A-Stability For Two Species Competition Diffusion Systems
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classifled information.
Tung Nguyen
Certiflcate of Approval:
Wenxian Shen, Co-Chair
Professor
Mathematics and Statistics
Georg Hetzer, Co-Chair
Professor
Mathematics and Statistics
Paul Schmidt
Professor
Mathematics and Statistics
A.J. Meir
Professor
Mathematics and Statistics
Pat Goeters
Professor
Mathematics and Statistics
Stephen L. McFarland
Dean
Graduate School
A-Stability For Two Species Competition Diffusion Systems
Tung Nguyen
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulflllment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
07 August 2006
A-Stability For Two Species Competition Diffusion Systems
Tung Nguyen
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Tung Nguyen was born on November 19, 1974 in Ho Chi Minh City, Vietnam. He is
the only child of Tam Nguyen and Le Mai. He attended the University of Ho Chi Minh City,
Vietnam, graduating in 1996 with a Bachelor of Science in Mathematics degree. In 1999,
he received a Diploma certiflcate in Mathematics at the International Center of Theoretical
Physics, Trieste, Italy. He entered the Ph.D. program at Auburn University in September
1999.
iv
Dissertation Abstract
A-Stability For Two Species Competition Diffusion Systems
Tung Nguyen
Doctor of Philosophy, 07 August 2006
(Diploma Degree, I.C.T.P. Trieste Italy, 1999)
(B.S., University of HoChiMinh city, 1996)
80 Typed Pages
Directed by Georg Hetzer and Wenxian Shen
My dissertation research focuses on establishing the structural stability of the attractor
(A-stability) via Morse-Smale property for difiusive two-species competition systems
8>
>>>
>>><
>>>>
>>>
:
@tu = k1?u+uf(x;u;v); x 2 ?;
@tv = k2?v +vg(x;u;v); x 2 ?;
Bu = Bv = 0; x 2 @?;
(0.0.1)
on a C1 bounded domain ? ? Rn; n ? 1; with either Dirichlet or Neumann boundary
conditions. Here u(t;x), v(t;x) are the densities of two competing species, k1; k2 are
difiusive constants and (f;g); f;g : ???R?R!R C2 functions satisfying
(H1) f(x;0;0) > 0; g(x;0;0) > 0 8x 2 ??;
(H2) @uf(x;u;v); @vf(x;u;v); @ug(x;u;v); @vg(x;u;v) < 0; 8 u;v ? 0; 8x 2 ??;
(H3) supx2??; v?0 limsupu!1f(x;u;v) < 0;
(H4) supx2??; u?0 limsupv!1g(x;u;v) < 0:
v
These hypotheses describe key features of competition models, and sinceu and v are the
densities of two species, we are only interested in nonnegative solutions (u;v). We therefore
consider (1.0.1) in the positive cone of some appropriate phase space. Our main result states
for the spatially one-dimensional case that if (0.0.1) is a Morse-Smale system on the positive
cone, it is structurally stable. We also establish that the set of functions (f;g) for which
(1.0.1) possess the Morse-Smale property, is open in the space of all pairs (f;g) satisfying
(H1){(H4) under the topology of C2-convergence on compacta. Moreover, we show as a
su?cient condition that if all critical elements of (1.0.1) are hyperbolic with one-dimensional
unstable manifolds in case of equilibria (except 0) and two-dimensional unstable manifolds
in case of periodic orbits, then system (1.0.1) has the Morse-Smale property. These results
will have signiflcant impact on the study of the asymptotic dynamics of various classes of
discretizations of (0.0.1).
The proof of the openness of the set of functions for which (1.0.1) is a Morse-Smale
system, is an adaption of an idea used in [23], to a positive cone setting. As for a su?cient
condition under which the system has the Morse-Smale property, we are able to prove the
transversality of unstable manifolds and local stable manifolds of critical elements under
the hypotheses mentioned before.
The proof of the main result is quite technically di?cult because we have to work in a
positive cone setting. This proof can be broken down into a few main steps. First of all, since
the long-term features of the dynamics of (1.0.1) are determined by the global attractor,
which lies inside a su?ciently large ball, we can reduce (1.0.1) to a flnite dimensional system
by means of Chow, Lu & Sell?s inertial manifold theorem (cf.[6]). Secondly, we prove that
the flnite-dimensional system obtained in flrst step is also a Morse-Smale system. Thirdly,
vi
we establish the A-stability of the global attractor (in the positive cone) of the flnite-
dimensional system. The proof is an adaption of the main idea of a corresponding result
in the monograph [13] by J. Hale, L. Magalh~aes, & W.Olivia. As mentioned before, their
result does not apply to our problem because we work on a subset of the positive cone
which cannot be considered to be a Banach manifold imbedded in a Banach space, the
setting underlying the work of J. Hale, L. Magalh~aes, & W.Olivia. Since the attractor of
the flnite-dimensional system is the orthogonal projection of the global attractor of (1.0.1)
(in the positive cone) to the phase space of the flnite-dimensional system, the flnal step
requires to derive the A-stability of the global attractor of (1.0.1) (in the positive cone)
from A-stability of the global attractor of flnite-dimensional system.
vii
Acknowledgments
I would like to express my deepest appreciation to Professor Georg Hetzer and Pro-
fessor Wenxian Shen for their patience, attentiveness, and generosity while directing me to
complete this dissertation. I also would like to thank the other members of my advisory
committee, Professors Paul Schmidt, Amnon Meir, and Pat Goeters and the outside reader
Professor Dongye Zhao for reading the draft and for their valuable comments. I would like
to thank the NSF for the flnancial support (NSF-DMS-0504166).
viii
Style manual or journal used Transactions of the American Mathematical Society
(together with the style known as \auphd"). Bibliograpy follows van Leunen?s A Handbook
for Scholars.
Computer software used The document preparation package TEX (speciflcally LATEX)
together with the departmental style-flle auphd.sty.
ix
Table of Contents
1 Introduction 1
2 Definitions, Notations and Main Results 9
2.1 General Semi ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Semi ows Generated by Competition Models . . . . . . . . . . . . . . . . . 12
2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Preliminary results 17
3.1 General Semi ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Single Species Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Two Species Competition Systems . . . . . . . . . . . . . . . . . . . . . . . 22
4 Morse-Smale Structure and A Sufficient Condition 24
4.1 Morse-Smale Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 A Su?cient Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 A-stability via Morse-Smale structure 49
5.1 Reduction to inertial manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Tubular Family Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 A-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Bibliography 68
x
Chapter 1
Introduction
A fundamental goal of theoretical ecology is to understand how the interactions of
individual organisms with each other and with the environment afiect the distribution and
structure of populations. One way to achieve this goal is to use mathematical models.
Among these models, reaction-difiusion models have received great attention from many
experts in the fleld such as Fisher (1939), Skellam (1951), Kierstead and Slobodkin (1953),
Fife (1979), Smoller (1982), Murray (1993), Grindrod (1996), Leung (1989), Hess (1991),
Pao (1992), Hassel (1994), Cantrell and Cosner (1996),....
The reaction part of such models is derived from Lotka-Volterra models which in turn
are frequently based on the logistic model of population growth. The latter was flrst sug-
gested by Verhulst (1838) to describe the growth of human populations and was later derived
independently by Pearl and Reed (1920) for modeling the population growth in the United
States. Logistic models are based on the assumption that the growth of a population is
determined by two factors, the reproduction rate (which is the difierence of birth and death
rates of individuals) and the limitation of the habitat?s resources. The simplest logistic
model of population growth leads to the following equation
_u = ru
?
1? uC
?
;
where u stands for the population density, r is the growth rate and C (which is called
carrying capacity) represents the maximum number of individuals that can be sustained
by the resources of the habitat. It is obvious that over a long period of time, the size of
1
the population will approach C. The flrst model of interacting species were introduced by
Lotka (1925) and Volterra (1931). Derived from the logistic model of population growth, a
Lotka-Volterra competition model for two species has the form
8>
>><
>>>:
_u = u(a1 ?b1u?c1v);
_v = v(a2 ?b2u?c2v);
where u; v denotes the population densities of the two species. The extra terms c1uv,
b2uv added to the logistic equation of each species represent the negative efiect which one
species has on the other one due to competition for the habitat?s resource. This type of
model has also been studied thoroughly, and in general, the model predicts either compet-
itive exclusion (extinction of the weaker competitor or the initially disadvantaged species)
or stable coexistence of the two competing species. It appears that the predictions of the
model correspond to biologically realistic situations although some of its assumptions are
rather idealized, for example, the assumption of spatial homogeneity, the linear competition
between the two species, ... However, this model has drawn enormous amount of empiri-
cal and theoretical research since its introduction because of its great value for ecological
understanding,
In reality, individuals are not distributed homogeneously in their habitat and typically
interact with both the physical environment and other individuals in their neighborhood.
Therefore, there is the need for expanding Lotka-Volterra models by taking spatial depen-
denceintoaccount. Itisanaturalflrstattempttoassumethatmigrationoccursfromregions
of higher population density towards regions of lower density, and Skellam (1951) has used
an random walk approach to argue that, in fact, the dispersal of a population should be
2
modeled by difiusion, which suggests the following extension of the above Lotka-Volterra
competition model
8>
>>>>
>><
>>>>
>>>
:
@tu = k1?u+u(a1 ?b1u?c1v); x 2 ?;
@tv = k2?v +v(a2 ?b2u?c2v); x 2 ?;
Bu = Bv = 0; x 2 @?:
Here, ? ? Rn, n ? 1, is a C1 bounded domain and B stands for either Dirichlet or
Neumann boundary conditions which describe certain restrictions of the movement of the
two species at the boundary of the analyzed region. E.g., homogeneous Neumann boundary
conditions apply to isolated habitats with no migration through the boundary.
The long-term behavior of such systems is well understood. For example, a study by
Brown (1980) for the Neumann case showed that if
a1c2 ?c1a2 > 0 and b1a2 ?a1b2 > 0
then any solution with positive initial value converges over time to the spatially homogenous
positive solution
a1c2 ?c1a2
b1c2 ?c1b2 ;
b1a2 ?a1b2
b1c2 ?c1b2
?
which means the two species coexist. On the other hand, if
a1c2 ?c1a2 > 0 but b1a2 ?a1b2 < 0
3
then solutions with positive initial value converges over time to the spatially homogenous
positive solution
a1
b1;0
?
which predicts the extinction of species 2. Clearly, reproduction rates, carrying capacities
and competition rates may also vary throughout the habitat and in time (seasonal efiects).
This suggests to study more general reaction terms, space and possibly time dependent
ones. Many interesting results have been obtained addressing such issues, and we mention
Leung (1980), Pao (1981), Pao & Zhou (1982), Smith & Thieme (2001), Dancer & Zhang
(2002),...
The starting point of my Ph.D. research work was the simulation of the long-term
behavior of two competing species which populate overlapping, but difierent spatially het-
erogeneous habitats. How trustworthy are such flndings considering that the issue is not
the approximation of a solution of an initial value problem on a flnite time interval, but the
asymptotic behavior on an inflnite time-interval. A paper [2] by S.M. Bruschi, A.N. Car-
valho and J.G. Ruas-Filho addresses this issue for one-dimensional parabolic equation. The
authors establish the dynamical equivalence of the ows on the attractor of the continuous
problem and the attractor of the spatially discretized problem, respectively.
There are quite a few obstacles to extending this result. The most signiflcant one which
needs to be addressed flrst, is the question of structural attractor stability. In the case of a
parabolic equation this issue is linked to the concept of gradient system, but this concept
does not apply to parabolic systems in general, and in particular, not to systems under
consideration.
4
My thesis research therefore focuses on establishing this structural stability of the
attractor (A-stability) for two-species competition systems with difiusion
8
>>>>
>>><
>>>
>>>>
:
@tu = k1?u+uf(x;u;v); x 2 ?;
@tv = k2?v +vg(x;u;v); x 2 ?;
Bu = Bv = 0; x 2 @?;
(1.0.1)
on a C1 bounded domain ? ?Rn with either Dirichlet or Neumann boundary conditions.
Here u(x;t), v(x;t) are the densities of two competing species, k1; k2 are difiusion constants
(calleddispersalratesintheecologicalliterature), andf; g aresmoothfunctionson ???R?R
satisfying certain properties preserving features of competition models. Ecologically, one is
only interested in nonnegative solutions (u;v).
It turns out that in order to understand the relationship between the dynamics of
(1.0.1) and of certain classes of discretizations for (1.0.1), it is essential to investigate the
structural stability of the global attractor of (1.0.1) on a positive cone. Obviously, one
cannot expect that numerical approximations re ect the asymptotic behavior of a solution
semi- ow, if the behavior is not \generically robust" against small perturbations.
The concept of structural stability was introduced by Andronov and Pontryagin (1937).
Since then, a systematic theory of structural stability for difieomorphisms and vector flelds
on manifolds has been well developed by M.M. Peixoto (1959), S. Smale (1967), D.V.
Asonov (1967), C.Pugh (1967), J. Moser (1969), J. Palis (1969), J. Palis & S. Smale (1970),
J.Robin (1971), C. Robinson (1976), S. T. Liao (1980), S. Newhouse (1980), J. Hale (1981),
R.Ma~n?e (1988), M. Hirsch (1990),... In 1984, the weaker notion of A?stability (attractor
5
stability) which is more suitable in the inflnite-dimensional case and for the numerical issue
mentioned before, was introduced by J. Hale L. Magalh~aes & W.Olivia in [13].
The concept of Morse-Smale structure emerges as a su?cient condition for structural
stability. Classically, Morse-Smale system refers to systems which have a flnite number of
critical elements, all of which are hyperbolic and satisfy a transversality condition when
their stable and unstable manifolds intersect.
One of the celebrated results due to the J. Palis & S. Smale states that if a Cr(r ?
1) difieomorphism on a compact C1 manifold without boundary is Morse-Smale, then it
is structurally stable. In [13], J. Hale L. Magalh~aes & W.Olivia proved that any f 2
KCr(B;B) which is Morse-Smale, is A-stable. Here, B is a Banach manifold imbedded in
a Banach space E and the choice of the classes KCr(B;B) depends on the problems under
consideration. Another important result in this context is due to Kennig Lu [22] who proved
the \structural stability on a neighborhood of the attractor" of scalar parabolic equations.
