# The Convergence of Attractors for Some Discrete Cahn-Hilliard Systems

Authors: Ruijing Wang; Chunqiu Li
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this article, we use a finite difference scheme to discretize the Cahn-Hilliard equation with the space step sizeh. We first prove that this semidiscrete system inherits two important properties, called the conservation of mass and the decrease of the total energy, from the original equation. Then, we show that the semidiscrete system has an attractor on a subspace of ℝN+1. Finally, the convergence of attractors is established as the space step size h of the semidiscrete Cahn-Hilliard equation tends to 0.


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## Body

## 1. Introduction

In this paper, we are concerned with some discrete forms of the following Cahn-Hilliard equation:(1)∂u∂t=∂2∂x2δGδu,x∈0,L,t>0,(2)δGδu=−u+u3−∂2u∂x2,associated with boundary conditions:(3)∂u∂x|x=0=∂u∂x|x=L=0,∂∂xδGδu|x=0=∂∂xδGδu|x=L=0,where ux,t denotes a distribution function of the concentration of one component of the binary mixture, the functional G (called the Ginzburg-Landau free energy) usually denotes a local free energy, and the notation δG/δu given by (1) is consistent with the variational derivative of(4)Gux,t=−12u2+14u4+12∂u∂x2.The Cahn-Hilliard equation was first proposed by Cahn and Hilliard [1] and is usually used to describe the complex phase separation phenomenon which occurs when two miscible substances are rapidly cooled to critical temperatures at high temperatures; see [2–5] for details.Nowadays, the Cahn-Hilliard equation has attracted much attention from researchers; see [3, 6–13], etc. Especially, Alikakos, Bates, and Fusco [6] established the existence of some extremely slowly evolving solutions of the Cahn-Hilliard equation. Later, Cholewa and Dlotko [7] proved the existence of global attractors for the Cahn-Hilliard equation. Based on this result, Li and Zhong [10] further established the existence of global attractors for the Cahn-Hilliard equation with fast growing nonlinearity. Recently, Furihata [3] proposed a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation. Specifically, they designed a new difference scheme which inherits some characteristic properties from the equation, including the conservation of mass and the decrease of the total energy.Inspired by these works mentioned above, we use a finite difference scheme (with respect tox∈0,L) to discretize the Cahn-Hilliard equations (1) and (2) with the space step size h, which can generate Nh+1 nodes. Thus, one can obtain the following equations:(5)dzkdt=δh,k2δGdδzk,k=0,1,…,Nh,where(6)δh,k2zk=1h2zk−1−2zk+zk+1,δGdδzk=−zk+zk3−δh,k2zk,associated with the initial conditions(7)zk0=zk,0,k=0,1,…,Nh.Moreover, the boundary conditions (3) can be discretized as(8)z−1=z1,z−2=z2,zNh−1=zNh+1,zNh−2=zNh+2.We are interested in the existence and the convergence of attractors for this semidiscrete Cahn-Hilliard equation. It is well-known that semidiscretization is a very effective technique in the finite element analysis of solid bodies and computational fluid mechanics; see [14, 15]. The semidiscretization of a partial differential equation means that the space is discretized and the time is continuous, which is widely studied by many researchers. For example, Castro and Micu [16] studied the controllability of a semidiscrete system with a boundary control at one extremity. Mai, Qin, and Zhang [17] studied the Turing instability of a two-dimensional semidiscrete Gierer–Meinhardt system and carried out a series of simulations. Concerning this topic of semidiscrete systems, the interested reader is referred to [18, 19, 20, 21, 22, 23] for some concrete examples.In this paper, we discuss the attractor of the semidiscrete Cahn-Hilliard system (5)–(8). We first develop some ideas in [3, 7] to give some properties of solutions and prove the existence of attractors. Then, we establish the convergence of the attractor with respect to the space step size h.This paper is organized as follows. In Section 2, we present two well-known properties of the system (5)–(8) called the conservation of mass and the decrease of the total energy. In Section 3, we prove the existence of attractors for the system (5)–(8) on some affined subspaces. In Section 4, we further establish the convergence of the attractor with respect to the space step size h; that is, the attractor of the system (5)–(8) tends to the attractor of the original system (1)–(3) as h⟶0.


