδ(c∗) Mε Mε M1× Cδ(c∗) Wc_(M 0× Cδ(c∗)) Wc_(M 1× Cδ(c∗)) N 1 (P1(w1), c∗) (w2+ δ, c∗) (w2, c∗) (w2− δ, c∗) (w1, c∗) 5

Figure 5: Part (1) of Proposition 3.2. In this graph, we denote Cδ(c) := (c− δ, c + δ) for some δ > 0

u v (w, c) Wu_(M 0× Cδ(c∗)) Wu_(M 1× Cδ(c∗)) E0 M0× Cδ(c∗) Mε Mε M1× Cδ(c∗) Wc_(M 0× Cδ(c∗)) Wc_(M 1× Cδ(c∗)) N 1 (w2, c∗) (w1, c∗) 6

Figure 6: Part (2) of Proposition 3.2. In this graph, we denote Cδ(c) := (c− δ, c + δ) for some δ > 0

and identify the (w, c)-plane with the vertical axis.

3.1.4 Admissible Conditions for Singular Orbits

In view of the results of exchange lemmas with turning points, not all singular orbits are shadowed by true orbits. To guarantee the shadowing property, we introduce some admissible conditions for the construction of singular orbits.

Let w = (w1, w2, ..., wn) with wi ∈ [0, 1] and s = (s1, s2, ..., sn+1) with s1 = 1, si ∈ {0, a, 1},

si 6= si+1 and sn+1∈ {0, 1}. For any two words w and s, we denote the singular orbit starting from

0 to sn+1 by 0 → s1 → ... → sn+1 such that the local path si → si+1 (part of the orbit from si to

si+1) occurring at the plane w = wi. Here we only discuss the case, for all w∈ (wi−1, wi), a(w) > 1 if

wi∈ H5(c∗) and a(w) < 0 if wi∈ H6(c∗). The singular orbit on the manifold Ma will pass a turning

point, or else. Because the manifold Ma doesn’t persist, the exchange lemma fails. As a result, we

                           wi∈ H1(c∗)∪ G1(c∗)\ {T01, T02, ..., T p 0}, when sisi+1= 01, wi∈ H2(c∗)∪ G2(c∗)\ {T11, T12, ..., T q 1}, when sisi+1= 10, wi∈ H3(c∗)∪ G3(c∗)\ {T01, T02, ..., T p

0}, when sisi+1= 0a,

wi∈ H4(c∗)∪ G4(c∗)\ {T11, T 2 1, ..., T

q

1}, when sisi+1= 1a,

wi∈ H5(c∗), when sisi+1= a0,

wi∈ H6(c∗), when sisi+1= a1,

(3.22)

for i = 1,_{· · · , n and the following conditions (A1)∼(A3) hold:} (A1) P1(0) > w1and    a(w) < 1,_{∀w ∈ [w}n, 1], if sn+1= 1, a(w) > 0,_{∀w ∈ [0, w}n], if sn+1= 0;

(A2) For si = 0, wi−1> wiand

P0(wi−1)    < wi, when wi∈ H1(c∗)∪ H3(c∗), = wi, when wi∈ G1(c∗)∪ G3(c∗)\ {T01, T02, ..., T p 0};

(A3) For si = 1, wi−1< wiand

P1(wi−1)    > wi, when wi∈ H2(c∗)∪ H4(c∗), = wi, when wi∈ G2(c∗)∪ G4(c∗)\ {T11, T12, ..., T q 1}.

Furthermore, we say that a word w = (w1, w2, ..., wn) is admissible with respect to c∗ if there is a

word s = (s1, s2, ..., sn+1) with s1 = 1, si ∈ {0, a, 1}, si 6= si+1 and sn+1 ∈ {0, 1}, such that w is

s-admissible with respect to c∗.

3.2

Existence of Traveling Waves

According to the exchange lemma and the admissible conditions illustrated in previous section, we state the main theorems in this section and prove them in next section. For a description of the statement of our main results, we give the following definition.

Definition 3.4 Let _{O be a singular orbit for some fixed c}∗ _{> 0. The singular orbit O is “weakly}

shadowed” if for any neighborhood _{U of the O, there is an ε}0> 0 such that, for all 0 < ε≤ ε0, there

is a full orbit _{O(ε) ∈ U of system (3.14) with c = c(ε) and (O(ε), c(ε)) → (O, c}∗) as ε _{→ 0 with} respect to the Hausdorff distance of sets. Furthermore, if c(ε) = c∗ for all 0 < ε_{≤ ε}0, then we say the

First, we consider the traveling wave solutions connecting (0,0,0) to (1,0,1). From Table 1, we know that such kind of traveling wave solutions exist only if s1 = 1 or s1= a(0). If s1 = 1 then it’s

required that λ1₋(0; c) < 0 and λ1₋(1; c) < 0 to guarantee the first and last connection. It’s easy to see that these two conditions are equivalent to a(0) < 1 and a(1) < 1 respectively. In addition, it could be seen that the structures of traveling wave solutions are dramatically different for different sign of a(0). Therefore, we will consider two situations a(0) > 0 and a(0) < 0 separately.