Typically, Morse-Smale systems have been deflned in the context of (Banach) manifolds
with or without boundaries (see [26], [27], [24], [25],...), but positive cones do not fall into
these categories. Therefore we flrst need to modify the classical concepts of Morse-Smale
system and structural stability in such a way that they apply to positive cone settings.
Since we are only interested in positive solutions, we will consider nonlinearities in the set
of pairs (f;g); f;g : ???R?R!R C2 functions satisfying
(H1) f(x;0;0) > 0; g(x;0;0) > 0 8x 2 ??;
(H2) @uf(x;u;v); @vf(x;u;v); @ug(x;u;v); @vg(x;u;v) < 0 8 u;v ? 0; 8x 2 ??;
(H3) supx2??; v?0 limsupu!1f(x;u;v) < 0;
6
(H4) supx2??; u?0 limsupv!1g(x;u;v) < 0:
Our main result states for the spatially one-dimensional case that if (1.0.1) is a Morse-
Smale system on a positive cone, it is structurally stable. We also provide the su?cient
condition under which the system has the Morse-Smale property. These results will have sig-
niflcant impact on the study of the asymptotic dynamics of various classes of discretizations
of (1.0.1).
To obtain these results, we view (1.0.1) as an evolution equation on a suitable fractional
power space. Under the conditions imposed on the nonlinearities, we have global existence
of solutions and hence existence of a global attractor. Since the long-term features of the
dynamics of the system are determined by the global attractor, which lies inside a su?ciently
large ball, we flrst reduce (1.0.1) to a flnite dimensional system by means of Chow, Lu &
Sell?s inertial manifold theorem [6]. Recall that an inertial manifold I is a subset of the
phase space satisfying the following properties
1. I is a flnite dimensional smooth manifold
2. I is invariant under the semi- ow generated by (1.0.1)
3. I is exponentially attracting solutions.
Next, we need to prove A-stability in the (flnite dimensional) inertial manifold setting. The
proof adapts the main idea, J. Hale, L. Magalh~aes, & W.Olivia used in [13]. As mentioned
before, their result cannot be applied to our problem because we work on a subset of the
positive cone and not on a Banach manifold which is imbedded in another Banach space.
7
After having established the structural stability of (1.0.1) in this dissertation, we intend
to investigate long-term aspects of various classes of numerical approximation schemes to
(1.0.1) in the future work.
Having solved one problem in mathematics leads usually to an array of new questions.
Let me just mention a few. Obviously, it will be an important task to address the same
questions as considered here for spatially higher dimensional cases. The key obstacle arises
from the fact that one cannot utilize C1-inertial manifolds as a reduction tool since they
rarely exist in higher dimensions.
Two important issues in connection with the original ecological problem arise: How
does the size of the overlapping region of the two habitats afiects coexistence and extinction.
Mathematically speaking, one is lead to a peculiar bifurcation problem for the steady state
system associated with (1.0.1). What is the impact of seasonal efiects, a question, which in
the general case leads to systems similar to (1.0.1), but with time-dependent reaction terms
and on domains which vary in time.
8
Chapter 2
Definitions, Notations and Main Results
2.1 General Semi ows
Deflnition 2.1.1. (local semi- ow, semi- ow)
(1) Let (Y;d) be a metric space. A map T : D(T) ?R+ ?Y ! Y is said to be a local
semi- ow on Y if for each y 2 Y, there is ?(y) > 0 such that [0;?(y))?fyg ? D(T),
(t;y) 62D(T) for any t ? ?(y) (i.e. [0;?(y)) is the maximal interval of existence of the
local semi- ow with initial condition y at t = 0), and the following hold,
(i) T0 = Id,
(ii) Given y 2 Y, and t1;t2 ? 0, if (t1 + t2;y) 2 D(T), then (t1;Tt2(y)) 2 D(T) and
Tt1+t2(y) = Tt1 ?Tt2(y),
(iii) Tt(x) is continuous in t;x for (t;x) 2D(T);
where Tt(y) = T(t;y): Furthermore, if for each y 2 Y, ?(y) = 1, then T is called a
semi- ow on Y.
(2) Let T be a (local) semi- ow on Y and r 2N. If Y is a Banach space and T satisfles
the following additional property
(iv) Tt(x) is continuous in t;x together with Fr?echet derivatives in x up through order
r for (t;x) 2D(T);
then T is called a (local) Cr semi- ow.
9
Deflnition 2.1.2. (hyperbolicity) Let (Y;d) be a metric space and T : D(T) ?R+?Y ! Y
be a local semi- ow on Y.
(i) fi 2 Y is a flxed point if ?(fi) = 1, Tt(fi) = fi for all t ? 0. A flxed point fi of
the semi- ow T is said to be hyperbolic if the spectrum (DTt(fi)) of the Fr?echet
derivative DTt(fi) is disjoint from the unit circle of the complex plane for all t > 0.
(ii) fi ? Y is a periodic solution of period if for any p 2 fi, ?(p) = 1, Tt+ (p) = Tt(p)
for all t ? 0, and fTt(p)j t 2 [0;1)g = fi. A periodic solution fi of period of the
semi- ow T is said to be hyperbolic if for any p 2 fi, ? = 1 is a simple eigenvalue
of DT (p) and the spectrum set (DT (p))nf1g of the Fr?echet derivative DT (p) is
disjoint from the unit circle of the complex plane.
(iii) A critical element of the semi- ow T is either a flxed point or a periodic solution.
Deflnition 2.1.3. Let (Y;d) be a metric space and T : D(T) ?R+?Y ! Y be a local semi-
ow on Y. We say that y 2 Y has a global backward extension with respect to T if there
exists a continuous function ? : (?1;?(y)) ! Y such that ?(0) = y and Tt(?(s)) = ?(s+t)
for all t > 0 and t + s < ?(y). If y has a global backward extension, we will write T?t(x)
for ?(?t); t > 0. The set S?2(?1;?(y)) ?(?) is called the global orbit of y with respect to T.
Deflnition 2.1.4. (stable, local stable, unstable, local unstable manifolds)
Let (Y;d) be a metric space and T : D(T) ?R+ ?Y ! Y be a local semi- ow on Y.
The stable, local stable, unstable, local unstable manifolds at a hyperbolic critical element
fi of the semi- ow T, denoted by Ws(fi), Wsloc(fi), Wu(fi), Wuloc(fi) are deflned as follows
Ws(fi) = fx 2 Y j ?(x) = 1; d(Tt(x);fi) ! 0 as t !1g,
Wsloc(fi) = fx 2 Ws(fi) j Tt(x) 2 B 8t ? 0g, B is some neighborhood of fi,
10
Wu(fi) = fx 2 Y j x has a global backward extension with respect to T
and d(T?t(x);fi) ! 0 as t !1g,
Wuloc(fi) = fx 2 Wu(fi) j T?t(x) 2 B 8t ? 0g, B is some neighborhood of fi.
Deflnition 2.1.5. (Tubular family for ows) Let W be a flnite dimensional Banach space
and r 2N. Consider a Cr semi- ow : D( ) ?R+ ?W ! W on W.
(1) Let fi be a flxed point of . A tubular family of Ws(fi), denoted by ?s(fi), is a
collection of disjointCr-submanifolds (called leaves) of W, denoted byf?sygorf?sy(fi)g
for clarity, indexed by y in an open neighborhood N of fi in Wu(fi) with the following
properties
a. ?s(fi) = Sy2N ?sy is an open set of W containing Ws(fi);
b. ?sfi = Ws(fi);
c. ?sy intersects N transversally at y,
d. The map ?s(fi) ! N, ?sy 7! y is continuous; the section s which sends x 2 ?sy into
the tangent space of ?sy at x is a continuous map from ?s(fi) into the Grassmann
bundle over ?s(fi).
(2) Let fi be a periodic orbit of with period and S be a cross-section of fi at p 2 fi.
S is called invariant if (U) ? S, where U is a neighborhood of p 2 S. Let k be
the restiction of to U, k := jU : U ! S, and ?sy; y 2 U be an invariant tubular
family of Ws(p) (with respect to k). The tubular family of Ws(fi) is now deflned by
?s( t(y)) = t(?sy) for y 2 U and all t ? 0.
11
(3) A system of tubular families of is a set of tubular families of flxed point(s) and
periodic orbit(s). It is compatible if given ?sx(fi) \ ?sy(fl) 6= ; (where fi; fl are two
difierent critical elements of ), then one submanifold contains the other.
2.2 Semi ows Generated by Competition Models
Deflnition 2.2.1. Let ? be a C1 bounded domain in Rn; n ? 1, and X ? Lp(?) (p > n)
be a fractional power space of ?? : D(?) ! Lp(?), see [15], satisfying X ,! C1(??), where
D(?) = fu 2 H2;p(?)j Bu = 0 on @?g. Here, H2;p(?); p ? 1 are the well-known Sobolev
spaces (cf.[1] ). Under the assumptions (H1)-(H4), (1.0.1) generates a (local) semi- ow on
X ?X (cf. [15]), we denote it by ?,
? : D(?) ?R+ ?X ?X ! X ?X
?t(u0;v0) := ?(t;u0;v0) = (u(t;?;u0;v0);v(t;?;u0;v0)); t 2 ?(u0;v0);
where (u(t;?;u0;v0);v(t;?;u0;v0)) is the solution of (1.0.1) with
(u(0;?;u0;v0);v(0;?;u0;v0)) = (u0;v0);
and [0;?(u0;v0)) is the maximal interval of existence of solution of (1.0.1) with initial
condition (u0;v0) at t = 0. For clarity, we may write ?fgt instead of ?t.
As mentioned in Chapter 1, due to the nature of (1.0.1), we are only interested in
nonnegative solutions of (1.0.1). Therefore we introduce the positive cone X+ ? X+ of
X ?X,
X+ ?X+ := f(u;v) 2 X ?Xj u ? 0; v ? 0 on ?g:
12
A solution (u(t;x);v(t;x)) is called nonnegative if (u(t;?);v(t;?)) 2 X+ ? X+ for t 2
[0;?(u(?;0);v(?;0)). It can be proved that if (u0;v0) 2 X+ ? X+, then ?(u0;v0) = 1
and ?t(u0;v0) 2 X+?X+ for all t ? 0 (see Proposition 3.3.1). Hence ?t (t ? 0); restricted
to X+ ?X+ is a semi- ow.
Therestriction toX+?X+ ofstable, localstable, unstable, andlocalunstable manifolds
of a critical element fi of ? are denoted by
Wu+(fi) = Wu(fi)\(X+ ?X+);
Wu+loc (fi) = Wuloc(fi)\(X+ ?X+);
Ws+(fi) = Ws(fi)\(X+ ?X+);
Ws+loc(fi) = Wsloc(fi)\(X+ ?X+):
Deflnition 2.2.2. (Global attractors and Non-wandering sets)
Let X and ?fg be deflned as in deflnition 2.2.1.
a. We deflne the global attractor of ?fg, denoted by A (or A(f;g)), to be the set
f(u;v) 2 X+ ?X+ j (u;v) has a bounded global orbitg:
b. An element (u;v) 2A is called a non-wandering point if , for any neighborhood U of
(u;v) in A and any T > 0, there exists t0 = t0(U;T) > T and (~u;~v) 2 U such that
?fgt0 (~u;~v) 2 U. The non-wandering set, denoted by ?(f;g), is the set of non-wandering
points.
13
Deflnition 2.2.3. (competition order) Given (u1;v1), (u2;v2) 2 X+ ?X+, we write
(u1;v1) ?2 (u2;v2) if u1 ? u2; v2 ? v1;
(u1;v1) <2 (u2:v2) if (u1;v1) ?2 (u2;v2); (u1;v1) 6= (u2;v2);
(u1;v1) ?2 (u2;v2) if (u2 ?u1;v1 ?v2) 2 IntX+ ?IntX+:
Deflnition 2.2.4. A hyperbolic critical element fi in X+?X+ of (1.0.1) is called a source
if Ws+(fi)\A = fi and a sink if Wu+(fi) = fi, otherwise fi is said to be a saddle.
Deflnition 2.2.5. (fundamental domains and fundamental neighborhoods)
Let fi 2 X+ ?X+ be a hyperbolic critical element of (1.0.1).
a. If fi is not a sink , a fundamental domain for Wu+loc (fi) is denoted by Gu+(fi) and is
deflned as Gu+(fi) = @B(fi) for some open disk B(fi) ? Wu+loc (fi) (that is B(fi) is a
disk centered at fi if fi is a flxed point or a tubular neighborhood of fi if fi is a periodic
orbit). Any neighborhood Nu+(fi) in A of Gu+(fi) such that Nu+(fi)\Ws+(fi) = ;
is called fundamental neighborhood for Wu+(fi).
b. If fi is not a source , a fundamental domain for Ws+loc(fi) is denoted by Gs+(fi) and is
deflned as Gs+(fi) = @B(fi)\A for some open disk B(fi) ? Wu+loc (fi) (that is B(fi) is a
disk centered at fi if fi is a flxed point or a tubular neighborhood of fi if fi is a periodic
orbit). Any neighborhood Ns+(fi) in A of Gs+(fi) such that Ns+(fi)\Wu+(fi) = ;
is called fundamental neighborhood for Ws+(fi).
Deflnition 2.2.6. (Morse-Smale structure) We say (1.0.1) has Morse-Smale structure if
14
a. the semi- ow ?tjX+?X+ has a flnite number of critical elements (i.e. flxed points or
periodic solutions), all of which are hyperbolic and their union coincides with the
non-wandering set ?(f;g),
b. if fi, fl are two critical elements of ?tjX+?X+, their unstable manifolds are flnite-
dimensional, and the global unstable manifold Wu+(fi) and the local stable manifold
Ws+loc(fl) either do not intersect or they intersect transversally (i.e. x 2 Wu+(fi) \
Ws+loc(fl) implies TxWu+(fi)'TxWs+loc(fl) = X ?X).