## 2. Two Well-Known Properties

In this section, we discuss two well-known properties of the semidiscrete Cahn-Hilliard system (5-8). For the purpose, we let Jh=0,1,2,…,Nh and Zh=zktk∈Jh and define the linear operators by(9)δh2Zhk=δh,k2zk=1h2zk−1−2zk+zk+1,δh4Zhk=δh,k2δh2Zhk=1h4zk−2−4zk−1+6zk−4zk+1+zk+2.Then, one can rewrite the system (5) as(10)dZhdt=−δh4Zh+δh2−Zh+Zh3,t>0.Without loss of generality, in the following, we first assume that the space step sizeh=1, hoping that there is no confusion.LetJ=0,1,2,…,N and U=Uktk∈J. Then, the system (5)–(8) can be simply written as the following system(11)dUdt=−δ4U+δ2−U+U3,t>0,with the initial value(12)U0=Uk0k∈J=zk,0k∈J,and boundary conditions(13)U−1=U1,U−2=U2,UN−1=UN+1,UN−2=UN+2,where δ2 and δ4 are linear operators defined by(14)δ2Uk=δk2Uk=Uk−1−2Uk+Uk+1,(15)δ4Uk=δk2δ2Uk=Uk−2−4Uk−1+6Uk−4Uk+1+Uk+2.In the following, we verify that the solution of the semidiscrete Cahn-Hilliard system (11)–(13) satisfies(16)ddt∑k=0NU″kt=0,(17)ddt∑k=0NG″dUk≤0,which corresponds to the following two properties:(18)∫0Lux,tdx=∫0Lux,0dx,(19)ddt∫0LGux,tdx≤0,(called the conservation of mass and the decrease of the total energy, respectively.) of the solutions to the Cahn-Hilliard system (1)–(3); see [3, 9] for details.In (16) and (17), ∑k=0N″ is a summation operator (see [3]), given by(20)∑k=0NU″k=12U0+∑k=1N−1Uk+12UN,(21)GdUk=−12Uk2+14Uk4+12δk+Uk2+δk−Uk22.is the discrete local free energy, where(22)δk+Uk=Uk+1−Uk,δk−Uk=Uk−Uk−1.Remark 1.
These two properties (18) and (19) play an important role in studying the Cahn-Hilliard equation and are widely investigated by researchers; see e.g., [24, 25]. On the other hand, it is clear to see from (17) that ∫0LG(u(x, t)) dx can be employed as a Lyapunov function of the Cahn-Hilliard system (see e.g., [26]). Similarly, one can regard ∑k=0NG″dUk as a Lyapunov function of (11) and (12).Proposition 1.
The solution of the semidiscrete Cahn-Hilliard system (11)–(13) satisfies the properties (16) and (17).Proof.
First, we show that the solution of (11)–(13) satisfies the conservation of mass (16). Indeed, by the definition of the summation operator and (5), we easily see that(23)ddt∑k=0NU″kt=∑k=0Nddt″Ukt=∑k=0Nδ″k2δGdδUk=0,where δGd/δUk=−Uk+Uk3−δk2Uk.
Now, let us prove that the solution of (11)–(13) satisfies (17). By the definition of GdUk, one has(24)ddt∑k=0N″GdUk=ddt∑k=0N″−12Uk2+ddt∑k=0N″14Uk4+ddt∑k=0N″14Uk+1−Uk2+Uk−Uk−12=∑k=0N″–UkdUkdt+∑k=0N″Uk3dUkdt−∑k=0N″Uk−1−2Uk+Uk+1dUkdt=∑k=0N″δGdδUkdUkdt=∑k=0N″δGdδUkδk2δGdδUk.
Thanks to the general identity (summation by parts), we have(25)∑k=0Nδk+Uk2+δk−Uk22″=sk1Ukδk1Uk|k=0N−∑k=0NU″kδk2Uk,where(26)sk1Uk=Uk+1+Uk−12,δk1Uk=Uk+1−Uk−12.
Thus, it follows from (24) and (25) that(27)ddt∑k=0N″GdUk=∑k=0N″δGdδUkδk2δGdδUk=14δGdδUk+12−δGdδUk−12|k=0N−12∑k=0N″δk+δGdδUk2+δk−δGdδUk2=−12∑k=0N″δk+δGdδUk2+δk−δGdδUk2≤0,which completes the proof of the proposition.


