1.3

1.4

3.2

3.3

3.4

4.1,4.3

4.2

NOTES

The term "sectorial operator" generalizes the usage of Kato [56].

H. B. Stewart (Trans. Am. Math. Soc. 199(1974), 141-162) studies

analytic semigroups in the uniform norm.

Fractional powers of operators are studied extensively in [63] and

[103]; the operators need not be sectorial. The connection with

interpolation of Hilbert spaces is developed in [71] - see in

particular the results of Grisvard described in [71, p. 107].

More general results are available [57, 63, 93].

More general results are available [93].

Analyticity has been proved for some more general linear equations;

see in particular [57].

Many of these notions apply to more general dynamical systems; see,for example, [38]. Sell [89] provides an introduction to recent

work for nonautonomous ODE's.

The argument follows Yoshizawa [135].

5.1

5.2

Stability of equilibrium for the Navier-Stokes equation was provedby Prodi [80]. Sattinger later proved stability and instability

results for weak (Hopf) solutions of the Navier-Stokes equations,

and some stability and instability results (of strong solutions)

are proved by Kirchgassner and Kie1hofer [59]. The instability

theorems 5.1.5, 5.1.6 were inspired by Th. 2.3 (Ch. 7) of Da1eckii

and Krein [114]; I am grateful to E. Poulson for bringing this

result to my attention.

The stable manifold was also studied by Crandall and Rabinowitz,

and these manifolds have been studied in more general context by

Hirsch, Pugh and Shub.

5.3 The work of Chafee and Infante [14] was done in 1971; our argu-

ments differ in detail. Some of the results on gradiant flows

were reported in Nonlinear Diffusion, Pitman (1976).

6.1 The proof of existence is modeled on [37, Ch. 7]. Another ap-proach to stability of the manifold is presented by Lykova [72],

but her differential equations involve only bounded operators.Smoothness of the invariant manifold may be proved along the lines

of [58], [66], [104], under plausible assumptions. Note the

example in [66], proving the manifold cannot be analytic, in

332

general, without some global bounds.

6.2 An example of van Strein (Math. :. 166(1979), 143-145) shows the

critical or center manifold need not be Coo:

·23x = -x + y, Y = -yz - y, z = O.

A center manifold which is C2N has the form

x = o(y,z):\

Lm=l

m1)' 2m/ IT (1 )m-.y +

j=l

is analytic

but not analytic,

If z < 0, Y -T o(y,z)

this function is COO

as y -T 0, for Z < 1/2N.

near y = O. When z = 0,

and when z '> 1/2N, it is not differentiable. My thanks

to Jack Carr for pointing out this example.

6.3 Results of this kind date back to Poincare, in the case of ordin-

ary differential equations. The fundamental reason for the rela-

tion between stability and the geometry of the bifurcation seems

to be the homotopy invariance of topological degree: this was

discovered by Gavalas [33] and Sattinger [86], although the paper

of Sattinger involves a (correctable) error, confusing two notions

of the simplicity of an eigenvalue. Some results of section 8.5

may be proved in this manner (assuming the Poincare map is com-

pact, i.e. A has compact resolvent.) Our work is modeled, in

part, on [41].

6.4 E. Hopf proved this result for analytic ODE's, and his work was

generalized by many people in particular Chafee [12] who also

studied this problem for retarded functional differential equa-

tion. Hale [41] extended the treatment to neutral FDE's, and our

work is roughly modeled on [41]. Marsden and McCracken present

many approaches in The Hopf Bifurcation and its Applications

(Springer, Appl. Math. Sciences 19, 1976).

7.2 The argument is modeled on the work of Stokes [98] for retarded

functional differential equations. These must also be some con-

nections with the work of Krein [63] on evolution of subspaces.

7.3 More general backward uniqueness results are available; see [29,

30, 64].

7.5 Thm. 3.4.8 can be used to prove "averaging" results, but the

direct proof of Thm. 7.5.2 gives sharper results.

7.6

8.2

8.4,8.5

333

Coppel's recent book [113] is a good source for dichotomies in

ODE theory. Many of the results were extended to infinite dimen­

sions (especially Hilbert space) for bounded operators by Daleckii

and Krein [115]. C. V. Coffman and J. J. Schaffer (Math. Ann.

172(1967), 139­166) discuss dichotomies for difference equations

(in a Banach space) on a half­line, emphasizing the connections

between dichotomies and admissibility, a connection studied by

Massera and Schaffer [128] for ODE's. A proof of the continuous­

time version of Thm. 7.6.5 might allow simplification of this

section. Theorems 7.6.12 and 7.6.14 seem to be new even for

ODE's.

Orbital stability was proved by 100ss [52]. Orbital instability

is apparently rarely proved even for ordinary differential equa­

tions.

Use of the Poincare map for ODE's is described in [68].