Deflnition 2.2.7. Let
CP := f(f;g)j f;g : ???R?R!R C2 functions; f;g satisfy (H1)-(H4)g:
We deflne a metric on CP, denoted by dCP, as follows
?N((f;g);(~f;~g)) := k(f;g)?(~f;~g)kC2(???[?N;N]?[?N;N]);
dCP((f;g);(~f;~g)) :=
1X
N=1
(?N((f;g);(~f;~g))
2N(1+?N((f;g);(~f;~g))):
Deflnition 2.2.8. We deflne the Morse-Smale set as follows
MS := f(f;g) 2CPj (1.0.1) has Morse-Smale structureg:
Deflnition 2.2.9. (A-stability) Given (f0;g0) 2 CP. System (1.0.1) is A-stable if there
exists an "0 > 0 such that for each (f;g) 2 CP, dCP((f;g);(f0;g0)) < "0, there exists a
homeomorphism
H : A(f0;g0) !A(f;g)
15
which takes trajectories of A(f0;g0) to trajectories of A(f;g) and preserves the sense of
direction in time.
2.3 Main Results
The following are the main results of the dissertation and are stated for the positive
cone setting.
Theorem A. The set MS (cf. Deflnition 2.2.8) is open (in CP).
Theorem B. Given (f;g) 2CP. Assume all the critical elements of (1.0.1) are hyperbolic
and their union coincides with the non-wandering set ?(f;g). Furthermore, suppose that
the dimension of the unstable manifold of an equilibrium solution in X+ ?X+ nf(0;0)g is
at most one and the dimension of the unstable manifold of a periodic solution is at most
two, then (1.0.1) has the Morse-Smale structure.
Theorem C. Let ? = (0;1). If (f0;g0) 2MS, then (1.0.1) is A-stable.
16
Chapter 3
Preliminary results
3.1 General Semi ows
Deflnition 3.1.1. Let (Y;d) be a complete metric space, ? be a metric space and T? :R?
Y ! Y be a continuous semi- ow for each ? 2 ?. The semi- ow T? is asymptotically smooth
if, for any nonempty, closed, bounded set B ? Y for which T?t (B) ? B 8t ? 0, there is a
compact set J?(B) ? B such that J?(B) attracts B under T?t , that is, d(Tt(B);J?(B)) ! 0
as t ! 1. We say the family of semi- ows fT?g?2?, is collectively asymptotically smooth
if S?2? J?(B) is compact.
Theorem 3.1.1. Let (Y;d) be a complete metric space, ? be a metric space and T? :
R?Y ! Y; t ? 0, is a semi- ow, for each ? 2 ?. Suppose that
(i) There is a bounded set B ? Y independent of ? such that B attracts points of Y
under T?t , that is, d(T?t (y);B) ! 0 as t !1, 8y 2 Y.
(ii) For any bounded set U, the set V := S?2?St?0 T?t U is bounded,
(iii) The family of semi- ows is collectively asymptotically smooth.
Then the global attractor A? of T? is upper semicontinuous in ?.
Proof. This is Theorem 3.5.3 in [12].
Lemma 3.1.2. (see [15]) Let A be a sectorial operator in a Banach space Y with Re (A) >
? > 0and Y fi; Y fl (0 ? fi ? fl < 1)denote thefractionalpowerspacesof Y andf : Y fi ! Y
17
be locally Lipschitzian. Denoted by w(t;t0;w0) the unique solution of
8>
<
>:
_w +Aw = f(w); t > t0;
w(t0) = w0 2 Y fi:
(3.1.1)
Then there exist constants C1; C2 depending only on fi; fl such that
kw(t;t0;w0)kY fl ? C1(t?t0)?(fl?fi)e??(t?t0)kw0kY fi+
C2Rtt0 (t?s)?fle??(t?s)kf(w(s))kY ds; 8t0 < t < ?(t0;w0);
where [t0;?(t0;w0)) is the maximal interval of existence o solution of (3.1.1) with initial
condition w0 at t = t0.
Proof. Using the variational representation of w, we have
w(t;t0;w0) = e?A(t?t0)w0 +
Z t
t0
e?A(t?s)f(w(s)) ds
Aflw(t;t0;w0) = Afle?A(t?t0)w0 +
Z t
t0
Afle?A(t?s)f(w(s)) ds
kw(t;t0;w0)kY fl ? kAfl?fie?A(t?t0)kYkAfiw0kY +
Z t
t0
kAfle?A(t?s)kYkf(w(s))kY ds
? C1(t?t0)?(fl?fi)e??(t?t0)kw0kY fi +
C2
Z t
t0
(t?s)?fle??(t?s)kf(w(s))kY ds; 8t0 < t < ?(t0;w0):
18
3.2 Single Species Equation
Consider the boundary value problem
8
>>>
<
>>>:
?t = k??+?h(x;?) on ?;
B? = 0 on @?:
(3.2.1)
on a C1 bounded domain ? ? Rn, n ? 1, with either Dirichlet or Neumann boundary
conditions. Here ?(x;t) is the density of certain species, k is difiusive constant, and h is a
C2 function h : ???R!R satisfying
(h1) h(x;0) > 0 8x 2 ??;
(h2) @?h(x;?) < 0 8 ? ? 0; 8x 2 ??;
(h3) supx2?? limsup?!1h(x;?) < 0:
Deflnition 3.2.1. Let L := fh : ???R!RC2 function j h satisfy (h1)-(h3)g. We deflne
a metric on L, denoted by dL, as following
dN(h;~h) := kh?~hkC2(???[?N;N]);
dL(h;~h) :=
1X
N=1
dN(h;~h)
2N(1+dN(h;~h)):
Deflnition 3.2.2. Let X ? Lp(?) (p > n) be a fractional power space of ?? : D(?) !
Lp(?), see [15], satisfying X ,! C1(??), where D(?) = f? 2 H2;p(?)j B? = 0 on @?g.
Under the assumptions (h1)-(h3), (3.2.1) generates a (local) semi- ow on X (cf. [15]), we
19
denote it by ?,
? : D(?) ?R+ ?X ! X
?t(?0) := ?(t;?0) = ?(t;?;?0) for t 2 [0;?(?0));
where ?(t;?;?0) is the solution of (3.2.1) with ?(0;?;?0) = ?0 and [0;?(?0)) is the maximal
interval of existence of solution of (3.2.1) with initial condition ?0) at t = 0.
Proposition 3.2.1. For any ?0 2 X+, the solution ?t(?0) of (3.2.1) with initial ?0 exists
and ?t(?0) 2 X+ for all t ? 0.
Proof. For any M > 0, let ~?t(M) be the solution of the following ODE
_? = ?~h(?)
here ~h(?) = maxx2?? h(x;?). By (h1) and (h3),
~h(0) > 0; ~h(M) < 0 for M 1:
Therefore ~?t(M) exists and ~?t(M) > 0 for all t > 0.
For given ?0 2 X+, let M0 1 be such that ?0(x) ? M0 for all x 2 ??. Observe that
h(x;?) ? ~h(?). Then by comparison principle for parabolic equations,
0 ? ?t(?0) ? ~?t(M0)
for all t 2 [0;?(?0)). Since ~?t(M) exists for all t > 0, ?(?0) = 1.
20
Proposition 3.2.2. There is a unique positive stationary solution ?h in C1(??) \ C2(?)
of (3.2.1) which satisfles k?(t;?;?0) ? ?h(?)kX ! 0 as t ! 1 for any ?0 2 X+; ?0 6? 0.
Moreover, if dL(hn;h) ! 0 as n !1, k?hn ??hkC1(??) ! 0 as n !1.
Proof. By (h3), there exist constants Mh > 0 and ? > 0 such that h(x;m) < ?? for
all x 2 ?? and for all m ? Mh. Clearly, 0 is a lower solution of (3.2.1) and Mh is an
upper solution of (3.2.1). By Theorem 3.4 in [29], (3.2.1) has a unique positive solution
?h 2 Cfi(??)\C2(?) and ?h ? Mh. Since ?h 2 Cfi(??) and h 2 C2(???R), the function
?hh(?;?h) 2 Lp(?) for any p ? 1. Therefore ?h = (??+Id)?1(?hh(?;?h)+?h) 2 H2;p(?)
for any p ? 1. By the imbedding theorem in [1], we have ?h 2 C1(??).
Now, we will prove that if dL(hn;h) ! 0 as n !1, k?hn ??hk? ! 0 as n !1. Since
dL(hn;h) ! 0 as n ! 1, there exists N such that khn ?hkC(??;[0;Mh+1]) < ?=2; 8n > N.
This and the fact that h(x;m) < ?? < 0 for all x 2 ?? and for all m ? Mh imply
hn(x;Mh + 1) < ??=2 < 0 for all x 2 ?? and for all n > N. By (h2), hn(x;m) ?
hn(x;Mh + 1) < ??=2 < 0 for all x 2 ??, for all m ? Mh + 1 and for all n > N. Let
M := maxfMh + 1;Mh1;:::;MhNg, we have h(x;m); hn(x;m) < 0 for all x 2 ??, for all
m ? M and for all n ? 1. From the fact that ?hn ? Mhn for all n ? 1, we have ?hn ? M
for all n ? 1. Hence, 0 ? max?? ?hnhn((?;?hn) + ?hn) ? K for some K > 0. This and
?hn = (??+Id)?1(?hnhn((?;?hn)+?hn)) 2 H2;p(?) for all p ? 1 imply k?hnkH2;p(?) < C
for all n ? 1 for some C > 0 because (??+Id)?1 is a bounded linear operator. Therefore
there exists a subsequence fhnkg of fhng such that ?hnk converges in C1(??) to some u0
( since the imbedding H2;p ,! C1(??) is compact). Hence ?k??hnk = ?hnkhnk(?;?hnk)
converges to u0h(?;u0) in Lp(?) for all p ? 1. Because ?k? (with boundary condition)
is a closed operator on Lp(?), we have u0 2 D(?k?) and ?k??hnk converges to ?k?u0.
21
Therefore, ?k?u0 = u0h(?;u0). Since u0 2 C1(??), we have u0h(?;u0)+u0 2 C1(?). Hence
u0 = (?k? + Id)?1(u0h(?;u0) + u0) 2 C2;fi(?). This means u0 is also a positive solution
of (3.2.1). By uniqueness of solution of (3.2.1), u0 = ?h. Hence k?hn ??hkC1(??) ! 0 as
n !1.
3.3 Two Species Competition Systems
Proposition 3.3.1. For any t ? 0, we have ?t(X+ ?f0g) ? X+ ?f0g, ?t(f0g?X+) ?
f0g?X+ and ?t(X+ ?X+) ? X+ ?X+:
Proof. Let ?t(u0) be the solution of (3.2.1) with initial u0 2 X+ and h(x;u) = f(x;u;0)
and ?t(v0) be the solution of (3.2.1) with initial u0 2 X+ and h(x;u) = g(x;0;v). Then by
(H2) and comparison principle for parabolic equations, we have
0 ? u(t;?;u0;v0) ? ?t(u0)
and
0 ? v(t;?;u0;v0) ? ?t(v0)
for t 2 [0;?(u0;v0)). By Proposition 3.2.1, ?(u0;v0) = 1.
Proposition 3.3.2. If (u1;v1), (u2;v2) 2 X+ ?X+ and (u1;v1) ?2 (u2;v2), then
?t(u1;v1) ?2 ?t(u2;v2) for any t ? 0:
Moreover, if (u1;v1) <2 (u2;v2) and (u1;v1) 62 X+ ? f0g, (u2;v2) 62 f0g ? X+, then
?t(u1;v1) ?2 ?t(u2;v2) for all t > 0.
22
Proof. See [21].
23
Chapter 4
Morse-Smale Structure and A Sufficient Condition
4.1 Morse-Smale Structure
In this section, we shall prove Theorem A, that is, the openness of Morse-Smale set
MS. We flrst show the upper semi-continuity of the global attractor A.
Theorem 4.1.1. Given (f0;g0) 2 CP. The global attractor A(f0;g0) of (1.0.1) is upper
semi-continuous (in X+ ?X+).
Proof. First, we will prove there exists a neighborhood ? of (f;g) in CP and a bounded set
B ? X+ ? X+ independent of (f;g) 2 ? such that B attracts points of X+ ? X+ under
?fgt for any (f;g) 2 ?. Let u0; v0; ~u; ~v be the unique positive solutions of (3.2.1) with
h = f0(?;?;0); g0(?;0;?); f(?;?;0); g(?;0;?) respectively. Repeating the argument used in
proving Proposition 3.2.2, we can flnd a constant M > 0 such that
ku0kC(??); kv0kC(??); k~ukC(??); k~vkC(??) ? M; (4.1.1)
provided dCP((f;g);(f0;g0)) is su?ciently small. Let "0 > 0 be small enough so that
dCP((f;g);(f0;g0)) < "0 )k(f;g)?(f0;g0)kC(???[0;3M]?[0;3M])?C(???[0;3M]?[0;3M]) ? 1:
We then deflne
? := f(f;g) 2CPj (4.1.1) holds and dCP((f;g);(f0;g0)) < "0g:
24
For any (u;v) 2 X+ ? X+; (u;v) 6= (0;0), we have (0;v) ?2 (u;v) ?2 (u;0). By lemma
3.3.2, we have
?fgt (0;v) ?2 ?fgt (u;v) ?2 ?fgt (u;0); 8t > 0; 8(f;g) 2 ?: (4.1.2)
Since ?fgt (u;0) ! (~u;0) and ?fgt (0;v) ! (0;~v) in X ?X ,! C(??)?C(??) as t !1, there
exists tfg0 (u;v) ? 1 such that
k?fgt (u;v)kC(??)?C(??) ? 3M; 8t > tfg0 (u;v); 8(f;g) 2 ?: (4.1.3)
Applying lemma 3.1.2 with Y := Lp(?); Y fi = Y fl := X, we have
k?fgt (u;v)kX?X ? C1e?tk(u;v)kX?X (4.1.4)
+C2
Z t
tfg0 (u;v)
(t?s)?fle?(t?s)kHfg(?fgs (u;v))kLp(?)?Lp(?) ds;
for all t > tfg0 (u;v), where
Hfg : X ?X ! Lp(?)?Lp(?)
(u;v) 7! [uf(:;u;v)+u;vg(:;u;v)+v];
By (4.1.1), we have kHfg(?fgt (u;v))kLp(?)?Lp(?) ? Kf0g0 some constant Kf0g0 depending
on (f0;g0). Hence,
k?fgt (u;v)kX?X ? C1e?tk(u;v)kX?X
+C2Kf0g0
Z 1
0
(t?s)?fle?(t?s) ds; 8t > tfg0 (u;v);
25
Since R10 (t?s)?fle?(t?s) ds < C3 for some constant C3, we then have
k?fgt (u;v)kX?X ? C1e?tk(u;v)kX?X +C2C3Kf0g0; 8t > tfg0 (u;v); (4.1.5)
Let B := f(u;v) 2 X+ ?X+j k(u;v)kX+?X+ ? 2C1 + C2C3Kf0g0g. By (4.1.5), B attracts
(u;v) under ?fgt for any (f;g) 2 ?.