## 3. Attractors of the Semidiscrete System

In this section, we aim to prove the existence of attractors for the semidiscrete Cahn-Hilliard system. We first give some properties of the solutions to (11)–(13). To this end, we use ℝN+1 to denote the N+1 Euclid space and equip it with the norm ⋅ defined by(28)U=∑k=0NUk21/2,U∈ℝN+1.We denote byCs a positive constant depending on s, whose value may change from line to line.Note that the equation (11) is finite-dimensional. By the basic knowledge on ODEs, one can easily see that the Cauchy problem of (11)–(13) is well-posed in ℝN+1. Specifically, for each U0∈ℝN+1, the system (11)–(13) has a unique solution Ut on 0,T for some T>0 with U0=U0.LetΦ denote the local semiflow generated by (11)–(13). Then, we have the following fundamental facts on the solutions.


### 3.1. The Boundedness of the Solution

Theorem 1.
LetUt be a solution of (11)–(13) with the initial data U0. Then,(29)Ut≤CU0,t>0.Proof.
We define a norm⋅d as follows:(30)Ud=∑k=0N−1δk+Uk2+∑k=0NU″k21/2.
Then, by (20) and a simple calculation, we have(31)Ud2=∑k=0N″δk+Uk2+δk−Uk22+∑k=0NU″k2≤2∑k=0N″12δk+Uk2+δk−Uk22+Uk2≤2∑k=0N″12δk+Uk2+δk−Uk22−12Uk2+14Uk4+94≤2∑k=0N″12δk+Uk2+δk−Uk22−12Uk2+14Uk4+92N=2∑k=0N″GdUk+92N.
Here, we used the inequalityx2≤−1/2x2+1/4x4+9/4 (see [3]). Thereby, we deduce from (17) that(32)Ud2≤2∑k=0NG″dUk0+92N.
On the other hand, it is trivial to show that(33)∑k=0NG″dUk=−12∑k=0N″Uk2+14∑k=0N″Uk4+12∑k=0N−1δk+Uk2≤14∑k=0NUk2+14∑k=0NUk4+2∑k=0NUk2≤CU2.
Thus, by the equivalence of norms, (32) and (33), we can conclude that (29) holds true.


