Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
Подождите немного. Документ загружается.
Attractor
relative
to
u,
236
Autocatalysis,
362
Autonomous,
170
Auxiliary
functions,
206
Anosov,
260
Axiom
A,
137,
261
Axiom
B,
137
Banach
manifold,
124
Banach
space,
123
Base
space,
83,
85
Basic
sets,
261
Basin
of
attraction,
202
Bifurcation
point,
268
Boundary,
42
Bracket
of
X
and
Y,
51
Brouwer
Fixed
Point
Theorem,
59
Bundle
map,
105
Bundle
of
bases,
87
Bundle
space,
85
Canonical
transformation,
97
catastrophe,
270
Center
manifold
theorem,
276
Chain,
42
Chaos,
280
ck
curve,
49
ck
perturbation,
243
Classifying
bundle,
106
Classifying
map,
107
Closed,
94
Closed
orbit,
157
Closed
set
of
X,
33
Closure
of
A,
33
Cobordant,
73
Cobound,
63
Codimension
one
bifurcations,
272
444
Collar
on
M,
108
Compact
set,
36
Compact
space,
36
Competitive
interacting
populations,
359
Complete
stability,
220
Completely
unstable,
175
Connected,
38
Conservative,
7
Continuous,
34
Continuous
at
p,
34
Converge
pointwise,
33
Converge
to,
30
Converges
locally
and
uniformly,
130
Cooperativity,
368
Coset
space,
74
Cosets,
26
Cotangent
space,
93
Covering
of
X,
36
Covering
space,
44
Critical
point,
57,
60
Critical
value,
57,
60
Cross-section,
86,
150
Cross-section
to
the
flow,
193
Damped
anharmonic
oscillator,
327
De Rham
cohomology
group,
94
Diffeomorphic,
48
Diffeomorphism,
46,
48,
49,
125
Differentiable,
46,
48,
124
Differentiable
conjugacy,
153
Differentiable
manifold,
47
Differentiable
structure,
46
Dispersive,
175
Dissipative,
8
Dynamical
systems,
146
Effective,
74
445
Elementary
transversality
theorem,
122
Elliptic
differential
operator
of
order
k,
141
Equi-asymptotically
stable,
203
Equi-attractive,
201
Equi-attractor,
233
Euler
characteristic
of
X,
67
Exact,
26,
94
Exotic,
261
Expansion,
184
Exponential
map,
79,
80
Exterior
derivative,
94
Exterior
product,
93
Faithful,
79
Feedback
control,
379
Fiber,
83
Fiber
bundle
associated
to
(B,M,G)
with
fiber
F,
88
Fiber
bundle
over
X
with
fiber
F,
85
Fiber
derivative,
101
Filippov
solution,
382
First
axiom
of
countability,
33
Flow,
55,
146
Flow
equivalence,
154,
156
Flow
map,
153
Fredholm
map,
142
Function
of
class
K,
207
G
acts
on
X
to
the
left,
74
Gauss-Bonnet
theorem,
68
Generalized
Liapunov
function,
227
Generated,
26
Generators,
26
Generic,
253
Globally
asymptotically
stable,
203
Globally
unstable,
205
Grassmann
manifold
of
k-planes
in
n-space,
106
Group,
25
446
Group
of
diffeomorphisms,
49
Hausdorff
space,
31
Hessian,
60,
191
Heteroclinic
orbits,
162
Homeomorphic,
35
Homeomorphism,
35
Homoclinic
cycles,
162
Homoclinic
orbits,
162
Homogeneous,
75
Homologous,
41,
42
Homology
group,
42
Homomorphism,
25
Homotopically
trivial,
43
Hopf
bifurcation,
273
Hyperbolic,
179, 188,
193
Hyperbolic
attractor,
250
Hyperbolic
linear
automorphisms,
182,
184
Hyperbolic
structure
with
respect
to
f,
250
Hyperbolic
subset
of
X,
251
Hyperbolic
toral
automorphism,
151
Image,
25
Imbedding,
51,
125
Immersion,
51,
125
Indecomposable,
261
Induced
bundle,
105
Inner
product,
95
Integrable,
56,
168
Integral
curve,
54,
168
Integral
flow,
168
Integral
manifold,
56
Interior
of
A,
28
Interior
point,
28
Invariant
k-form,
98
Invariant
set,
160
Involutive,
56
447
Isomorphic,
26,
185
Isomorphism,
26
Isotropy,
108
Isotropy
group,
74
Isotropy
subgroup,
152
Jacobi
identity,
51
Jacobian
matrix,
51
k-jet
bundle
of
M
and
N,
114
k-jet
extension
map,
134
k-jet
of
f
at
p,
113
k-jet
of
mappings,
114
Kernel,
25
Lagrange
stability,
200
Lagrange
stable,
167
Lagrange
unstable,
175
Left
invariant,
76
Left
transformation
group,
74
Left
translation
by
g,
76
Liapunov
functions,
206
Liapunov-like