If a(0) > 0, then there exists a unique c∗with a1(c∗) = a(0) (in fact c∗= (1−2a(0))/

√

2) such that system (3.17) has a heteroclinic orbit connecting from (0,0,0) to (1,0,0) approaching (0,0,0) backward along the eigenvector associated to λ+₀(0, c∗).

Theorem 3.5 Assume 0 < a(0) < 1, a(1) < 1 and c∗_{∈ S be the unique value such that a}1(c∗) = a(0).

(1) If w = (w1, w2, ..., wn) is admissible with respect to c∗, then the associated singular orbit is

weakly shadowed.

(2) If w = (w1, w2, ..., wn) is not admissible with respect to c∗, then the associated singular orbit is

not weakly shadowed.

If a(0) _{≤ 0, from Table 1, system (3.17) possess a heteroclinic orbit connecting from (0,0,0) to} (1, 0, 0) only if a(0)_{≤ a}6(c), or equivalent to c∈ Λ := {c : c ≥ (1 + a(0))/

√ 2_}. Theorem 3.6 Assume a(0) < 0, a(1) < 1 and c∗_{∈ Λ ∩ S.}

(1) If w = (w1, w2, ..., wn) is admissible with respect to c∗, then the associated singular orbit is

strongly shadowed.

(2) If w = (w1, w2, ..., wn) is not admissible with respect to c∗, then the associated singular orbit is

not weakly shadowed.

Next, we consider the traveling wave solutions connecting (0,0,0) to (0,0,0), i.e. traveling pulse solutions. By Table 1, we know that such kind of traveling wave solutions exist only if s1 = 1 or

s1 = a(0). If s1 = 1, then it’s required that λ1−(0; c) < 0 and λ0−(0; c) < 0 to guarantee the first

and last connection. Both conditions are equivalent to 0 < a(0) < 1. If 0 < a(0) < 1 then there exists a unique c∗ with a1(c∗) = a(0) (in fact c∗ = (1− 2a(0))/

√

2) such that system (3.17) has a heteroclinic orbit from (0,0,0) to (1,0,0) approaching (0,0,0) backward along the eigenvector associated to λ+₀(0, c∗).

(1) If w = (w1, w2, ..., wn) is admissible with respect to c∗, then the associated singular orbit is

weakly shadowed.

(2) If w = (w1, w2, ..., wn) is not admissible with respect to c∗, then the associated singular orbit is

not weakly shadowed.

3.3

Proof of the Existence Results

3.3.1 Melnikov function and Transversality of Manifolds

To prove the main results in the last subsection , we first detect the transversality of invariant manifolds for connecting orbits by investigating the Melnikov function.

First, we recall the results for the formula of Melnikov function, [5, 20, 27, 34]. Lemma 3.8 Consider the plane system:

y0= R0(y) + ¯εR1(y, ¯ε), (3.23)

where ¯ε_{≥ 0 and R}0, R1 ∈ Cr with r≥ 2. Suppose that y01 and y20 are two different hyperbolic saddle

points of (3.23)_|ε=0¯ , and there exists a heteroclinic orbit y0(t) of (3.23)|¯ε=0connecting from y10 to y02.

Then the Melnikov function of (3.23) is M(y0) = Z ∞ −∞ e−R0tσ(s)dsD(t)dt (3.24) where σ(t) = tr∂R0 ∂y (y0(t)) and D(t) = R0(y0(t))∧ R1(y0(t), 0).

According to formula (3.24), we can compute the Melnikov function of system (3.17) in the fol-lowing.

Lemma 3.9 Suppose, for some c0 and w0, system (3.17) has a heteroclinic orbit Γ := r(t; w0) =

(u0(t; w0), v0(t; w0)).

(1) For fixed w = w0 and varying c, the Melnikov function with respect to the heteroclinic orbit Γ is

given by M(c0) = Z ∞ −∞ e−c0t_v2 0(t; w0)dt. In particular,_M(c0)6= 0.