### 3.2. Properties of the Stationary Solution

As noted in the Remark1, one can see ∑k=0NG″dUk is a Lyapunov function. Next, we present some basic results on ∑k=0NG″dUk.Lemma 1.
IfU=Ukk∈J is a solution of the system (11)–(13) satisfying(34)∑k=0NG″dUk=constant,t>0,then U is a stationary (time independent) solution.Proof.
By (27), we have(35)ddt∑k=0NG″dUk=∑k=0Nδk+δGdδUk2+δk−δGdδUk2″=0,from which it can be seen that(36)δGdδUk−1=δGdδUk=δGdδUk+1,∀k∈J.
Thereby,(37)δk2δGdδUk=δGdδUk−1−2δGdδUk+δGdδUk+1=0,which together with (11) gives that(38)dUkdt=0,∀k∈J.
Hence, the result of Lemma1 holds true.Lemma 2.
LetU be a solution of (11)–(13). Then, U is a stationary solution if and only if there is a constant γ such that(39)δGdδUk=γ,∀k∈J,where δGd/δUk=−Uk+Uk3−δk2Uk.Proof.
LetU be a solution of (11)–(13). Then, if U is a stationary solution, we see from (6) and (11) that(40)δk2δGdδUk=0,∀k∈J.
By the boundary conditions (13), one has(41)δGdδU0=δGdδU1=δGdδU2=⋯=δGdδUN:=constant.
Conversely, if there is a constantγ such that δGd/δUk=γ, then(42)δk2δGdδUk=dUkdt=0,which completes the proof of what we desired.
Letγ be the constant given by Lemma 2. Assume U is a stationary solution of (11)–(13) such that (39) holds. Then,(43)∑k=0NδGdδU″k=∑k=0N−Uk+Uk3″+∑k=0N−δk2Uk″=∑k=0N−Uk+Uk3″=Nγ,which shows that the γ can be expressed as(44)γ=1N∑k=0N−Uk+Uk3″.
LetU¯ denote the spatial average of U, i.e.,(45)U¯t=1N∑k=0NU″kt.
Then, one can see from (44) that γ=−U+U3¯. Write(46)D=U∈ℝN+1:δGdδUk=−U+U3¯andDα=U∈D:U¯≤α,where α>0. Then, one can deduce that D is actually the set of all stationary solutions of the system (11)–(13). In the following, we show that Dα is bounded for each α>0, which plays an important part in constructing the attractor.Theorem 2.
For each fixedα>0, Dα is a bounded set of ℝN+1.Proof.
LetU∈Dα. We infer from Lemma 2 that(47)γ∑k=0NU″k=∑k=0NU″kδGdδUk=∑k=0N−Uk2+Uk4−Ukδk2Uk″,where γ is given by (44).
By (14), one has(48)∑k=0N−Ukδk2Uk″​​=∑k=0NUk+1−Uk″2+12U1−U02−12UN−1−UN2≥∑k=0NUk+1−Uk″2−12∑k=0NUk+1−Uk″2=12∑k=0NUk+1−Uk″2.
Thus, it follows from (47) and (48) that(49)12∑k=0NUk+1−Uk″2≤γ∑k=0NU″k−∑k=0N−Uk2+Uk4″=U¯∑k=0N−Uk+Uk3″−∑k=0N−Uk2+Uk4″.
Here, we have used the definition ofγ (see (44)).
Now, by Young inequality, one can deduce that for everyr>0, there exists Cr such that(50)−Uk+Uk3≤rUk4+Cr.
It is trivial to pick positive constantsk0 and k1 satisfying(51)Uk2−Uk4≤−k0Uk4+k1.
Substituting (50) and (51) into (49), we obtain that(52)12∑k=0NUk+1−Uk″2≤U¯∑k=0N′′rUk4+Cr+∑k=0N−k0Uk4+k1″=U¯r−k0∑k=0NUk″4+NU¯Cr+k1.
Taker=k0/U¯+1 in the above estimate, then(53)∑k=0NUk+1−Uk″2≤2NU¯Cr+k1.
Now, we define a new norm⋅H as follows:(54)UH=∑k=0NUk+1−Uk″2+1N∑k=0NU″k21/2.
Then, by (53), we find UH≤CU¯. Therefore, by the equivalence of the norms, one can immediately conclude that(55)U≤CU¯≤Cα,which completes the proof of the theorem.


### 3.3. The Existence of the Attractors

Inspired by much literature (see e.g., [7, 10]) on the global attractor for the Cahn-Hilliard dynamical system (1)–(3), we consider the attractor on the space ℋα=U∈ℝN+1:U¯≤α, where α>0.Lemma 3.
There exists a bounded subsetℬ of ℋα attracting each point of ℋα.Proof.
Define the setℬ by(56)ℬ=∪U∈ℋαωU,where ωU denotes the ω-limit set of U.
For eachU0∈ℬ, there is W0∈ℋα such that U0∈ωW0. Let Ttt≥0 be the family of operators which form a continuous semigroup satisfying Vt=TtV0 on the space ℋα, where Vt is the solution of (11) with the initial value V0. By (17), we know that ∑k=0NG″dTtW0k is decreasing along the solution TtW0=TtW0kk∈J and bounded below. Thus, there exists β∈ℝ such that(57)limt⟶+∞∑k=0NG″dTtW0k=β.
Then, one can deduce that(58)∑k=0NG″dUkt=β,∀t≥0,where Uktk∈J=TtU0. Hence, by (16) and Lemma 1, we see that U0 is a stationary solution of the system (11)–(13) belonging to Dα. Consequently, one can conclude from Theorem 2 that ℬ is bounded.
On the other hand, by Theorem1, one finds that the semiflow Φ generated by the system (11)–(13) takes bounded sets into bounded sets. Moreover, by the definition of ℋα, we see that Ttt≥0 is compact. Hence, ℬ attracts each point of ℋα.
According to [14], Theorem 4.2.4, we can obtain the existence of the attractor for the semifolw Φ on ℋα.Theorem 4.
The semiflowΦ generated by the system (11)–(13) possesses a connected global attractor on a metric space ℋα.