Next, we prove that for any bounded set U ? X+?X+, the setS(f;g)2?St?0 ?fgt (U) is
bounded. Because U is bounded in X?X ,! C(??)?C(??), there exists a constant K > M
such that
(0;K) ?2 ?fgt (u;v) ?2 (K;0); 8t ? 0; 8(u;v) 2 U; 8(f;g) 2 ?:
Hence
kHfg(?fgt (u;v))kLp(?)?Lp(?) < L(K;f;g); 8t ? 0; 8(u;v) 2 U: (4.1.6)
where L(K;f;g) is a constant depending on K;f;g. Using (4.1.6) and the fact that
dCP((f;g);(f0;g0)) < "0 )k(f;g)?(f0;g0)kC2(???[0;K]?[0;K])?C2(???[0;K]?[0;K]) < 2K"0;
we have
kHfg(?fgt (u;v))kLp(?)?Lp(?) < ~L(K;f0;g0); 8t ? 0; 8(u;v) 2 U; 8(f;g) 2 ?: (4.1.7)
26
where ~L(K;f0;g0) is a constant depending only on K, f0 and g0. Applying lemma 3.1.2
with Y := Lp(?); Y fi = Y fl := X, we have
k?fgt (u;v)kX?X ? C1e?tk(u;v)kX?X (4.1.8)
+C2
Z t
0
(t?s)?fle?(t?s)kHfg(?fgs (u;v))kLp(?)?Lp(?) ds; 8t > 0:
From (4.1.7) and (4.1.8), we have
k?fgt (u;v)kX?X ? C1e?tk(u;v)kX?X +C2~L(K;f0;g0)
Z 1
0
(t?s)?fle?(t?s) ds (4.1.9)
for all t ? 0, for all (u;v) 2 U and for all (f;g) 2 ?. Because R10 (t?s)?fle?(t?s) ds < C3,
we then have
k?fgt (u;v)kX?X ? C1e?tk(u;v)kX?X +C2C3~L(K;f0;g0); 8t ? 0; 8(u;v) 2 U; 8(f;g) 2 ?:
(4.1.10)
It is clear from (4.1.10) that S(f;g)2?St?0 ?fgt (U) is bounded in X ?X.
Finally, we prove that the family of semigroups f?fgt ; t ? 0g; (f;g) 2 ? is collectively
asymptotically smooth. Fix any (f;g) 2 ?. For any bounded, closed set B ? X+ ? X+
for which ?fgt (B) ? B; 8t ? 0, we deflne Jfg(B) = ?fg1 (B) (the closure is taken in
X ?X). Since B is closed, we have Jfg(B) ? B. We also have ?fgt (B) = ?1(?fgt?1(B)) ?
?fg1 (B) ? Jfg(B); 8t > 1. This means Jfg(B) attracts B under ?fgt . Applying lemma
3.1.2 with Y := Lp(?); Y fi := X; Y fl := ~X, where ~X is another fractional power space of
27
? : D(?) ? H2;p(?) ! Lp(?) and fl > fi is chosen so that ~X ,! X is compact, we have
k?fg1 (u;v)k~X? ~X ? C1e?1k(u;v)kX?X (4.1.11)
+C2
Z 1
0
(1?s)?fle?(1?s)kHfg(?fgs (u;v))kLp(?)?Lp(?) ds:
Because B is a bounded set in X+ ?X+, we can use same argument used for the bounded
set and get
kHfg(?fgt (u;v))kLp(?)?Lp(?) < ~~L(B;f0;g0); 8t ? 0; 8(u;v) 2 B; 8(f;g) 2 ?: (4.1.12)
where ~L(K;f0;g0) is a constant depending only on B, f0 and g0. From (4.1.12) and (4.1.12),
we have
k?fg1 (u;v)k~X? ~X ? C1e?1k(u;v)kX?X +C2~~L(B;f0;g0)
Z 1
0
(1?s)?fle?(1?s); (4.1.13)
for all (u;v) 2 B and for all (f;g) 2 ?. (4.1.13) and the compact imbedding ~X ,! X imply
S
(f;g)2? Jfg(B) is a compact set in X ? X. Hence, the family of semigroups f?
fg
t ; t ?
0g; (f;g) 2 ? is collectively asymptotically smooth. By lemma 3.1.1, A(f;g) is upper
semi-continuous.
Lemma 4.1.2. Let e 2 X?X be an critical point of (1.0.1). Let Eu(e) and Es(e) denote for
the linear spaces spanned by eigenfunctions which correspond to eigenvalues with negative
and positive real parts of the linearization of (1.0.1) at e. Deflne Eur(e) = Eu(e) \ Br(e)
and Esr(e) = Es(e)\Br(e) where Br(e) is a su?ciently small neighborhood of e (in X?X).
28
Then there are two C1 maps
'1 : Eur(e) ! Esr(e)
(u;v) 7! (k(u;v);l(u;v))
and
'2 : Esr(e) ! Eur(e)
(u;v) 7! (m(u;v);n(u;v)):
such that the local unstable and stable manifolds of e, Wuloc;r(e) and Wsloc;r(e), are the
graphs of '1 +e and '2 +e respectively. Moreover, '1(0) = 0, '2(0) = 0, D'(0) ? 0 and
D?(0) ? 0. Here, D'1 and D'2 are the Fretchet derivatives of '1 and '2.
Proof. This is a corollary of theorem 6.1 in [31].
Lemma 4.1.3. Let e 2 X be a critical element of boundary value problem
8>
<
>:
?t ?k1?? = ?q(x;?); x 2 ?
Bv = 0; x 2 @?
(4.1.14)
where B is either Dirichlet or Neumann boundary condition, q : ?? ?R ! R is a C2-
function. Let Ou(e) and Os(e) denote for the linear spaces spanned by eigenfunctions
which correspond to eigenvalues with negative and positive real parts of the linearization
of (4.1.14) at e. Deflne Our(e) = Ou(e)\Qr(e) and Osr(e) = Os(e)\Qr(e) where Qr(e) is a
su?ciently small neighborhood of e (in X). Then there are two C1 maps
~h1 : Our(e) ?! Osr(e)
? 7?! ~h1(?)
29
and
~h2 : Osr(e) ?! Our(e)
(u;v) 7?! ~h2(?):
such that the local unstable and stable manifolds of e, Wuloc;r(e) and Wsloc;r(e), are the
graphs of ~h1 + e and ~h2 + e respectively. Moreover, ~h1(0) = 0, ~h2(0) = 0, D~h1(0) ? 0 and
D~h2(0) ? 0. Here, D~h1 and D~h2 are the Fretchet derivatives of ~h1 and ~h2 respectively.
Proof. This is a corollary of theorem 6.1 in [31].
Proposition 4.1.4. (representation of stable, unstable manifolds in positive cone)
Let e be an critical point of (1.0.1) on @(X+ ?X+). We have
Wu+loc (e) = e+f(u;v)+(k(u;v);l(u;v))j (u;v) 2 Eur(e)\(X+ ?X+)g;
Ws+loc(e) = e+f(u;v)+(m(u;v);n(u;v))j (u;v) 2 Esr(e)\(X+ ?X+)g;
where Eur, Esr and k;l;m;n are from lemma 4.1.2.
Proof. (1.0.1) has three flxed points on @(X+ ?X+): (0;0), (u0;0) and (0;v0).
Case 1: e = (0;0). By lemma 4.1.2 , we have
Wuloc(e) = f(u;v)+(k(u;v);l(u;v))j (u;v) 2 Eur(e)g:
We will prove
Wu+loc (e) := Wuloc(e)\(X+ ?X+)
= f(u;v)+(k(u;v);l(u;v))j (u;v) 2 Eur(e)\(X+ ?X+)g:
30
By lemma 4.1.3, we can write the unstable manifold at 0 of the semi- ow generated
by the boundary value problems
8
><
>:
ut ?k1?u = uf(x;u;0); x 2 ?;
Bu = 0; x 2 @?:
and 8
><
>:
vt ?k1?v = vg(x;0;v); x 2 ?;
Bv = 0; x 2 @?:
as
Wuloc(0) = fu+h1(u)j u 2 Uur (0)g and fWuloc(0) = fv + ~h1(v)j v 2 V ur (0)g:
For any u 2 Uur (0), we have (u+h1(u);0) 2 Wuloc(e). Therefore, there exists (~u;~v) 2
Eur(e) such that (~u;~v)+(k(~u;~v);l(~u;~v)) = (u+h1(u);0). Hence
8
><
>:
~u+k(~u;~v) = u+h1(u)
~v +l(~u;~v) = 0
)
8>
<
>:
~u?u = h1(u)?k(~u;~v)
~v = ?l(~u;~v)
)
8>
<
>:
~u?u = h1(u)?k(~u;~v) = 0
~v = ?l(~u;~v) = 0
because (~u?u;~v) 2 Eur(e), (h1(u)?k(~u;~v);?l(~u;~v)) 2 Esr(e) and Eur(e)\Esr(e) =
(0;0). Hence l(u;0) = 0 for all u 2 Uur (0). A similar argument yields k(0;v) = 0; 8v 2
31
V ur (0). For (u;v) 2 Eur(e), put ?(?) = k(?u;v); 0 ? ? ? 1. We have
k(u;v) = ?(1)??(0) =
Z 1
0
?0(?)d? =
Z 1
0
@1k(?u;v)d?
?
u:
A similar argument yields
l(u;v) =
Z 1
0
@2l(u;?v)d?
?
v:
Therefore
(u;v)+(k(u;v);l(u;v)) = (u(K(u;v)+1);v(L(u;v)+1))
where
K(u;v) =
Z 1
0
@1k(?u;v)d? and L(u;v) =
Z 1
0
@2l(u;?v)d?:
By lemma 4.1.2, we have K(u;v);L(u;v) ? 1 for u;v ? 1. Therefore, we have
Wu+loc (e) := Wuloc(e)\(X+ ?X+)
= f(u;v)+(k(u;v);l(u;v))j (u;v) 2 Eur(e)\(X+ ?X+)g:
Case 2: e = (u0;0). By lemma 4.1.2 , we have
Wuloc(e) = f(u;v)+(k(u;v);l(u;v))j (u;v) 2 Eur(e)g:
32
We will prove
Wu+loc (e) := Wuloc(e)\(X+ ?X+)
= f(u;v)+(k(u;v);l(u;v))j (u;v) 2 Eur(e)\(X+ ?X+)g:
By lemma 4.1.3, we can write the unstable manifold at u0 of the semi- ow generated
by the boundary value problem
8>
<
>:
ut ?k1?u = uf(x;u;0); x 2 ?;
Bu = 0; x 2 @?:
as
Wuloc(u0) = fu+h1(u)j u 2 Uur (u0)g:
For any u 2 Uur (u0), we have (u+h1(u);0) 2 Wuloc(e). Therefore, there exists (~u;~v) 2
Eur(e) such that (~u;~v)+(k(~u;~v);l(~u;~v)) = (u+h1(u);0). Hence
8>
<
>:
~u+k(~u;~v) = u+h1(u)
~v +l(~u;~v) = 0
)
8
><
>:
~u?u = h1(u)?k(~u;~v)
~v = ?l(~u;~v)
)
8
><
>:
~u?u = h1(u)?k(~u;~v) = 0
~v = ?l(~u;~v) = 0
because (~u?u;~v) 2 Eur(e), (h1(u)?k(~u;~v);?l(~u;~v)) 2 Esr(e) and Eur(e)\Esr(e) =
(0;0). Hence l(u;0) = 0 for all u 2 Uur (u0). For (u;v) 2 Eur(e), put ?(?) =
33
l(u;?v); 0 ? ? ? 1. We have
l(u;v) = ?(1)??(0) =
Z 1
0
?0(?)d? =
Z 1
0
@2l(u;?v)d?
?
v:
Therefore
(u;v)+(k(u;v);l(u;v)) = (u+k(u;v);v(L(u;v)+1))
where
L(u;v) =
Z 1
0
@2l(u;?v)d?:
By Lemma 4.1.2, we have L(u;v) ? 1 for u?u0;v ? 1. Therefore, we have
Wu+loc (e) := Wuloc(e)\(X+ ?X+)
= f(u;v)+(k(u;v);l(u;v))j (u;v) 2 Eur(e)\(X+ ?X+)g:
Proposition 4.1.5. Given (f0;g0) 2 CP. Let e0 2 X+ ? X+ be a hyperbolic critical
element of f?f0g0t ; t ? 0g. Then there exist a neighborhood eO of e0 in (X+ ?X+) and a
neighborhood V(f0;g0) of (f0;g0) (in CP) such that given (f;g) 2V(f0;g0) there exists a
unique homeomorphism
? := ?(f;g) : e0 ! ?(e0) =: e 2 eO
close to the inclusion i : e0 ! (X+ ? X+) in the C0-topology, and e 2 X+ ? X+ is a
hyperbolic critical element of f?fgt ; t ? 0g. Moreover, the map (f;g) 2V(f0;g0) 7! ?(f;g)
34
is continuous, Wu+loc (e), Ws+loc(e) depend continuously on (f;g) 2 V(f0;g0) which yields
dimWu+loc (e) = dimWu+loc (e0) for all (f;g) 2V(f0;g0).