(2) For fixed c = c0 and varying w, the Melnikov function is given by

M(w0) = a0(w0)

Z ∞

−∞

e−c0t_v

Proof. (1) For fixed w = w0, let F (u, v; c) be the vector field of system (3.17), i.e., F (u, v; c) =

(v, cv_{− u(u − a(w}0)(1− u))) and define Fc(u, v; c) := (0, v). Applying Lemma 3.8 by taking ¯ε = c,

the Melnikov function is M(c0) = Z ∞ −∞ e−R0ttrDF (r(s;w0);c0)ds_{(F (r(t; w} 0); c0)∧ Fc(r(t; w0); c0)) dt = Z ∞ −∞ e−c0t_v2 0(t; w0)dt6= 0.

(2) For fixed c = c0, we have F (u, v; w) = (v, c0v− u(u − a(w)(1 − u))). Denote Fw(u, v; w) :=

(0, u(1_{− u)a}0(w)). Applying Lemma 3.8 by taking ¯ε = w, the Melnikov function is M(w0) = Z ∞ −∞ e−R0ttrDF (r(s;w0);w0)ds_{(F (r(t; w} 0); w0)∧ Fw(r(t; w0); w0)) dt = a0(w0) Z ∞ −∞ e−c0t_v 0(t; w0)u0(t; w0)(1− u0(t; w0))dt.

The proof is complete.

Base on the results of Lemma 3.9, we now compute the first order Melnikov function of system (3.17) when w varies in different parameter regions.

Lemma 3.10 Assume a0(w0) 6= 0, then M(w0) 6= 0 for any c0 ∈ (0, 1/

√

2) and w0 ∈ Hi(c0), i =

1._{· · · 6.}

Proof. (1) If w0∈ H1(c0)∪ H2(c0) then u0(t; w0)∈ (0, 1) for all t. By Lemma 3.9, we have

M(w0) = a0(w0) Z ∞ −∞ e−c0t_v 0(t; w0)u0(t; w0)(1− u0(t; w0))dt6= 0. (2) If w0 ∈ H5(c0) then a(w0) = 2− √

2c0∈ (1, 2). According to Remark 2.1, the heteroclinic orbit

(u0(t; w0), v0(t; w0)) can be represented explicitly in the following:

u0(t; w0) = a(w0)(1 + ea(w0)t/ √ 2₎−1 _{and v} 0(t; w0) = u00(t; w0). Thus e−ct = (a(w0)− u0(t; w0))`u−`0 (t; w0), where 0 < ` := 1− 2 a(w0) < 1/2. We can compute equation (3.25) by

M(w0) a0_(w 0) = Z 0 a(w0) u1−`(a(w0)− u)`(1− u)du = Z a(w0) 0 u2−`(a(w0)− u)`du− Z a(w0) 0 u1−`(a(w0)− u)`du = a3(w0) Z 1 0 t2−`(1<sub>− t)</sub>`dt_{− a}2(w0) Z 1 0 t1−`(1<sub>− t)</sub>`dt = a3(w0)B(1 + `, 3− `) − a2(w0)B(1 + `, 2− `) = a3(w0) (Γ(1 + `)Γ(3− `)/Γ(4)) − a2(w0) (Γ(1 + `)Γ(2− `)/Γ(3)) = a2(w0)(a(w0)− 1)Γ(1 + `)Γ(2 − `)/Γ(4) > 0,

where B(x, y) and Γ(x) are the Beta function and the Gamma function respectively. Note that B(x, y) = Γ(x)Γ(y)/Γ(x + y).

(3) If w0∈ H3(c0) then a(w0) = 2 +√2c0∈ (2, 3). Similar to the proof of part (2), the heteroclinic

orbit (u0(t; w0), v0(t; w0)) satisfies

u0(t; w0) = a(w0)− a(w0)(1 + ea(w0)t/ √

2₎−1_,

e−ct = (a(w0)− u0(t; w0))`u0(t; w0) −`

, where 0 < ` < 1/3. After simple computation, we can obtain

M(w0) = a0(w0)a2(w0)(1− a(w0))Γ(1 + `)Γ(2− `)/Γ(4) < 0.

(4) Similarly, if w0 ∈ H4(c0) then a(w0) = −1 −

√

2c0 ∈ (−2, −1) and the heteroclinic orbit

(u0(t; w0), v0(t; w0)) satisfies

u0(t; w0) = a(w0) + (1− a(w0))(1 + e(1−a(w0))t/ √

2

)−1, e−ct = (u0(t; w0)− a(w0))γ(1− u0(t; w0))−γ,

where γ := 1_{− 2(1 − a(w}0))−1. Therefore 0 < γ < 1/3 and

M(w0) =−a0(w0)a(w0)(1− a(w0))2Γ(1 + γ)Γ(2− γ)/Γ(4) > 0.