## 3.1. The Boundedness of the Solution

Theorem 1.
LetUt be a solution of (11)–(13) with the initial data U0. Then,(29)Ut≤CU0,t>0.Proof.
We define a norm⋅d as follows:(30)Ud=∑k=0N−1δk+Uk2+∑k=0NU″k21/2.
Then, by (20) and a simple calculation, we have(31)Ud2=∑k=0N″δk+Uk2+δk−Uk22+∑k=0NU″k2≤2∑k=0N″12δk+Uk2+δk−Uk22+Uk2≤2∑k=0N″12δk+Uk2+δk−Uk22−12Uk2+14Uk4+94≤2∑k=0N″12δk+Uk2+δk−Uk22−12Uk2+14Uk4+92N=2∑k=0N″GdUk+92N.
Here, we used the inequalityx2≤−1/2x2+1/4x4+9/4 (see [3]). Thereby, we deduce from (17) that(32)Ud2≤2∑k=0NG″dUk0+92N.
On the other hand, it is trivial to show that(33)∑k=0NG″dUk=−12∑k=0N″Uk2+14∑k=0N″Uk4+12∑k=0N−1δk+Uk2≤14∑k=0NUk2+14∑k=0NUk4+2∑k=0NUk2≤CU2.
Thus, by the equivalence of norms, (32) and (33), we can conclude that (29) holds true.


## 3.2. Properties of the Stationary Solution

As noted in the Remark1, one can see ∑k=0NG″dUk is a Lyapunov function. Next, we present some basic results on ∑k=0NG″dUk.Lemma 1.
IfU=Ukk∈J is a solution of the system (11)–(13) satisfying(34)∑k=0NG″dUk=constant,t>0,then U is a stationary (time independent) solution.Proof.
By (27), we have(35)ddt∑k=0NG″dUk=∑k=0Nδk+δGdδUk2+δk−δGdδUk2″=0,from which it can be seen that(36)δGdδUk−1=δGdδUk=δGdδUk+1,∀k∈J.
Thereby,(37)δk2δGdδUk=δGdδUk−1−2δGdδUk+δGdδUk+1=0,which together with (11) gives that(38)dUkdt=0,∀k∈J.
Hence, the result of Lemma1 holds true.Lemma 2.
LetU be a solution of (11)–(13). Then, U is a stationary solution if and only if there is a constant γ such that(39)δGdδUk=γ,∀k∈J,where δGd/δUk=−Uk+Uk3−δk2Uk.Proof.
LetU be a solution of (11)–(13). Then, if U is a stationary solution, we see from (6) and (11) that(40)δk2δGdδUk=0,∀k∈J.
By the boundary conditions (13), one has(41)δGdδU0=δGdδU1=δGdδU2=⋯=δGdδUN:=constant.
Conversely, if there is a constantγ such that δGd/δUk=γ, then(42)δk2δGdδUk=dUkdt=0,which completes the proof of what we desired.
Letγ be the constant given by Lemma 2. Assume U is a stationary solution of (11)–(13) such that (39) holds. Then,(43)∑k=0NδGdδU″k=∑k=0N−Uk+Uk3″+∑k=0N−δk2Uk″=∑k=0N−Uk+Uk3″=Nγ,which shows that the γ can be expressed as(44)γ=1N∑k=0N−Uk+Uk3″.
LetU¯ denote the spatial average of U, i.e.,(45)U¯t=1N∑k=0NU″kt.
Then, one can see from (44) that γ=−U+U3¯. Write(46)D=U∈ℝN+1:δGdδUk=−U+U3¯andDα=U∈D:U¯≤α,where α>0. Then, one can deduce that D is actually the set of all stationary solutions of the system (11)–(13). In the following, we show that Dα is bounded for each α>0, which plays an important part in constructing the attractor.Theorem 2.
For each fixedα>0, Dα is a bounded set of ℝN+1.Proof.
LetU∈Dα. We infer from Lemma 2 that(47)γ∑k=0NU″k=∑k=0NU″kδGdδUk=∑k=0N−Uk2+Uk4−Ukδk2Uk″,where γ is given by (44).
By (14), one has(48)∑k=0N−Ukδk2Uk″​​=∑k=0NUk+1−Uk″2+12U1−U02−12UN−1−UN2≥∑k=0NUk+1−Uk″2−12∑k=0NUk+1−Uk″2=12∑k=0NUk+1−Uk″2.
Thus, it follows from (47) and (48) that(49)12∑k=0NUk+1−Uk″2≤γ∑k=0NU″k−∑k=0N−Uk2+Uk4″=U¯∑k=0N−Uk+Uk3″−∑k=0N−Uk2+Uk4″.
Here, we have used the definition ofγ (see (44)).
Now, by Young inequality, one can deduce that for everyr>0, there exists Cr such that(50)−Uk+Uk3≤rUk4+Cr.
It is trivial to pick positive constantsk0 and k1 satisfying(51)Uk2−Uk4≤−k0Uk4+k1.
Substituting (50) and (51) into (49), we obtain that(52)12∑k=0NUk+1−Uk″2≤U¯∑k=0N′′rUk4+Cr+∑k=0N−k0Uk4+k1″=U¯r−k0∑k=0NUk″4+NU¯Cr+k1.
Taker=k0/U¯+1 in the above estimate, then(53)∑k=0NUk+1−Uk″2≤2NU¯Cr+k1.
Now, we define a new norm⋅H as follows:(54)UH=∑k=0NUk+1−Uk″2+1N∑k=0NU″k21/2.
Then, by (53), we find UH≤CU¯. Therefore, by the equivalence of the norms, one can immediately conclude that(55)U≤CU¯≤Cα,which completes the proof of the theorem.