Proof. For any l > 0, we deflne
Rl : CP ! C2(??[0;l]?[0;l])?C2(??[0;l]?[0;l])
(f;g) 7! (fj???[0;l]?[0;l];gj???[0;l]?[0;l]):
It is clear that Rl is a continuously linear map. By (H3) and (H4), there exist constants
Kf0g0 > 0 and ? > 0 such that
f0(x;m;n) < ??; g0(x;m;n) < ?? for all x 2 ??; for all m; n ? Kf0g0: (4.1.15)
Now, flx a neighborhood U(f0;g0) of (f0;g0) in CP such that
k(f;g)?(f0;g0)kC2(???[0;Kf0g0+1]?[0;Kf0g0+1])?C2(???[0;Kf0g0+1]?[0;Kf0g0+1]) < ?=2: (4.1.16)
From (4.1.15) and (4.1.16), we have
f(x;Kf0g0 +1;Kf0g0 +1); g(x;Kf0g0 +1;Kf0g0 +1) < ??=2; 8x 2 ??; 8(f;g) 2 U(f0;g0):
By (H2), we then have
f(x;m;n) < ??=2; g(x;m;n) < ??=2; 8x 2 ??; 8m;n ? Kf0g0 +1; 8(f;g) 2 U(f0;g0):
35
Hence, all critical elements of f?f0g0t ; t ? 0g and f?fgt ; t ? 0g take values inside the
contracting rectangle [0;Kf0g0+1]?[0;Kf0g0+1]. Therefore, we can consider the restriction
of the semi- ows f?f0g0t ; t ? 0g and f?fgt ; t ? 0g to the Banach manifold
f(u;v) 2 X ?Xj u(?);v(?) ? (0;Kf0g0 +1)g:
If e 2 int(X+?X+), apply proposition 2.12 in [23] with B := f(u;v) 2 X?Xj u(?);v(?) ?
(0;Kf0g0 + 1)g and ~F := C2(???[0;Kf0g0 + 1]?[0;Kf0g0 + 1])?C2(???[0;Kf0g0 + 1]?
[0;Kf0g0 + 1]); there exists a neighborhood O of e0 in B and a neighborhood Q(f0;g0) of
(f0;g0) in ~F such that given (f;g) 2Q(f0;g0) there exists a unique homeomorphism
? := ?(f;g) : e0 ! ?(e) =: e 2O
close to the inclusion i : e0 !B in the C0-topology and e is a hyperbolic critical element of
?fgt . Since ? is close to the inclusion i, e must be in B ? int(X+ ?X+) and therefore ? is
close to the inclusion i : e0 ! (X+ ?X+). Put eO = O\(X+ ?X+). Then e 2O. Because
Wu+loc (e) := Wuloc(e)\(X+?X+) and Wuloc(e) depends continuously on (f;g) (by proposition
2.12 in [23]), Wu+loc (e) also depends continuously on (f;g). Similar reason implies Ws+loc(e)
depends continuously on (f;g). Because RKf0g0+1 is a continuous map, R?1Kf0g0+1(Q(f0;g0))
is an open neighborhood of (f0;g0) in CP. Then V(f0;g0) := U(f0;g0)\R?1Kf0g0+1(Q(f0;g0))
is the desired neighborhood of (f0;g0) in CP.
If e 2 @(X+ ?X+), Due to the conditions (H1)-(H4), we have three cases e = (0;0),
e = (u0;0) and e = (0;v0) where u0; v0 are the unique stationary solutions of (3.2.1)
with h(?;u) = f0(?;u;0) and h(?;v) = g0(?;0;v). Deflne ?(0;0) = (0;0); ?(u0;0) =
36
(~u;0); ?(0;v0) = (0;~v) where ~u; ~v are solutions of (3.2.1) with h(?;u) = f(?;u;0) and
h(?;v) = g(?;0;v). By proposition 3.2.2 and lemma 4.1.2, we have Ws(u0;0) and Ws(~u;0)
are C1 close. Hence, Ws+loc(u0;0) and Ws+loc(~u;0) are C1 close. Similar argument yields the
conclusion for the (0;v0) and (0;0).
Proposition 4.1.6. Given (f0;g0) 2 CP. Let e0 2 X+ ?X+ be a critical element (e0 is
not a sink ) of f?f0g0t ; t ? 0g and Nu+(e0) be a fundamental neighborhood of Wu+(e0).
Then there exists a neighborhood V(f0;g0) of (f0;g0) in CP and a neighborhood ~O of e0
in X+ ? X+ such that Nu+(e0) is also a fundamental neighborhood of Wu+(e), where
e := ?(e0) is the unique hyperbolic critical element in ~O of f?fgt ; t ? 0g; (f;g) 2V(f0;g0).
Here, ? is the homeomorphism in proposition 4.1.5. Moreover, there exists a neighborhood
~B of e0 such that for any (f;g) 2V(f0;g0), we have ~B ?St?0 ?fg?t(Nu+(e0))[Ws+loc(e):
Proof. If e0 2 int(X+ ? X+), we use the same argument used in proposition 4.1.5 and
hence the result is direct from proposition 2.14 of [23]. We only need to consider the cases
e0 2 @(X+ ? X+). By proposition 2.14 of [23], there exists a neighborhood V(f0;g0) of
(f0;g0) in CP and a neighborhood O of e0 in X ? X such that Nu(e0) is also a funda-
mental neighborhood of Wu(e), e := ?(e0), is the unique hyperbolic critical element in O
of f?fgt ; t ? 0g; (f;g) 2 V(f0;g0). Here ? is the homeomorphism in proposition 2.12
of [23]. By proposition 4.1.5, e is in X+ ? X+. Hence, e is the unique hyperbolic crit-
ical element in ~O := O \ (X+ ? X+) of f?fgt ; t ? 0g; (f;g) 2 V(f0;g0). It is also
obvious that Nu+(e0) := Nu(e0) \ (X+ ? X+) is also a fundamental neighborhood of
Wu(e)\(X+ ?X+) =: Wu+(e). Finally, we will prove the existence of ~B. By proposition
37
2.14 of [23], there exists a neighborhood B of e0 such that
B ?
[
t?0
?fg?t(Nu(e0))[Wsloc(e):
Therefore,
B \(X+ ?X+) ?
2
4[
t?0
?fg?t(Nu(e0))\(X+ ?X+)
3
5[Ws+loc(e):
Due to the fact that X+ ?X+ is invariant under the semi- ow f ?fgt ; t ? 0g, it can be
verifled that
[
t?0
?fg?t(Nu(e0))\(X+ ?X+) =
[
t?0
?fg?t(Nu(e0)\(X+ ?X+)) =
[
t?0
?fg?t(Nu+(e0)):
Let ~B := B \(X+ ?X+). Then we have
~B ? [
t?0
?fg?t(Nu+(e0))[Ws+loc(e):
For the cases e = (0;0), e = (u0;0) and e = (0;v0) where u0; v0 are the unique stationary
solutions of (3.2.1) with h(?;u) = f0(?;u;0) and h(?;v) = g0(?;0;v), the conclusion is clear
from the C1 closeness of the local stable, unstable manifolds of (u0;0) and (~u;0) as well as
the local stable, unstable manifolds of (0;v0) and (0;~v).
Proposition 4.1.7. The set of all critical hyperbolic elements of (1.0.1) in X+ ?X+ has
a partial order structure ?3 deflned by e1 ?3 e2 ifi Wu+(e2)\Ws+loc(e1) 6= ;.
38
Proof. Firstly, we have e ?3 e because Wu+(e)\Ws+loc(e) = feg. Second, suppose e1 ?3 e2,
e2 ?3 e1 and e1 6= e2. Since Wu+(e2) \ Ws+loc(e1) 6= ; (e1 ?3 e2), we have Wu(e2) \
Wsloc(e1) 6= ;. By proposition 3.4 of [23], there is a submanifold of Wu(e2) "-C1 close to
B(e1)\Wuloc(e1) (" is small enough and B(e1) is an appropriate open neighborhood of e1).
Since Wu+(e1) \Ws+loc(e2) 6= ; (e2 ?3 e1), we can choose p 2 B(e1) \Wu+loc (e1) such that
?t0(p) 2 Ws+loc(e2) for some t0 > 0. Then for q 2 Wu+(e2) (q 6= e2) close enough to p, we
must have ?t0(q) 2 Ws+loc(e2) (because of transversality). So, q 2 Wu+(e2)\Ws+loc(e2) which
is a contradiction. This implies e1 = e2. Thirdly, suppose e1 ?3 e2 and e2 ?3 e3. Using
the similar argument as above, we have Wu+(e3) \ Ws+loc(e1) 6= ; which means e1 ?3 e3.
Therefore, ?3 is partial order.
Deflnition 4.1.1. Let Crit(f;g) denote the set of all critical elements in X+ ? X+ of
(1.0.1). For e1; en 2 Crit(f;g), e1 6= en, the sequence e1 ?3 e2 ?3 ::: ?3 en (if exists)
is called a chain from e1 to en of length n ? 1. We also write beh(e1jen) = n ? 1 if the
maximum length of chains from e1 to en is equal to n?1.
Proposition 4.1.8. Let (f0;g0) 2 CP. Then there exist a neighborhood V ? X+ ?X+ of
the attractor A(f0;g0) such that if dCP((f0;g0);(f;g)) is small enough, we have
?(f;g)\V = ?(Crit(f0;g0)\V);
where ? is the homeomorphism in proposition (4.1.5).
Proof. We construct V by induction. Let e0i be any sink of Crit(f0;g0) and ei := ?(e0i)
where ? is the homeomorphism in proposition (4.1.5). We have ei 2 Crit(f;g) and e0i, ei
are close. Because e0i, ei are close, there exists a neighborhood V0(e0i) ? Wsloc(e0i) such that
39
ei 2 V0(e0i) ? Wsloc(ei) where Br(e0i)(f0;g0) is a su?ciently small neighborhood of (f0;g0)
in CP. Put V0 := Si V0(e0i) and r0 := minifr(e0i)g. We then have ?(Crit(f0;g0)\V0) ?
?(f;g) \ V0. In the other hand, if x0 2 ?(f;g) \ V0, then x0 2 V0 and x0 2 Crit(f;g).
Since x0 2 V0, there exists ei := ?(e0i); e0i 2 Crit(f0;g0) \ V0, such that x0 2 Wsloc(ei).
Because x0 2 ?(f;g), x0 must be ei. Therefore, x0 2 ?(Crit(f0;g0) \ V0). This implies
?(f;g)\V0 = ?(Crit(f0;g0)\V0) for all (f;g) 2 Br0(f0;g0):
Now, suppose we have constructed Vk, rk corresponding to critical elements of ?fgt
whose behaviors with respect to sinks are ? k, that is,
?(f;g)\Vk = ?(Crit(f0;g0)\Vk); (f;g) 2 Brk(f0;g0):
Let ek+1 2 Crit(f0;g0) be a saddle of ?fgt whose behavior with respect to sinks is less than
or equal k+1. For each x 2 Gu+(ek+1), we have ?fgt (x) ! e for some e 2 Crit(f0;g0)\Vk.
Therefore, there exists tx > 0 such that ?fgtx (x) 2 Vk. Let Ox ? Vk be a neighborhood
of ?fgtx (x) (in Vk). Then U(x) := ?fg?tx(O) is a neighborhood of x in X+ ? X+. O can
be chosen small enough such that U(x) \ Ws+(ek+1) = ;. Since Gu+(ek+1) is compact,
there exists fxjgn1 such that Gu+(ek+1) ? Snj=1 U(xj). Then Nu+(ek+1) := Snj=1 U(xj)
is a fundamental neighborhood of Wu+(ek+1). Put t(ek+1) = maxftxjgn1. Then for any
x 2 Nu+(ek+1), we have ?t(x) 2 Vk for some t < t(ek+1). By proposition 4.1.6, there exist
a neighborhood Br(ek+1)(f0;g0) of (f0;g0) in C and a neighborhood ~B(ek+1) of ek+1 such
that for any (f;g) 2 Br(ek+1)(f0;g0), we have
~B(ek+1) ? Ws+loc(e0k+1)[h[t?0?f0g0?t (Nu+(ek+1))i; (4.1.17)
40
where e0k+1 := ?(ek+1), ? is the homeomorphism in proposition 4.1.5. Deflne Bk+1 :=
S ~B(e
k+1), rk+1 = minfrk;minfr(ek+1)gg, tk+1 = maxft(ek+1)g and
Vk+1 = Vk [
?
[0?t?tk+1?fg?t(Vk)
?
[Bk+1:
We will prove for any (f;g) 2 Brk+1(f0;g0),
?(f;g)\Vk+1 = ?(Crit(f0;g0)\Vk+1):
First, we prove ?(Crit(f0;g0)\Vk+1) ? ?(f;g)\Vk+1. If x0 2 ?(Crit(f0;g0)\Vk+1), then
x0 = ?(x); x 2 Crit(f0;g0) \ Vk+1. Because x 2 Crit(f0;g0), we have x0 2 Crit(f;g) ?
?(f;g) by proposition (4.1.5). Since
Crit(f0;g0)\
h
Vk [
?
[0?t?tk+1?fg?t(Vk)
?i
= Crit(f0;g0)\Vk;
we have
Crit(f0;g0)\Vk+1 = (Crit(f0;g0)\Vk)[(Crit(f0;g0)\Bk+1):
If x 2 Crit(f0;g0)\Vk, then from the induction assumption, we have x0 2 ?(f;g)\Vk ?
?(f;g) \ Vk+1. If x 2 Crit(f0;g0) \ Bk+1 then x 2 Crit(f0;g0) \ ~Bek+1 for some ek+1 2
Crit(f0;g0). By (4.1.17), we then have x 2 Ws+loc(e0k+1) [
h
[t?0?fg?t(Nu+(ek+1))
i
. Since
x is a critical element, it can not be in [t?0?fg?t(Nu+(ek+1)). Therefore, x 2 Ws+loc(e0k+1).
Because ?(x) is a critical element of f?fgt ; t ? 0g, ?(x) must be e0k+1 2 Vk+1 which implies
x0 := ?(x) 2 Crit(f;g)\Vk+1. So, we have ?(Crit(f0;g0)\Vk+1) ? ?(f;g)\Vk+1.
41
Now we prove ?(f;g)\Vk+1 ? ?(Crit(f0;g0)\Vk+1). If x0 2 ?(f;g)\Vk+1, then x0 2
Vk+1 := Vk [
?
[0 0 and ? 2 Wu+loc (e) such
that fl = ?t0(?). Let !(~e) be the normalized (in X ? X) principle eigenfunction of the
linearization problem of system (1.0.1) at ~e 6= (0;0). By the Krein-Rutman theorem (cf.
[16]), we have !(~e) 2 (0;0). By lemma 4.1.2, we have either
lim
?!~e; ?2Wu(~e);
? ? ~e
k? ? ~ekX?X = !(~e) (4.2.1)
or
lim
?!~e; ?2Wu(~e)
? ? ~e
k? ? ~ekX?X = ?!(~e): (4.2.2)
Applying (4.2.1) for ~e = e, there exists ? 2 Wuloc(e) such that
? ?e = k? ?ekX?X
!(e)+ o(k? ?ekX?X)k? ?ek
X?X
?