(5) Finally, if w0∈ H6(c0) then a(w0) =−1+

√

2c0∈ (−1, 0) and the heteroclinic orbit (u0(t; w0), v0(t; w0))

satisfies

u0(t; w0) = 1− (1 − a(w0))(1 + e(1−a(w0))t/ √

2₎−1_,

where_{−1 < γ < 0. Then we have}

M(w0) = a0(w0)a(w0)(1− a(w0))2Γ(1 + γ)Γ(2− γ)/Γ(4) < 0.

The proof is complete. However, if a0_(w

0) = 0 in Lemma 3.10 then M(w0) = 0, and there give no information about

the transversality of the invariant manifolds. Therefore we need to compute the higher order term of Melnikov function to detect the transversality of the invariant manifolds. The following we only investigate the second-order term of Melnikov function_M2(w0).

Lemma 3.11 Suppose, for some small c0 and w0, a0(w0) = 0 and system (3.17) has a heteroclinic

orbit Γ0:= (u0(t; w0), v0(t; w0)). For fixed c = c0and varying parameter w, the second order Melnikov

function is given by M2(w0) = a00(w0) Z ∞ −∞ e−c0t_v 0(t; w0)u0(t; w0)(1− u0(t; w0))dt.

Proof. Without lost of generality, we may assume w0= 0. For such fixed w near w0, let’s write the

system (3.17) in the following vector form d dt   u(t; w) v(t; w)  =   v(t; w)

c0v(t; w)− u(t; w)(1 − u(t; w))(u(t; w) − a(w))

  (3.26) = R0(r(t; w)) + R1(r(t; w))w + R2(r(t; w))w2+ O(w3), where r(t; w) := (u(t; w), v(t; w))T_, R0(r(t; w)) =   v(t; w)

c0v(t; w) + u(t; w)3− (a(0) + 1))u(t; w)2+ a(0)u(t; w)

 , R1(r(t; w)) =   0 a0(0)(u(t; w)_{− u}2_{(t; w)))}  , R2(r(t; w)) =   0 a00(0)(u(t; w)_{− u}2_{(t; w))/2}  .

As w = 0, system (3.17) has a heteroclinic orbit r(t; w) connecting two equilibria, E1

0 and E02. Let

L be a line segment transversal to r(t; 0) at r(0; 0). For sufficiently small w, there exists a unique bounded solution ru_{(t; w) for t}

≤ 0 such that ru_{(t; w) in the unstable manifold of one equilibrium E}1 w

and ru_{(0; w)}

∈ L. For t ≤ 0, let’s define zu(t) := ∂

∂wr

u_{(t; w)}

|w=0, 4u(t) := zu(t)∧ R0(ru(t; 0)),

Differentiating equation (3.26) with respect to w, we have d dt ∂ ∂wr u_{(t; w) =} ∂R0 ∂r (r u_{(t; w))} ∂ ∂wr u_{(t; w) +}∂R1 ∂r (r u_{(t; w))} ∂ ∂wr u_{(t; w)w + R} 1(ru(t; w)) +∂R2 ∂r (r u_{(t; w))} ∂ ∂wr u_{(t; w)w}2_{+ 2R} 2(ru(t; w))w + O(w2). Thus d dty u_{(t) =} ∂R0 ∂r (r u_{(t; 0))y}u_{(t) + (} ∂ ∂w ∂R0 ∂r (r u_{(t; w))} |w=0)zu(t) +2∂R1 ∂r (r u_{(t; 0))z}u_{(t) + 2R} 2(ru(t; 0)), d dt u_{(t) = (}d dty u_(t)) ∧ R0(ru(t; 0)) + yu(t)∧ ( ∂R0 ∂r (r u_{(t; 0))R} 0(ru(t; 0))) = tr∂R0 ∂r  u_{(t) + 2R} 2∧ R0+ ( ∂ ∂w ∂R0 ∂r (r u_{(t; w))} |w=0)zu(t)∧ R0(ru(t; 0)) +2∂R1 ∂r (r u_{(t; 0))z}u_(t) ∧ R0(ru(t; 0)) = _σ(t)u(t) + D2(t), where σ(t) = tr∂R0 ∂r and D2(t) = v0u0(1− u0)a00(0) + 2v0[3u0− a(0) − 1]( ∂u ∂w|w=0) 2_{= v} 0u0(1− u0)a00(0), since ∂u ∂w|w=0= ∂u ∂aa

0_{(0) = 0. Our purpose is to compute }u_{(0). By the variation of constant formula,}

u(t) = eR0tσ(τ )dτ  u(0) + Z t 0 e−R0τσ(s)dsD₂(τ )dτ  , for t < 0. Since lim t→−∞e −Rt 0σ(τ )dτu(t) = 0, we have u(0) = Z 0 −∞ e−R0τσ(s)dsD₂(τ )dτ.