## 3.3. The Existence of the Attractors

Inspired by much literature (see e.g., [7, 10]) on the global attractor for the Cahn-Hilliard dynamical system (1)–(3), we consider the attractor on the space ℋα=U∈ℝN+1:U¯≤α, where α>0.Lemma 3.
There exists a bounded subsetℬ of ℋα attracting each point of ℋα.Proof.
Define the setℬ by(56)ℬ=∪U∈ℋαωU,where ωU denotes the ω-limit set of U.
For eachU0∈ℬ, there is W0∈ℋα such that U0∈ωW0. Let Ttt≥0 be the family of operators which form a continuous semigroup satisfying Vt=TtV0 on the space ℋα, where Vt is the solution of (11) with the initial value V0. By (17), we know that ∑k=0NG″dTtW0k is decreasing along the solution TtW0=TtW0kk∈J and bounded below. Thus, there exists β∈ℝ such that(57)limt⟶+∞∑k=0NG″dTtW0k=β.
Then, one can deduce that(58)∑k=0NG″dUkt=β,∀t≥0,where Uktk∈J=TtU0. Hence, by (16) and Lemma 1, we see that U0 is a stationary solution of the system (11)–(13) belonging to Dα. Consequently, one can conclude from Theorem 2 that ℬ is bounded.
On the other hand, by Theorem1, one finds that the semiflow Φ generated by the system (11)–(13) takes bounded sets into bounded sets. Moreover, by the definition of ℋα, we see that Ttt≥0 is compact. Hence, ℬ attracts each point of ℋα.
According to [14], Theorem 4.2.4, we can obtain the existence of the attractor for the semifolw Φ on ℋα.Theorem 4.
The semiflowΦ generated by the system (11)–(13) possesses a connected global attractor on a metric space ℋα.


## 4. The Convergence of Attractors

Notice that we considered a special case where the space step sizeh equals 1 in the above arguments. By using the same arguments as Theorem 3.6, one can easily verify that the system (11)–(13) has an attractor Ah with respect to the general space step size h.Thanks to [11, Theorem 3], we deduce that the solutionUt of (11)–(13) converges to the solution ux,t of (1)–(3) as the space step size h⟶0. Based on this fact, in what follows we further show that the attractor Ah in ℋα tends to the attractor A of the original system (1-3) as h⟶0.Theorem 3.
LetA be the attractor of the original system (1)–(3) given by [7]. Then,(59)distℝNh+1A,Ah⟶0ash⟶0+,where distℝNh+1⋅,⋅ denotes the Hausdorff semidistance in ℝNh+1.Proof.
We argue by contradiction. If the conclusion is not true, then there existsε0>0 and a sequence xn∈Ahn with hn⟶0 (as n⟶0) such that(60)distℝNhn+1A,xn≥ε0.
On the other hand, we infer from the invariance ofAhn that there exists a bounded complete solution γn of Φ in Ahn such that γn0=xn. By a standard argument, it can be easily shown that γn has a subsequence converging some bounded full trajectory γ0 of Φ in A uniformly on any compact interval of ℝ. Hence,(61)distℝNhn+1γ0,γn⟶0ashn⟶0.
Especially,distℝNhn+1γ00,xn⟶0asn⟶∞, which contradicts (60) and completes the proof of the theorem.



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