2 (0;0): (4.2.3)
This implies
?t(?)??t(e) = ?t(?)?e 2 (0;0); 8 t ? 0: (4.2.4)
43
Since fl 2 Ws+loc(e0), we have
?t(?) ! e0 (in X ?X) as t !1: (4.2.5)
Letting t ! 1 in (4.2.4) and using (4.2.5), we have e0 ?2 e. By proposition 3.3.2, we have
?t(e0) 2 ?t(e); 8t > 0 but this means e0 2 e. Now, suppose Wu+(e0) 6= fe0g. Applying
(4.2.2) for ~e := e0, there exists ? 2 Wuloc(e0) such that
? ?e0 = k? ?e0kX?X
?!(e0)+ o(k? ?e
0kX?X)
k? ?e0kX?X
?
?2 (0;0): (4.2.6)
This implies
?t(?)??t(e0) = ?t(?)?e0 ?2 (0;0); 8 t ? 0: (4.2.7)
Since e0 2 e, we can choose ? and ? closed enough to e and e0 (respectively) such that
e ?2 ? ?2 ? ?2 e0. This and ?t(?) ! e0 (in X ? X) as t ! 1 imply ?t(?) ! e0 (in
C(??)?C(??)) as t ! 1. This and the fact that the !-limit set of ? is relative compact in
X ?X implies ?t(?) ! e0 (in X ?X) as t ! 1. Hence ? 2 Ws+(e0) which contradicts to
the fact ? 2 Wu+(e0). Therefore Wu+(e0) = fe0g. This implies Ws+loc(e0) = X ?X. Then, it
is clear that we have TflWu+(e)'TflWs+loc(e0) = X ?X.
Case 2: Let e 6= (0;0) be an flxed point, e0 be a periodic orbit with period and
suppose that 9 fl 2 Wu+(e)\Ws+loc(e0). Using the same argument as in the flrst part of case
1, there exists ? 2 Wu+loc (e) such that fl = ?t0(?) and
?t(?) 2 e; 8 t ? 0: (4.2.8)
44
Since fl 2 Ws+loc(e0), we have
?t(?) ! e0 (in X ?X) as t !1: (4.2.9)
Let p0 2 e0 be a limit point of f?t(?); t ? 0g. We have an increasing sequence tn ! 1
as n ! 1 such that ?tn(?) ! p0. From (4.2.8), we have ?tn(?) 2 e; 8 t ? 0. Let
n !1, we have p0 ?2 e. By proposition 3.3.2, we have ? (p0) 2 ? (e) which is p0 2 e
Now, suppose Wu+(e0) 6= fe0g. This means Wu+(e0) is two-dimensional, D? (p0) has one
eigenvalue with real part greater than 1. By Krein-Rutman theorem, this eigenvalue has a
strongly positive eigenfunction !(p0). Let
W = f? 2 X ?Xj liminfn!1 dX?X(?n (?);e0) > 0g:
Then we have
lim
k??p0kX?X!0; ?2W
? ?p0
k? ?p0kX?X = ?!(p0): (4.2.10)
This implies
? ?p0 = k? ?p0kX?X
?!(p0)+ o(k? ?p0kX?X)k? ?p
0kX?X
?
?2 (0;0): (4.2.11)
Since p0 2 e, we can choose ? and ? closed enough to e and p0 (respectively) such that
e ?2 ? ?2 ? ?2 p0. Thus, ?tn(?) ?2 ?tn(?) ?2 ?tn(p0). The set f?tn(p0)g is compact.
So there exists a subsequence tnkg of ftng and p1 2 e0 such that ?tnk(p0) ! p1 as n ! 1.
If p0 6= p1, then we have p0 ?2 p1. Again, by proposition 3.3.2, we have p0 ?2 p1. By
theorem 2.3 in [32], this can not happen. Therefore, p1 ? p0. Then we have ?tnk(?) ! p0
45
as n !1 which is a contradiction to the fact ? 2 W. Hence, Wu+(e0) = fe0g. This implies
Ws+loc(e0) = X ?X. Then, it is clear that we have TflWu+(e)'TflWs+loc(e0) = X ?X.
Case 3: Let e be a periodic orbit with period , e0 be an equilibrium and suppose that
9 fl 2 Wu+(e)\Ws+loc(e0). Since fl 2 Wu+(e), there exist p?0 2 e and ? 2 Wu+loc (e) such that
fl = ?t0(?) and p?0 ?2 ?. Since ?t(?) ! e0 as t ! 1, we have p?0 ?2 e0. By proposition
3.3.2, we have p?0 ?2 e0. Now, suppose Wu+(e0) 6= fe0g. Applying (4.2.2) for ~e := e0, there
exists ? 2 Wuloc(e0) such that
? ?e0 = k? ?e0kX?X
?!(e0)+ o(k? ?e
0kX?X)
k? ?e0kX?X
?
?2 (4.2.12)
This implies
?t(?)??t(e0) = ?t(?)?e0 ?2 (0;0); 8 t ? 0: (4.2.13)
Since p?0 ? e0, we can choose ? and ? closed enough to e0 and p?0 (respectively) such that
? ?2 ? ?2 e0. This and ?t(?) ! e0 as t !1 in X?X imply ?t(?) ! e0 (in C(??)?C(??))
as n ! 1. This and the fact that the !-limit set of ? is relative compact in X ?X imply
?tn(?) ! p0 (in X ? X) as n ! 1. This is a contradiction to the fact ? 2 Wu+(e0).
Therefore Wu+(e0) = fe0g. This implies Ws+loc(e0) = X ?X. Then, it is clear that we have
TflWu+(e)'TflWs+loc(e0) = X ?X.
Case 4: Let e, e0 be periodic orbits with periods , 0 respectively. Suppose that
9fl 2 Wu+(e) \ Ws+loc(e0). Since fl 2 Wu+(e), there exist p?0 2 e and ? 2 Wu+loc (e) such
that fl = ?t0(?) and p?0 ?2 ?. Let p0 2 e0 be a limit point of f?t(?); t ? 0g. We have
an increasing sequence tn ! 1 as n ! 1 such that ?tn(?) ! p0. Hence, p0 2 p?0.
Now, suppose Wu+(e0) 6= fe0g. This means Wu+(e0) is two-dimensional, D? (p0) has one
46
eigenvalue with real part greater than 1. By Krein-Rutman theorem, this eigenvalue has a
strongly positive eigenfunction !(p0). Let
W = f? 2 X ?Xj liminfn!1 dX?X(?n (?);e0) > 0g:
Then we have
lim
k??p0kX?X!0; ?2W
? ?p0
k? ?p0kX?X = ?!(p0): (4.2.14)
This implies
? ?p0 = k? ?p0kX?X
?!(p0)+ o(k? ?p0kX?X)k? ?p
0kX?X
?
?2 (0;0): (4.2.15)
Since p0 2 p?0, we can choose ? and ? closed enough to p?0 and p0 (respectively) such that
? ?2 ? ?2 p0. Thus, ?tn(?) ?2 ?tn(?) ?2 ?tn(p0). The set f?tn(p0)g is compact. So
there exists a subsequence tnkg of ftng and p1 2 e0 such that ?tnk(p0) ! p1 as n ! 1.
If p0 6= p1, then we have p0 ?2 p1. Again, by proposition 3.3.2, we have p0 ?2 p1. By
theorem 2.3 in [32], this can not happen. Therefore, p1 ? p0. Then we have ?tnk(?) ! p0
as n !1 which is a contradiction to the fact ? 2 W. Hence, Wu+(e0) = fe0g. This implies
Ws+loc(e0) = X ?X. Then, it is clear that we have TflWu+(e)'TflWs+loc(e0) = X ?X.
Case 5: Suppose 9 fl 2 Wu+(0;0) \ Ws+loc(e); e 6= (0;0). If Wu+(e) = feg, then the
transverality is clear. Suppose Wu+(e) 6= feg. Let !(e) be the principle eigenfunction of
the linearization problem of system (1.0.1) at ~e 6= (0;0). We have !(e) 2 (0;0). Since
TeWs+(e) and TflWs+(e) are close if fl 2 Ws+loc(e), we have TflWs+(e)'spanf!(fl)g = X?X
with !(fl) is close to !(e). Let ? 2 Wu+loc (0;0) such that ?t0(?) = fl for some t0 > 0. The
linearization of (1.0.1) at (0;0) is decoupled and we have two positive eigenfunctions !u and
47
!v. The linear space spanned by f!u; !vg is in the tangent space T(0;0)Wu+(0;0). Since
? is close to (0;0), TflWu+(0;0) and T(0;0)Wu+(0;0) are close. We can chose a strongly
positive vector !(?) in the linear space spanned by f!u; !vg so that ?t0(!(?)) = !(fl).
Since ? is close to (0;0), TflWu+(0;0) and T(0;0)Wu+(0;0) are close. Hence TflWu+(0;0) =
spanf!(fl)g. So, we have TflWs+(e)'TflWu+(0;0))g = X ?X.
48
Chapter 5
A-stability via Morse-Smale structure
5.1 Reduction to inertial manifold
We consider (1.0.1) for the one-dimensional case ? = (0;1).
8>
>>>>
<
>>>
>>:
ut = k1uxx +uf(x;u;v); x 2 (0;1);
vt = k2vxx +vg(x;u;v); x 2 (0;1);
Bu = Bv = 0:
(5.1.1)
Let
A1 : D(A1) ? L2(?) ! L2(?)
(u;v) 7! ?k1uxx +u;
A2 : D(A2) ? L2(?) ! L2(?)
(u;v) 7! ?k2vxx +v;
A = (A1;A2); D(A) = D(A1)?D(A1):
where D(A1) = D(A2) = f? 2 H2;2(?)j B? = 0g, B is either Neumann or Dirichlet
boundary condition. It is easy to see that the the fractional power spaces generated by A1=21
and A1=22 are same Hilbert space with difierent inner products
D
A1=21 ?;A1=21 ?
E
L2(?)?L2(?)
and
D
A1=22 ?;A1=22 ?
E
L2(?)?L2(?)
, respectively. For simplicity, we denote both by X. The inner
49
product in X?X is thenh?;?iX?X =
D
A1=21 ?;A1=21 ?
E
L2(?)?L2(?)
+
D
A1=22 ?;A1=22 ?
E
L2(?)?L2(?)
.
It is implicitly understood that k?kX = kA1=21 ?kL2(?) and k?kX = kA1=22 ?kL2(?) depending
on whether ? is the flrst or second component of (u;v). The normalized (in L2(?)?L2(?))
eigenfunctions of A + Id which correspond to the eigenvalues f?ui = 1 + k1(i?)2; ?vj =
1+k2(j?)2j i;j 2N[f0g g are wui (x) = (p2cos(i?x);0); wvi (x) = (0;p2cos(j?x)); i;j 2
N[f0g if B is the Neumann boundary condition and are wui (x) = (p2sin(i?x);0); wvi (x) =
(0;p2sin(j?x)); i;j 2N if B is Dirichlet boundary condition. We arrange the eigenvalues
as an increasing sequence ?1 < ?2 < ::: < ?N < ?N+1 < :::. Let WN be the linear space
spanned by the f?igN1 and W?N be the orthogonal complement of WN in L2(?) ? L2(?).
Let eFfg(u;v) = [uf(?;u;v)+u;vg(?;u;v)+v]. Clearly, eFfg : X ?X ! H1;2(?)?H1;2(?).
Moreover, if B is the Dirichlet boundary condition, then eFfg : X ?X ! X ?X because
B(F(u;v)) = 0. If B is the Neumann boundary condition, it is well-known that X =
H1;2(?). Note that D(??+Id) is dense in H1;2(?) under the norm induced by the inner
product h?;?iH1;2(?)?H1;2(?) = h???;?iL2(?)?L2(?) + h?;?iL2(?)?L2(?). Therefore, we can
consider eFfg : X ?X ! X ?X in both cases.
Lemma 5.1.1. Let (f;g) 2 CP. Assume (u;v) ? 0; (u;v) 2 D(A) and k(u;v)kX?X ? R.
If R is large enough, then there exists m0 > 0 (depending on R and (f;g)) such that
D
A(u;v)? eFfg(u;v);(u;v)
E
L2(?)?L2(?)
? 2m0:
50
Proof. We have
D
A(u;v)? eFfg(u;v);(u;v)
E
L2(?)?L2(?)
= h?k1uxx ?uf(?;u;v);uiL2(?) +h?k2vxx ?vg(?;u;v);viL2(?)
=
Z
?
[k1u2x ?u2f(?;u;v)]dx+
Z
?
[k2v2x ?v2g(?;u;v)]dx
=
k1
Z
?
u2x dx+k2
Z
?
v2x dx
?
?
Z
?
u2f(?;u;v) dx?
Z
?
v2g(?;u;v) dx:
On the other hand, we have
k(u;v)k2X?X =
D
A1=21 u;A1=21 u
E
L2(?)
+
D
A1=22 v;A1=22 v
E
L2(?)
= hA1u;uiL2(?) +hA2v;viL2(?)
= k1
Z
?
u2x dx+k2
Z
?
v2x dx+
Z
?
u2 dx+
Z
?
v2 dx
?
= k1
Z
?
u2x dx+k2
Z
?
v2x dx+k(u;v)k2L2(?)?L2(?): (5.1.2)
Therefore
D
A(u;v)? eFfg(u;v);(u;v)
E
L2(?)?L2(?)
?k(u;v)k2X?X ?k(u;v)k2L2(?)?L2(?) ?
Z
?
u2f(?;u;v) dx?
Z
?
v2g(?;u;v)dx: (5.1.3)
From (H3) and (H4) there exists a positive constant R0 such that
f(?;u;v) < ?1=2 for all v ? 0; x 2 (0;1); u ? R0;
g(?;u;v) < ?1=2 for all u ? 0; x 2 (0;1); v ? R0:
(5.1.4)
51
Let
M0 = maxf sup
f0?u?R0g\fv?0g
f(?;u;v); sup
f0?v?R0g\fu?0g
g(?;u;v)g:
M0 exists and flnite because of (H1) and (H2). Choose R su?ciently large so that
2m0 = 12R2 ?R20 ?2M0R20 > 0:
From (5.1.3) and (5.1.4), we have
D
A(u;v)? eFfg(u;v);(u;v)
E
L2(?)?L2(?)