Similarly, there exists a unique bounded solution rs_{(t; w) for t}

≥ 0 such that rs_{(t; w) in the unstable}

manifold of the other equilibrium E2

w and rs(0; w)∈ L. For t ≥ 0, let’s define

ys(t) := ∂

2

∂2_wr s_{(t; w)}

|w=0 and s(t) := ys(t)∧ R0(rs(t; 0)).

By the similar computation, we have s(0) = Z 0 ∞ e−R0τσ(s)dsD₂(τ )dτ. Hence M2(0) = u(0)− s(0) = a00(0) Z ∞ −∞ e−c0t_v 0(t)u0(t)(1− u0(t))dt.

The proof is complete.

Corollary 3.12 Let’s assume the same assumptions as stated in Lemma 3.11. If a00(w0)6= 0 then

M2(w0)6= 0 for any c0∈ (0, 1/

√

2) and w0∈ Hi(c0), i = 1.· · · 6.

For more higher order terms of Melnikov function, the computation is similar but more complicated. The following we only illustrate the general result, and skip the proof.

Lemma 3.13 Suppose, for some small c0 and w0, a(i)(w0) = 0 for all 1≤ i < k, where k is a given

positive integer, and system (3.17) has a heteroclinic orbit Γ0:= (u0(t; w0), v0(t; w0)). For fixed c = c0

and varying parameter w, the kth order Melnikov function is given by Mk(w0) = a(k)(w0)

Z ∞

−∞

e−c0t_v

0(t; w0)u0(t; w0)(1− u0(t; w0))dt.

As a consequence of previous lemmas and corollary, we have the following conclusions for the transver-sality of invariant manifolds.

Lemma 3.14 Let M t N be in the sense that manifolds M and N intersect transversally, and Cδ(c) := (c− δ, c + δ).

(1) Consider system (3.17) with c = c0∈ (0, 1/

√

2). The transversality of various invariant mani-folds of (3.17) along Γ(w) are illustrated in Table 2.

Region of w transversality of manifolds along Γ(w) w_{∈ H}1(c0) W0u(M0δ(w)) t W0c(M1δ(w)) w_{∈ H}2(c0) W0u(M1δ(w)) t W0c(M0δ(w)) w_{∈ H}3(c0) W0u(Maδ(w)) t W0c(M0δ(w)) w_{∈ H}4(c0) W0u(M1δ(w)) t W0c(Maδ(w)) w_{∈ H}5(c0) W0u(Maδ(w)) t W0c(M0δ(w)) w_{∈ H}6(c0) W0u(Maδ(w)) t W0c(M1δ(w)) w_{∈ G}1(c0) W0cu(0, 0, w) t W0c(M1δ(w)) w_{∈ G}2(c0) W0cu(1, 0, w) t W0c(M0δ(w)) w_{∈ G}3(c0) W0cu(0, 0, w) t W0c(Maδ(w)) w_{∈ G}4(c0) W0cu(1, 0, w) t W0c(Maδ(w))

Table 2: Transversalities of manifolds Wc

0(M0,1,aδ (w)), W0u(M0,1,aδ (w)), W0cu(0, 0, w) and

Wcu

0 (1, 0, w).

(2) Consider (2.10) with c = c0 ∈ (0, 1/

√

Region of w transversality of manifolds along Γ(w)_{× {c}0} w_{∈ H}1(c0) W0u(M0δ(w))× {c0} t W0c(M1δ(w)× Cδ(c0)) w_{∈ H}2(c0) W0u(M1δ(w))× {c0} t W0c(M0δ(w)× Cδ(c0)) w_{∈ H}3(c0) W0u(M δ 0(w)× {c0}) t W0c(M δ a(w)× Cδ(c0)) w_{∈ H}4(c0) W0u(M1δ(w))× {c0} t W0c(Maδ(w)× Cδ(c0)) w_{∈ H}5(c0) W0u(M δ a(w)× {c0}) t W0c(M δ 0(w)× Cδ(c0)) w_{∈ H}6(c0) W0u(Maδ(w))× {c0} t W0c(M1δ(w)× Cδ(c0)) w_{∈ G}1(c0) W0cu(0, 0, w, c0) t W0c(M δ 1(w)× Cδ(c0)) w_{∈ G}2(c0) W0cu(1, 0, w, c0) t W0c(M0δ(w)× Cδ(c0)) w_{∈ G}3(c0) W0cu(0, 0, w, c0) t W0c(M δ a(w)× Cδ(c0)) w_{∈ G}4(c0) W0cu(1, 0, w, c0) t W0c(Maδ(w)× Cδ(c0))