Krasnoselski [61] studies some PDE problems by applying topologi­

cal degree theory to the Poincare map.

8.5 Extensive recent work on "generic" properties of maps should

have applications in this field: see [9,66,76,85]. Appli­

cability of this work to parabolic equations, in particular the

Navier­Stokes equations, has been widely suspected, but was ap­

parently not proved before [66, 85].

9.1 Thm. 9.1.1 was inspried by Coppel and Palmer [114], but is more

general even for ODE's, the generality arising from the use of

Thm. 7.6.12.

9.2 The coordinate system generalizes the construction of Urabe and

Hale [37]. Changing variables in an equation with unbounded opera­

tors is complicated, but the complications here seem excessive.

There must be an easier way. We could avoid the assumption of

"stable triviality" by developing the theory of parabolic equa­

tions on infinite dimensional manifolds rather than Banach spaces.

The cost seems high for this application, but it would also

allow a natural treatment of nonlinear boundary conditions.

10.1 The problem is formulated following Fleming [118].

334

10.2 Our "small parameter" E is not quite that of Sattinger [132],

so comparison of the results needs more care than is displayed

in the text. Parter, Stein and Stein (Studies in Appl. Math.

54(1975), 293-314) examined the equilibrium problem in more de-

tail. Their results do not prove there are no more than three

equilibria, but the bounds are such that this seems very likely.

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INDEX

Asymptotically autonomous, 96

Analytic semigroup,

definition of, 20

generator of, 20

Analyticity, 11

Averaging, 66,68,69,221,292-294

Bifurcation,

of equilibria, 178, 263

from a focus, 181-187

to a circle in diffeomorphisms, 266-274

to a torus, 266-274

transfer of stability at, 178-263

Burger's equation, 134-135

Center manifold,

approximation of, 171

existence and stability, 168

Center-stable manifold, 116

Center-unstable manifold, 262

Channel flow, swelling of, 117

Characteristic multiplier, 197

Chemical reactions, 43,82,167

344

Index - continued

Combustion theory, 101, 158, 319-330.

Contraction principle 12,13

Dichotomies, exponential,

definition of, 224

discrete, 229

implications of, 227

perturbation of, 237,238,240,242,245

Dynamical system, 82

Embedding theorems 9, 35-40

Equilibrium point, 83

Essential spectrum,

characterization of, 140

definition of, 136

examples of, 140-142

Fractional powers,

definition of, 24

examples of, 32-40.

Frechet derivative, 10

Gradient flow,

an example of, 118-128

domains of attraction in, 123-125

maximal invariant set of, 126-127.

Heat equation, 16,41

Hodgekin-Huxley equation, 42

Implicit function theorem, 15

345

Index - continued

Instability,

criterion for, 102,104,105,107,109

Integral inequalities, 188-190

Invariance principle, 92, 93

Invariant manifold,

coordinate system near some, 304

definition of, 143, 154

existence of, 143,161,164,165,275,297

extensions of local, 154,156

further properties of, 289-291

smoothness of, 152,278

stability of, 147,150,277,297

Invariant set, 91

Liapunov function, 84

Loo-bounds, 74

Maximum principle, 60,61,109

equation, 45, 79

Nirenberg-Gagliardo inequalities, 37

Nonautonomous linear equations, 190

adjoint of, 204

backward uniqueness, 208,209

density of range of solution operator of, 207

estimates on solution of, 191

nonhomogeneous, 193

periodic, 197

slowly varying coefficients of, 213

Index - continued

Nuclear reactors, 44,62,95

Omega limit set, 91

Operator,

fractional powers of, 24,25

products of, 28

sectorial, 18

spectral set of, 30

sums 0 f, 19,27

Periodic linear systems, 197

Floquet theory of, 198

Fredholm alternative for, 206

stability of, 200

Periodic orbit, 83

coordinate system near, 299

families of, 309

instability of, 254

perturbation of, 256, 257

stability of, 251

Periodic solutions,

instability of, 249

perturbation of, 255

stability of, 247

Poincare map, 258-261

Population genetics, 43, 314-319

Predator-prey systems, 201, 312

346

347

Index - continued

Saddle-point property, 112

Sectorial operator

definition of, 18

fractional powers of, 24,25

properties of, 19,23

Semiconductors, 42

Solution of differential equation,

classical, 75-76

compactness of, 57

continuation of, 55

continuous dependence of, 62

definition of, 53

differentiability of, 64,71,72

existence of, 54,59

uniqueness of, 54,59

Spaces Xa,

definition of, 29

properties of, 29

Spectral set, 30

Stability,

converse theorem of, 86,90

criteria for, 84,98,100

definitions of, 83

invariance principle for, 92,93

of families, 108

orbital, 84

under constant disturbances, 88,90

Index - continued

Traveling waves,

existence, 129

stability of, 130 -133

348