? 12k(u;v)k2X?X ? 12k(u;v)k2L2(?)?L2(?) ?
Z
?
u2f(?;u;v) dx?
Z
?
v2g(?;u;v)dx
? 12R2 ? 12k(u;v)k2L2(?)?L2(?) ?
Z
fu?R0g
u2f(?;u;v) dx?
Z
fv?R0g
v2g(?;u;v) dx
?
Z
f0?u 0 such
that A(f0;g0) ? f(u;v) 2 X+ ? X+j k(u;v)kX?X < Rg. By theorem 4.1.1, A(f0;g0) is
52
upper semi-continuous in X ?X. Hence
A(f;g) ?f(u;v) 2 X+ ?X+j k(u;v)kX?X < Rg; (5.1.5)
provided dCP((f;g);(f0;g0)) is small enough. From now on, we flx R > 0 such that (5.1.5)
holds and lemma 5.1.1 is true for (f0;g0): We deflne a C1 function ? : [0;1]?[0;1] ! [0;1]
with the following properties
8
>>>>
>>>
>>>
>><
>>>
>>>>
>>>
>>:
? ? 1 on [0;R]?[0;R];
? ? 0 on f(s1;s2) 2R2j maxfs1;s2g? 2Rg;
sups1?0k?s1(s1;s2)k? 1; 8 s2 ? 0;
sups2?0k?s2(s1;s2)k? 1; 8 s1 ? 0:
Let Ffg(u;v) = ?(kukX;kvkX)eFfg(u;v). Since f;g are C2 functions, Ffg has the following
properties
1. F : X?X ! X?X is bounded in C1 norm. Note that the bound depends on (f;g),
2. F(u;v) = eF(u;v) for all k(u;v)kX?X ? R,
3. F(u;v) = 0 for all k(u;v)kX?X ? 2R.
Therefore, rather than studying system (5.1.1), we can study the modifled one
8>
>>>
><
>>>
>>:
ut = k1uxx +?(kukX;kvkX)uf(x;u;v); x 2 (0;1);
vt = k2vxx +?(kukX;kvkX)vg(x;u;v); x 2 (0;1);
Bu = Bv = 0:
(5.1.6)
53
For simplicity, we will sometimes write F instead of Ffg. Let ? = (u;v), we can rewrite
system (5.1.6) as
?t +A? = F(?); ? 2D(A): (5.1.7)
Lemma 5.1.2. Given (f0;g0) 2CP and "0 > 0. Deflne
? := f(f;g) 2CPj dCP((f;g);(f0;g0)) < "0g:
Then there exists positive constants M0; M1 depending on "0; f0; g0 such that
kFfg(u;v)kX?X ? M0; 8(u;v) 2 X ?X; 8(f;g) 2 ?; (5.1.8)
kDFfg(u0;v0)(?;?)kX?X ? M1k(?;?)kX?X; 8 (u0;v0); (?;?) 2 X ?X; 8(f;g) 2 ?;
(5.1.9)
and
Supp Ffg ?f(u;v) 2 X ?X : k(u;v)kX?X ? 2Rg; 8(f;g) 2 ?: (5.1.10)
Proof. It is clear from the deflnition of Ffg that
Supp Ffg ?f(u;v) 2 X ?X : k(u;v)kX?X ? 2Rg; 8(f;g) 2 ?:
For (5.1.8) and (5.1.9), the existence of the constant M0 and M1 is guaranteed by (5.1.2)
and the fact that
f(u;v) 2 X ?Xj k(u;v)kX?X ? 2Rg?f(u;v) 2 X ?Xj k(u;v)kC(??)?C(??) ? 2RCg;
54
where C is the embedding constant X ,! C(??). Therefore , we can choose a su?ciently
small neighborhood ? of (f0;g0) such that
k(f;g)?(f0;g0)kC2(???[?2RC;2RC]?[?2RC;2RC]) < 1; 8(f;g) 2 ?:
Lemma 5.1.3. (Gap condition) Let K0 = 16M21; K1 = 4M1 where M1 is the positive
constant in proposition 5.1.2. Then there exists N 2N such that
?N > K0 and ?N+1 ??N > 2K1:
Proof. Since the eigenvalues of A can be rearranged as a strictly increasing sequence con-
verging to 1, we always have ?N > K0. Because ?N+1 ? ?N = O(N), it is clear that
?N+1 ??N > 2K1 if N is su?ciently large.
Proposition 5.1.4. Given (f0;g0) 2CP. Let ? be deflned as in proposition 5.1.2. Choose a
natural number N such that the gap condition in lemma 5.1.3 holds. Let W := WN. Then,
for each (f;g) 2 ?, there exists a C1 map 'fg : W ! W? \(X ?X); 'fg = ('ufg;'vfg)
with the following properties
(i) Supp 'fg ?f(u;v) 2 X ?X : k(u;v)kX?X ? 2Rg; 8(f;g) 2 ?,
(ii) k'fg(w)kX?X ? L0, 8w 2 W, 8(f;g) 2 ? where L0 = M0e?1=22?N+1 (M0 is the positive
constant in proposition 5.1.2),
(iii) The manifold Ifg :=Graph 'fg is invariant under the ow generated by (5.1.7) and
attracts all solutions of (5.1.7) exponentially,
55
(iv) kD'fg(w)kL(W;X?X) ? L1; 8w 2 W, 8(f;g) 2 ?, where L1 ? 2M1?N+1??N (M1 is the
positive constant proposition 5.1.2),
(v) 'fg; D'fg are continuous in (f;g).
Proof. Apply theorem 2.1 in [27] with fi = fl = 1=2; k = 1; C0 = M0; C1 = M1. For
simplicity, we will write 'fg as ' when no confusion should arises.
From now on, let W be the linear space deflned in proposition 5.1.4 with N chosen
large enough such that
L20 ?kFf0g0k0L0 < m0; (R+kFk0)L0L1 < m0=2 (5.1.11)
where kFf0g0k0 = sup(u;v)2X?X kF(u;v)kX?X. Let P be the orthogonal projection of
L2(?) ? L2(?) to W and Q = Id ? P. By applying P and Q to (5.1.7), we obtain the
system
wt +Aw = (P ?F)(w +w?); (5.1.12)
w?t +Aw? = (Q?F)(w +w?); w 2 W: (5.1.13)
By proposition 5.1.4, we can write (5.1.12) as
wt +Aw = (P ?F)(w +'(w)); w 2 W: (5.1.14)
Lemma 5.1.5. Given (f0;g0) 2CP. Let
?R = fw 2 Wj kw +'f0g0(w)kX?X = R; w +'(w) ? 0g:
56
Then
hAw?(P ?Ff0g0)(w +'f0g0(w));?(w)iL2(?)?L2(?) ? m0 > 0; 8w 2 ?R; (5.1.15)
where ?(w) is the outer normal vector at w and m0 (depending on f0;g0) is the positive
constant deflned in proposition 5.1.1.
Proof. For simplicity, we will write '; F instead of 'f0g0; Ff0g0. By lemma 5.1.1, we have
hA(w +'(w))?F(w +'(w));w +'(w)iL2(?)?L2(?) ? 2m0; 8w 2 ?R: (5.1.16)
On the other hand, we have
hA(w +'(w))?F(w +'(w));w +'(w)iL2(?)?L2(?)
= hAw +A'(w)?(P ?F)(w +'(w))?(Q?F)(w +'(w));w +'(w)iL2(?)?L2(?)
= hAw?(P ?F)(w +'(w));wiL2(?)?L2(?)
+hA'(w)?(Q?F)(w +'(w));'(w)iL2(?)?L2(?) ; 8w 2 W:
Therefore,
hAw?(P ?F)(w +'(w));wiL2(?)?L2(?) (5.1.17)
? 2m0 ?hA'(w)?(Q?F)(w +'(w));'(w)iL2(?)?L2(?)
= 2m0 ?hA'(w);'(w)iL2(?)?L2(?) +h(Q?F)(w +'(w));'(w)iL2(?)?L2(?)
? 2m0 ?k'(w)k2X?X ?k(Q?F)(w +'(w))kL2(?)?L2(?)k'(w)kL2(?)?L2(?)
? 2m0 ?k'(w)k2X?X ?kF(w +'(w))kL2(?)?L2(?)k'(w)kL2(?)?L2(?); 8w 2 ?R:
57
Using property (ii) in proposition 5.1.4 and (5.1.11), we have
hAw?(P ?F)(w +'(w));wiL2(?)?L2(?)
? 2m0 ?k'(w)k2X?X ?kFk0k'(w)kX?X
? 2m0 ?L20 ?kFk0L0 ? m0; 8w 2 ?R: (5.1.18)
Deflne
H : W ! R
p 7! hD'(w)(p);'(w)iX?X :
It is clear that H is a continous linear functional on (W;k?kX?X). By the Riesz represen-
tation (cf. [37]), there exists w? 2 W such that hw?;wi = H(w); 8w 2 W and
kw?kX?X = kHkL(W;R) ?kD'(w)kL(W;X?X)k'(w)kX?X ? L0L1 < 1; (5.1.19)
where L0; L1 are constants in proposition (5.1.4). The outer normal vector ?(w) at w 2 ?R
then has the representation ?(w) = 2w +2w?: Therefore,
hAw?(P ?F)(w +'(w)); ?(w)iL2(?)?L2(?)
= hAw?(P ?F)(w +'(w));2w +2w?iL2(?)?L2(?)
= hAw?(P ?F)(w +'(w));2wiL2(?)?L2(?) +
hAw?(P ?F)(w +'(w));2w?iL2(?)?L2(?) ; 8w 2 ?R: (5.1.20)
58
From (5.1.18) and (5.1.20), we have
hAw?(P ?F)(w +'(w)); ?(w)iL2(?)?L2(?)
? 2m0 ?2kAwkL2(?)?L2(?)kw?kL2(?)?L2(?)
?2k(P ?F)(w +'(w))kL2(?)?L2(?)kw?kL2(?)?L2(?)
? 2m0 ?2kAwkL2(?)?L2(?)kw?kL2(?)?L2(?) (5.1.21)
?2kFk0kw?kL2(?)?L2(?); 8w 2 ?R:
We have kAwkL2(?)?L2(?) =
qP
N
1 c2i?2i for w 2 WN, w =
PN
1 ciwi. If w 2 ?R, we have
kwkX?X ?kw +'(w)kX?X = R. But kwkX?X =
qP
N
1 c2i?i. Therefore,
kAwkL2(?)?L2(?) ?
q
?N?N1 c2i?i ?
p
?N R: (5.1.22)
From (5.1.11), (5.1.19), (5.1.21) and (5.1.22), we have
hAw?(P ?F)(w +'(w)); ?(w)iL2(?)?L2(?) ? 2m0 ?2(R+kFk0)L0L1 ? m0:
Lemma 5.1.6. Given (f0;g0) 2CP and "0 > 0. If dCP((f;g);(f0;g0)); (f;g) 2CP, is small
enough so that kFfg ?Ff0g0kX?X < m04(R+C) where C is the constant from the embedding
X ,! C(??). Then
hAw?(P ?Ffg)(w +'f0g0(w));?(w)iL2(?)?L2(?) ? m0=2 > 0; 8w 2 ?R; (5.1.23)
59
where ?(w) is the outer normal vector at w.
Proof. We have
hAw?(P ?Ffg)(w +'f0g0(w));?(w)iL2(?)?L2(?) (5.1.24)
= hAw?(P ?Ff0g0)(w +'f0g0(w));?(w)iL2(?)?L2(?)
+h(P ?Ffg)(w +'f0g0(w))?(P ?Ff0g0)(w +'f0g0(w));?(w)iL2(?)?L2(?) ; 8w 2 W:
From (5.1.15) and (5.1.24), we have
hAw?(P ?Ffg)(w +'f0g0(w));?(w)iL2(?)?L2(?)
? m0 +h(P ?Ffg)(w +'f0g0(w))?(P ?Ff0g0)(w +'f0g0(w));?(w)iL2(?)?L2(?)
? m0 ?k(Ffg ?Ff0g0)(w +'f0g0(w))kL2(?)?L2(?)k?(w)kL2(?)?L2(?)
? m0 ?kFfg ?Ff0g0k0(2kwkL2(?)?L2(?) +2kw?kL2(?)?L2(?))
? m0 ?kFfg ?Ff0g0k0(2R+2C) ? m0 ? m02 = m0=2; 8w 2 ?R:
Let WR = fw 2 Wj kw + 'f0g0(w)kX?X ? R; w + 'f0g0(w) ? 0g. The boundary of
WR (in W) includes ?R; ?uR and ?vR where
?uR := fw 2 Wj u+'uf0g0(u;v) ? 0; v +'vf0g0(u;v) = 0g;
?vR := fw 2 Wj u+'uf0g0(u;v) = 0; v +'vf0g0(u;v) ? 0g:
60
Proposition 5.1.7. Given (f0;g0) 2CP. Let ?(f0;g0) be a su?ciently small neighborhood
of (f0;g0) in CP. The semi- ows f?fgt jW; t ? 0g; (f;g) 2 ?(f0;g0) are invariant on WR.
Proof. Clearly, the semi- ows f?fgt jW; t ? 0g; (f;g) 2 ?(f0;g0) are invariant on ?uR; ?vR.
On ?R, by lemma 5.1.6 guarantees inward ows.
Proposition 5.1.8. If (f;g) 2MS then (5.1.14) has Morse Smale property (in WR).
Proof. For simplicity, we write I and ?t for Ifg and ?fgt . Let e0 2 X+ ?X+ be a critical
element of (5.1.7). By applying P to (5.1.7), we have P(e0) is a critical element of (5.1.14)
in WR. Now, let w 2 WR be a critical element of (5.1.14). Because I is invariant under
the semi- ow f?t; t ? 0g, we have ?t(w(0)+'(w(0))) 2I; 8t ? 0. Therefore, there exists
~w(t); t ? 0 such that ?t(w(0)+'(w(0))) = ~w(t)+'(~w(t));8t ? 0. Since ?t(w(0)+'(w(0))
is the solution of (5.1.7) with initial w(0) + '(w(0)), P(?t(w(0) + '(w(0)))) = P(~w(t) +
'(~w(t))) = ~w(t) is a solution of (5.1.14) with initial w(0). Because of unique solvability
of (5.1.14), we have ~w = w. Hence, w + '(w) 2 X+ ?X+ is a critical element of (5.1.7).