Table 3: Transversalities of manifolds Wc

0(M0,1,aδ (w)× Cδ(c0)), W0u(M0,1,aδ (w))× {c0},

Wcu

0 (0, 0, w, c0) and W0cu(1, 0, w, c0).

Now we begin the proof of the main theorems. Proof of Theorem 3.5.

We only prove the first part of the theorem. The proof for the second part of the theorem is the same as the proof of Theorem 2.3 of [33] and omitted.

Assume w = (w1, ..., wn) is s-admissible with respect to c∗, where s = (s1, ..., sn+1). We first claim

that the singular local orbit 0_{−−−→ 1}w=0 w=w1

−−−−→ s2 is weakly shadowed by a true local orbit.

According to Table 1 and the assumption a(0) = a1(c∗), we know that 0∈ H1(c∗) and there exists a

singular local orbit 0_{−→ 1 at w = 0. For the following local path 1 → s}2, the admissible conditions lead

to s2= 0 or a. Therefore, there exists a singular local orbit 1−→ s2at w = w1if w1∈ H2(c∗)∪ G2(c∗)

or H4(c∗)∪ G4(c∗) (in fact, w1∈ H2(c∗) or H4(c∗)). Take M0= W0u((0, 0, 0)× Cδ(c∗)). According to

Lemma 3.14, we have

M0_{t W0}c(M₁δ(0)_{× C}δ(c∗)).

Let N0 be their intersection. Since the phase space of system (2.10) is R4and dimensions of M0and

Wc

0(M1δ(0)×Cδ(c∗)) are 2 and 3 respectively, then dimN0= 2 + 3−4 = 1. We now apply the exchange

lemma 3.2 to the vicinity of the slow manifold M1× Cδ(c∗) along the slow orbit from (1, 0, 0, c∗) to

(1, 0, w1, c∗). Taking Mε = Wεu((0, 0, 0)× Cδ(c∗)) be such that Mε → M0 as ε → 0. By condition

(A1) and part (1) of Lemma 3.2, a portion Mε p0 of M

the vicinity of slow orbit close to Wu

ε(M1δ(w1)× Cδ(c∗)). Note that dimW0u(M1δ(w1)× Cδ(c∗)) = 3.

Thus, the singular orbit 0_{−−−→ 1}w=0 w=w1

−−−−→ s2 is weakly shadowed by a true orbit.

Next, we claim that the singular local orbit 0_{−−−→ 1}w=0 w=w1

−−−−→ s2 w=w2

−−−−→ s3 is weakly shadowed by

a true local orbit. Two cases for s2 = 0 or a are considered. For the case s2 = 0, a portion Mpε1 of

Wu

ε((a(w1), 0, w1)× Cδ(c∗)) will proceed near the singular orbit and leave the vicinity of slow orbit

close to Wu

ε((s3, 0)× (w2− δ, w2+ δ)× Cδ(c∗)) or Wεcu((s3, 0)× {w2} × {c∗}). The details of proof

can be found in [33] by using the Exchange Lemma with turning point (Lemma 3.2) and omitted. For the case s2= a, the admissible conditions imply s3= 1 or 0. Thus there exists a singular local orbit

s2−→ s3 at w = w2 if w2∈ H5(c∗) or H6(c∗). From the admissible condition (A4), there is no turning

point lying between w1 and w2. It’s also easy to see that W0u((1, 0)× (w1− δ, w1+ δ)× (Cδ(c∗)) and

W₀c((a(w1), 0)× (w1− δ, w1+ δ)× Cδ(c∗)) intersect transversally. Then, by Lemma 3.3, a portion

Mpε1 of W

u

ε((a(w1), 0)× (w1− δ, w1+ δ)× Cδ(c∗)) will proceed near the singular orbit and leave the

vicinity of slow orbit close to Wεu((s3, 0)× (w2− δ, w2+ δ)× Cδ(c∗)). Note that dimW0u(M1δ(w1)×

Cδ(c∗)) =dimW0cu((s3, 0)× {w2} × Cδ(c∗)) = 3.

According to the above discussions, we can conclude inductively that the singular local orbit 0_{−−−→ 1}w=0 w=w1

−−−−→ · · · w=wn−1

−−−−−−→ snis weakly shadowed by a true local orbit. Generally, for 2 < i < (n+1),

we consider the following two cases.

(1) Assume there exists a turning point lying between wi−1 and wi. By Lemma 3.2, a portion

Mε

pi−1 of W

u

ε((si, 0, wi−1)× Cδ(c∗)) will proceed near the singular orbit and leave the vicinity of slow

orbit close to Wu

ε((si+1, 0)× (wi− δ, wi+ δ)× Cδ(c∗)) or Wεcu((si+1, 0, wi)× Cδ(c∗)).

(2) Assume there is no turning point lying between wi−1and wi. By Lemma 3.3, a portion Mpεi−1

of Wεu((si, 0, wi−1)× Cδ(c∗)) will leave the vicinity of slow orbit close to Wεu((si+1, 0)× (wi− δ, wi+

δ)_{× C}δ(c∗)).

Furthermore, we have dimWu

0((si+1, 0)× (wi− δ, wi+ δ)× Cδ(c∗)) = 3 and dimW0cu((si+1, 0, wi)×

Cδ(c∗)) = 3.

Finally, we prove that the true orbits obtained by above arguments are C1 _{O(ε)-close to the}

unstable manifold Wu

0((sn+1, 0, wn)× Cδ(c∗)). Since sn+1 = 1, the admissible conditions lead to

wn ∈ H1(c∗)∪ G1(c∗) or H6(c∗). Thus, there exists a singular local orbit sn −→ 1 at w = wn. By

condition (A1), we have a(w) < 1 for all w_{∈ [w}n, 1] and the singular orbit will approach to (1, 0, 1) as

time goes infinity. Moreover, W₀u((sn, 0, wn)× {c∗}) intersects W0c(M δ

1(wn)× Cδ(c∗)) transversally.

O(ε)-close to the unstable manifold Wu

0((sn+1, 0, wn)×Cδ(c∗)). The proof of the theorem is complete.

Proof of Theorem 3.6.

The proof of Theorem 3.6 is similar to that of Theorem 3.5. Let w = (w1, ..., wn) is s-admissible

for c = c∗ and s = (s1, ..., sn+1). We first claim that the singular local orbit 0 w=0

−−−→ 1 w=w1

−−−−→ s2 is

strongly shadowed by a true orbit.

According to Table 1 and the assumption a(0) _{≤ a}6(c∗), we know that 0 ∈ G1(c∗) and there

exists a singular local orbit 0 _{−→ 1 at w = 0. For the following local path 1 → s}2, the admissible

conditions lead to s2 = 0 or a. Therefore, there exists a singular local orbit 1 −→ s2 at w = w1 if

w1 ∈ H2(c∗)∪ G2(c∗) or H4(c∗)∪ G4(c∗) (in fact, w1∈ H2(c∗) or H4(c∗)). Take M0= W0u(0, 0, 0).

According to Lemma 3.14, we have

M0_{t W0}c_{M1δ(0)_}.

Let N0 be their intersection. Since the phase space of system (2.3) is R3 and both dimensions of

M0 _{and W}c

0(M1δ(0)) are 2, then dimN0 = 2 + 2− 3 = 1. We now apply the exchange lemma

3.2 to the vicinity of the slow manifold M1 along the slow orbit from (1, 0, 0) to (1, 0, w1). Taking

Mε= W_εu((0, 0, 0)) be such that Mε_{→ M}0 as ε_{→ 0. By condition (A1) and part (1) of Lemma 3.2,} a portion Mpε0 of M

ε _{will proceed near the singular orbit and leave the vicinity of slow orbit close}

to Wεu(M1δ(w1)). Note that dimW0u(M1δ(w1)) = 2. Thus, the singular orbit 0 w=0

−−−→ 1 w=w1

−−−−→ s2 is

strongly shadowed by a true orbit.

Next, we claim that the singular local orbit 0_{−−−→ 1}w=0 w=w1

−−−−→ s2 w=w2

−−−−→ s3 is strongly shadowed by

a true local orbit. Two cases for s2 = 0 or a are considered. For the case s2 = 0, a portion Mpε1 of

Wu

ε((a(w1), 0, w1)) will proceed near the singular orbit and leave the vicinity of slow orbit close to

Wu

ε((s3, 0)×(w2−δ, w2+ δ)) or Wεcu((s3, 0)×{w2}). The details of proof can be found in [33] by using

the exchange lemma with turning point (Lemma 3.2) and omitted. For the case s2= a, the admissible

conditions imply s3= 1 or 0. Thus there exists a singular local orbit s2−→ s3at w = w2if w2∈ H5(c∗)

or H6(c∗). From the admissible condition (A4), there is no turning point lying between w1 and w2.