Therefore, if (f;g) 2 MS, then (5.1.14) has flnitely many critical elements in WR. Next,
suppose w0 2 WR is a critical element of (5.1.14), we have w0 + '(w0) 2 X+ ? X+ is a
critical element of (5.1.7). Let IR+fg = Ifg\f(u;v) 2 X+?X+j k(u;v)kX?X ? Rg and Pfg
be the restriction of the orthogonal projection P to IR+fg . Clearly, Pfg is a homeomorphism
from IR+fg to WR and t = Pfg ? ?t ?P?1fg 8t ? 0 on WR, where f t; t ? 0g is the semi-
ows generated by (5.1.14). Therefore, (D t)(w) = (Pfg ? D?t ? P?1fg )(w) which implies
that ? is an eigenvalue of (D t)(w) with eigenfunction ? if and only if it is an eigenvalue
of D?t(w + '(w)) with eigenfunction ? + '(?). Since (f;g) 2 MS, w0 + '(w0) is a
hyperbolic critical element of (5.1.7). Hence, w0 is a hyperbolic critical element of (5.1.14).
Finally, we prove the transversal intersection of stable and unstable manifolds of critical
61
elements (in WR) of (5.1.14). Given two critical elements w1; w2 of (5.1.14). Suppose
~Wu(w1)\ ~Wsloc(w2) 6= ; where ~Wu(w1); ~Wsloc(w2) are unstable and local stable manifolds
of w1 and w2. Let ei = wi + '(wi); i = 1;2: Clearly, ~Wu(w1) = fw 2 Wj w + '(w) 2
Wu+(e1)g. Let fl 2 ~Wu(w1) \ ~Wsloc(w2). We now prove Tfl ~Wu(w1) ' Tfl ~Wsloc(w2) = W.
Since ? = fl + '(fl) 2 Wu+(e1) \ Ws+(e2), we have T?Wu+(e1) ' T?Ws+(e2) = X ? X
(because (f;g) 2MS). By the invariant foliation theory in [5], there exists a unique C1 leaf
J with codimension N which passes through ? and is transversal to the inertial manifold
I. Let Ws? := fw +'(w)j w 2 ~Ws(w2)g. Then we have that T?Ws+(e2) = T?Ws? 'T?J.
Hence Wu+(e1) is transversal to Ws? and the dimension of T?Wu+(e1)'T?Ws? is N. Since
k'kL(W;W?\X?X) < 1, the map w 7! w + '(w) is a difieomorphism from W to I and
it maps ~Wu(w1) to Wu+(e1), ~Ws(w2) to Ws?. Hence, ~Ws(w1) is transversal to ~Ws(w2)
because T?Wu+(e1)'T?Ws?.
5.2 Tubular Family Theorem
Lemma 5.2.1. Given an ordered chain fi1 ?3 fi2 ?3 ::: ?3 fin. Let Gu(fin) be a funda-
mental domain of Wu(fin) and Ws(fii), 1 ? i ? n, be the stable manifold of fii. Then we
have
@Ws(fi1)\Gu(fin) ?
[
2?i?n?1
Ws(fii)\Gu(fin):
Proof. Let x 2 @Ws(fi1) \ Gu(fin). There exists a sequence fxjg ? Ws(fi1), xj ! x
as j ! 1. We also have k?fgt (xj) ? ?fgt (x)k ? k?fgt kkxj ? xk for any t > 0. Notice
that ?fgt (xj) 2 Ws(fi1) for all j and for all t > 0. Thus, for any given " > 0, ?fgt (x) 2
B"(Ws(fi1)) for all t > 0. Suppose ?fgt (x) ! fl, then fl must be one of the fii; 2 ? i ? n?1.
So x 2S2?i?n?1 Ws(fii)\Gu(fin).
62
Theorem 5.2.2. ( Tubular family Theorem) Let (f;g) 2 MS. There exists a compatible,
invariant system of tubular families f?ig, each ?i being a tubular family of Ws(fii), where
fii is a critical element of fgt , where fgt is the ow introduced in the proof of proposition
(5.1.8).
Proof. Deflne L0 = ffi j fi is a sinkg, L1 = ffi j beh(fljfi) = 1; fl 2 L1g,
Lk = ffi j beh(fijfl) = k; fl 2 L1g. For each fi 2 L1, deflne ?fi = Ws(fi). In this proof,
?? denotes a tubular family of ?
Assume that ?fi has been constructed for all fi 2 S0?i?k?1 Li. We will construct
?fi; fi 2 Lk. Let fi 2 Lk be a periodic orbit, p 2 fi, Gu(p) be the fundamental do-
main of Wuloc(p) and fl 2 Lk?1, Wu(fi)TWsloc(fl) 6= ;. From lemma 5.2.1, we have
Gu(p)T@Ws(fl) = ;. Here, the boundary is relative with respect to Ws(fl). Now we
apply lemma 2.5 in [27] with C = Gu(p), W = Wu(p), U0 = ;, U = Ufl = ?fl \ Nu(p)
(Nu(p) is a properly chosen fundamental neighborhood of Wuloc(p)), we get a continuous
retraction rfl : Ufl ! Wu(p) which satisfles (i);(ii);(iii) of lemma 2.5 in [27]. Since ?fl are
mutually separated, we can deflne rk?1 = Sfl2L
k?1; Wu(fi)\Wsloc(fl)6=;
rfl. That is,
rk?1 : (
[
fl2Lk?1; Wu(fi)\Wsloc(fl)6=;
?fl)\Nu(p) ! Wu(p)
Next , we will extend rk?1 to S?2L
k?2; Wu(fi)\Wsloc(?)6=;
??. Let
? 2 Lk?2; Wu(fi)
\
Wsloc(?) 6= ;
63
From lemma 2.1.5, we have
@Ws(?)\Gu(p) ?
[
fl2Lk?1;Wu(fi)\Wsloc(fl)6=;
Ws(fl)\Gu(p) ?
[
fl2Lk?1;Wu(fi)\Wsloc(fl)6=;
?fl \Nu(p)
Applying lemma 2.5 in [27] with C = Gu(p), W = Wu(p), r0 = rk?1, U = U? = ?fl\Nu(p),
U0 = Sfl2L
k?1; Wu(fi)\Wsloc(fl)6=;
?fl \Nu(p), we get a continuous retraction r? : U? ! Wu(p)
which satisfles (i);(ii);(iii) of lemma 2.5 in [27]. Deflne rk?2 = S?2L
k?1; Wu(fi)\Wsloc(?)6=;
r?,
that is,
rk?2 : (
[
?2Lk?2; Wu(fi)\Wsloc(?)6=;
?fl)\Nu(p) ! Wu(p):
Continuing this process through Lk?3; Lk?4;:::; L1, we have a continuous retraction r :
Nu(p) ! Wu(p). Now, let N be an open neighborhood of p 2 fi in Wu(p) with @N = Gu(p).
For each y 2 N there is a unique xy and a unique txy 2 Gu(p) such that y = fg?txy(xy). We
deflne Tfiy = fg?txy(r?1(xy)) and Tfip = Ws(p). Then fTfiy gy2N is a invariant tubular family
of Ws(p) under k = ?t(?;:)jN (? is the period of fi). The tubular family of Ws(fi) is now
deflned as in deflnition (2.1.5).
5.3 A-stability
We shall prove Theorem C in this section.
Theorem 5.3.1. Let (f0;g0) 2 CP. The semi- ows f f0g0t jW
R
; t ? 0g is A-stable (the
notion of t was introduced in the proof of proposition 5.1.8).
Proof. Let p2 be a critical element of f f0g0t jW
R
; t ? 0g with behavior ? 1 with respect
to sources. Consider source p1 such that beh(p1jp2) = 1. Put ?pi = ?(pi) where ? =
?(f;g); (f;g) 2CP, is the homeomorphism in Proposition (4.1.5). Since ~Wu(p1) is C1 close
64
to ~Wu(?p1) on compact sets, ~Wu(p1) \ ~Gs(p1) is compact, there exists a difieomorphism
h2 : ~Wu(p1) \ ~Gs(p2) ! ~Wu(?p2) \ ~Wsloc(?p2), h2 is close to the identity map. For any
y 2 ~Wu(p1)\ ~Wsloc(p2), there exists a unique positive ( or negative) time ? = ?(y) such that
fg? (y) 2 ~Wu(p1)\ ~Gs(p2). Therefore we can extend h2 to ~Wu(p1)\ ~Ws(p2) by using the
ows
h2 = fg?? ?h2 ? f0g0? : ~Wu(p1)\ ~Ws(p2) ! ~Wu(?p1)\ ~Ws(?p2):
We do the same for all other sources p1 of which beh(p1jp2) = 1. Then we can extend h2
to A0WR \ ~Ws(p2) where A0WR is the attractor of f f0g0t jW
R
; t ? 0g. Note that A0WR =
P(A(f0;g0)). Repeat the same procedure for other p2 with behavior ? 1 with respect to
the sources to extend h2 to Sbeh(pjsources)=1(A0WR \ ~Ws(p)).
The next step is to consider p3 with behavior ? 2. For the sources of behavior 1
with respect to p3, the procedure is similar to the one we just use above. Let p1 be
a source with beh(p1jp3) = 2. There exists at least a sequence p3 ?3 p2 ?3 p1 such
that beh(p1jp2) =beh(p2jp3) = 1. Since beh(p2jp3) = 1, we can deflne a difieomorphism
?h3 on ~Wu(p2) \ ~Gs(p3) similarly to the way we deflne h2. The existence of compatible
system of unstable foliations guarantees that ~Wu(p1) intersects ~Wsloc(p3). For each leaf of
~Wu(p1) \ ~Gs(p3) which is near ~Wu(p2) \ ~Gs(p3), there corresponds a unique point y 2
~Wu(p1)\ ~Wsloc(p2) near p2 such that t
0(y) belongs to that leaf for some t0 > 0. We index
these leaves as Jy; y 2 ~Wu(p1)\ ~Wsloc(p2). Since Jy is near ~Wu(p2)\ ~Gs(p3), there exists a
difieomorphism iy : Jy ! ~Wu(p2)\ ~Wsloc(p3). The same happens for the perturbed system,
so we also have a difieomorphism ih2(y) : Jh2(y) ! ~Wu(?p2)\ ~Wsloc(?p3). Both iy and ih2(y) are
close to identity map. The composition map ?h3;y = i?1h2(y) ??h3 ?iy is a difieomorphism from
Jy to Jh2(y). Using ?h3;y; y 2 ~Wu(p1)\ ~Wsloc(p2), we can extend ?h to a small neighborhood
65
U of ~Wu(p2)\ ~Gs(p3) in
? ~
Wu(p1)[ ~Wu(p2)
?
\ ~Gs(p3) as follows
?h3(x) = ~h3;y(x); x 2Jy:
To extend ?h3 to
? ~
Wu(p1)[ ~Wu(p2)
?
\ ~Gs(p3), we apply the Isotopy Extension Theorem
(cf. [13]). Let N be the relative boundary in
? ~
Wu(p1)[ ~Wu(p2)
?
\ ~Gs(p3) of a small
neighborhood U1 ? U in the domain of ?h3, M = f0g0(??;?)(
? ~
Wu(p1)[ ~Wu(p2)
?
\ ~Gs(p3)nU2),
where U2 ? U1. Since ~Wu(p1) \ ~Gs(p3) and ~Wu(?p2) \ ~Wsloc(?p3) are C1 close, there exists
a difieomorsphism d which maps ~Wu(p1) \ ~Gs(p3) to ~Wu(?p2) \ ~Wsloc(?p3). Let j = d?1 ?
?h : N ! M. Applying the Isotopy Extension Theorem for j, we obtain an extension
? : M ! M of j such that ?jN = j, ?(x) = x for all x outside a neighborhood V
of j(N). Let ??h3 = d ? ?. Deflne h3(x) = ?h3(x) if x 2 U1 and h3(x) = ??h3(x) if x 2
h? ~
Wu(p1)[ ~Wu(p1)
?
\ ~Gs(p3)
i
nU1. Then h3 is an extension of ?h3 to
? ~
Wu(p1)[ ~Wu(p1)
?
\
~Gs(p3), h3 : ? ~Wu(p1)[ ~Wu(p1)?\ ~Gs(p3) ! ? ~Wu(p1)[ ~Wu(p1)?\ ~Wsloc(?p3). Using again
the ows to extend h3 to
? ~
Wu(p1)[ ~Wu(p2)
?
\ ~Ws(p3). For other possible sequence(s)
p3 ?3 p2 ?3 ?p1, we repeat the same procedure to extend h3 to A0WR \ ~Ws(p3). The last step
shows the full induction procedure. Because there are only flnitely many critical elements,
the induction is completed when we reach the sinks. Since [p?crit: elementA0WR \ ~Ws(p) =
A0WR, and (A0WR \ ~Ws(pi))\(A0WR \ ~Ws(pj)) = ;; i 6= j, we can deflne
h : A0WR(f;g) ?!A0WR(f0;g0)
by h = h1 [ h2 [ h3 [ :::. The flnal step is to check the continuity of h. For any x 2
A0WR(f;g), there exists pi such that x 2 A0WR \ ~Ws(pi). Notice that it is su?cient to
66
prove the continuity of h at those x 2 A0WR \ ~Wsloc(pi). If pi is a source or a sink, the
continuity is trivial because A0WR \ ~Wsloc(pi) is either pi or A0WR. Assume pi is a saddle.
Let xn ! x; xn 2Juxn(pi). We have h(xn) 2Juhi(xn)(p0i). By a property of tubular families,
Juhi(xn)(p0i) converges to Juhi(x)(p0i). Therefore the set of accumulation points of fh(xn)g
is contained in ~Wsloc(p0i) \ Juhi(x)(p0i) = fhi(x)g. Since the set of accumulation points of
fh(xn)g has only one single element hi(x), h(xn) must converge to hi(x) = h(x). Thus h is
continuous. Hence h is a homeomorphism.
Theorem 5.3.2. If (f0;g0) 2MS, then f?f0g0t ; t ? 0g is A-stable.
Proof. Let h : A0WR !AWR be the homeomorphism in theorem 5.3.1. Deflne H = P?1fg ?h?
Pf0g0. Then H : A(f0;g0) ! A(f;g) is a homeomorphism taking trajectories of A(f0;g0)
to trajectories of A(f;g) and preserves the sense of direction in time.
67
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