It’s also easy to see that Wu

0((1, 0)× (w1− δ, w1+ δ)) and W0c((a(w1), 0)× (w1− δ, w1+ δ)) intersect

transversally. Then, by Lemma 3.3, a portion M_pε₁ of W_εu((a(w1), 0)× (w1− δ, w1+ δ)) will proceed

near the singular orbit and leave the vicinity of slow orbit close to W_εu((s3, 0)× (w2− δ, w2+ δ)). Note

that dimW0u(M1δ(w1)) =dimW0cu((s3, 0)× {w2}) = 2.

According to the above discussions, we can conclude inductively that the singular local orbit 0_−−−→w=0 1 w=w1

−−−−→ · · · w=wn−1

−−−−−−→ sn is strongly shadowed by a true local orbit. Generally, for 2 < i < (n + 1), we

(1) Assume there exists a turning point lying between wi−1 and wi. By Lemma 3.2, a portion

M_pε_i−1 of W_εu((si, 0, wi−1)) will proceed near the singular orbit and leave the vicinity of slow orbit

close to Wεu((si+1, 0)× (wi− δ, wi+ δ)) or Wεcu((si+1, 0, wi)).

(2) Assume there is no turning point lying between wi−1and wi. By Lemma 3.3, a portion Mpεi−1

of Wu

ε((si, 0, wi−1)) will leave the vicinity of slow orbit close to Wεu((si+1, 0)× (wi− δ, wi+ δ)).

Furthermore, we have dimWu

0((si+1, 0)× (wi− δ, wi+ δ)) = dimW0cu((si+1, 0, wi)) = 2.

Finally, we prove that the true orbits obtained by above arguments are C1_{O(ε)-close to the}

unsta-ble manifold Wu

0((sn+1, 0, wn)). Since sn+1= 1, the admissible conditions lead to wn ∈ H1(c∗)∪G1(c∗)

or H6(c∗). Thus, there exists a singular local orbit sn −→ 1 at w = wn. By condition (A1), we have

a(w) < 1 for all w _{∈ [w}n, 1] and the singular orbit will approach to (1, 0, 1) as time goes infinity.

Moreover, W₀u((sn, 0, wn)) intersects W0c(M δ

1(wn)) transversally. As a result, the true orbit will

ap-proach a neighborhood of (1, 0, 1), near the singular orbit and C1O(ε)-close to the unstable manifold W0u((sn+1, 0, wn)). The proof of the theorem is complete.

4

Wazewski Theorem and Lasalle Invariance Principle

4.1

Wazewski theorem

We now recall a variant of the Wazewski Theorem which is a formalization and extension of the shooting method in higher dimension (see [9], p. 563). Let usconsider the differential equation:

y0(s) = f (y(s)) (4.27) where f : Rn

→ Rn _{is a Lipschitz continuous function. Suppose y(s; y}

0) is the unique solution of

(4.27) with initial value y(0) = y0. For convenience, set y(s; y0) = y0· s and let Y · S be the set of

points y_{· s, where y ∈ Y ⊂ R}n _{and s}

∈ S ⊂ R. Given W ⊆ Rn_{, define the immediate exit set W}− _of

W by

W−=_{y0∈ W : ∀s > 0, y0· [0, s) * W }.

Given Σ_{⊆ W , let}

Σ0=_{y0∈ Σ : ∃s0> 0 such that y0· s0∈ W }./

For y0∈ Σ, define the exit time T (y0) of y0by

T (y0) = sup{s : y0· [0, s] ⊂ W }.

Note that y0· T (y0)∈ W− and T (y0) = 0 if and only if y0∈ W−. The Wazewkski Theorem is then

stated as the following.

Theorem 4.1 Consider (4.27) and suppose that

(i) if y0∈ Σ and y0· [0, s] ⊆ c`(W ), then y0· [0, s] ⊆ W ;

(ii) if y0∈ Σ, y0· s ∈ W , y0· s /∈ W−, then there is an open set Vsabout y0· s disjoint from W−;

(iii) Σ = Σ0_{, Σ is a compact set and intersects a trajectory of y}0 _{= f (y) only once.}

Then the mapping F (y0) = y0·T (y0) is a homeomorphism from Σ to its image on W−. A set W ⊆ Rn

satisfying the conditions (i) and (ii) is called a Wazewski set

4.2

LaSalle’s Invariance Principle