function,
381
Lie
algebra,
76
Lie
derivative,
95
Lie
group,
75
Lie
subalgebra,
78
Lie
subgroup,
78
Limit,
30
Limit
point,
31
Linear
approximation,
188
Linear
contraction,
184
Linear
differential
operator,
129,
134
Linearization
of
a
non-linear
differential
operator
P,
138
Linearly
conjugate,
179
Linearly
equivalent,
180
Lipschitz,
172
448
Lipschitz
map,
172
Local
Hamiltonian,
98
Local
integral,
168
Local
one-parameter
group
of
transformations
T,
55
Local
stable
set,
246
Locally
Hamiltonian,
98
Locally
Lipschitz,
172
Lorenz
equations,
288
Manifold
molded
on
v,
124
Measure
zero,
58
Melnikov
function,
256
Metric
space,
27
Minimal
set
of
a
dynamical
system,
161
Monomorphism,
104
Morphism,
102
Morse
function,
65
Multijet
bundle,
122
Neat
imbedding,
109
Neat
submanifold,
109
Negatively
Lagrange
stable,
167
Negatively
Lagrange
unstable,
175
Negatively
Poisson
stable,
163
Negatively
prolongation.,
17
5
Neighborhood
of
p,
29
Non-degenerate,
96
Non-degenerate
critical
point
of
f,
60
Non-linear
differential
operator
of
order
k,
135
Non-singular,
51
Non-wandering,
164
Non-wandering
set,
162
Norm, 123
Normal
bundle,
89,
90
Nullity,
64
One-form,
93
449
One-parameter
subgroup
of
G,
79
Open
chart,
46
Open
covering
of
X,
36
Open
set,
28
Optical
Instabilities,
341
Orbit,
55,
146
Orbit
space,
147
Orbital
stability,
200
Order
of
a
linear
differential
operator
P,
129
Orientable,
96
Origin
of
attraction,
203
Orthogonal
complement,
90
p-complex,
41
p-dimensional
de
Rham
cohomology
group
of
M,
94
p-dimensional
distribution,
56
p-simplex,
41
p-th
Betti
number,
42
Paracompact,
45
Parallelizable,
91,
177
Parametric
expression
for
the
operator
P,
136
Partial
flow,
121,
155
Partial
tubular
neighborhood
of
M,
108
Partition
of
unity,
47
Path,
37
Permanence,
367
Permanent,
368
Phase
space,
146
Poincare
duality,
67
Poincare
map,
194
Poisson
bracket,
98
Poisson
stable,
163
Poisson
unstable,
175
Positive
definite,
207
Positive
(negative)
prolongation,
175
Positive
(negative)
prolongational
limit
set,
175
Positively
Lagrange
stable,
167
450
Positively
Lagrange
unstable,
175
Positively
Poisson
stable,
163
Positively
prolongation,
175
Positively
recursive,
163
Principal
bundle
of
M,
84
Principal
fiber
bundle,
85
Product
set,
28
Prolongations,
114
Pullback,
105
Qualitative
theory
of
diffferential
equations,
145
Quasi-linear
differential
opeartor
of
order
k,
136
Quotient,
26
Quotient
space
bundle,
89
Quotient
system,
151
r-form,
93
Reaction-diffusion
equations,
351
Recurrent,
166
Recurrent
synaptic
feedback,
384
Region
of
attraction,
202,
217
Region
of
instability,
205
Region
of
uniform
attraction,
202
Region
of
weak
attraction,
201
Regular,
157
Regular
point,
189
Regular
value,
57
Representation
of
G,
79
Restriction
of
the
flow,
34
Retraction,
59
Reverse
flow,
150
Right
translation
by
g,
76
s-fold
k-jet
bundle,
122
Saddle
point,
148
Section,
178
Self
positively
recursive,
163
451
Semi-asymptotically
stable,
233
Semi-attractor,
233
Semi-weak
attractor,
233
Sheets,
44
Simply-connected,
44
Singularity,
157
Smooth
dynamical
systems,
145
Smooth
flow,
12
Smooth
manifold
modeled
on
B,
124
Spectral
decomposition,
261
Spherical
modification,
62
Split,
104
Stability-additive,
238
Stable,
179, 198,
201, 202,
315
Stable
at
t
0
,
199
Stable
manifold,
185,
246
Stable
manifold
of
f
at
p,
247,
250
Stable
relative
to
u,
236
Stable
set,
179,
246
Stable
set
of
size
(b
a)
of
f
at
x,
Stable
summand,
185
Stable
with
respect
to
measurement,
Stable
with
respect
to
x,
Strange,
261
Strange
attractor,
202
Strange
saddles,
308
Strong
transversality,
261
209
251
383
Strongly
stable
under
perturbations,
229
Structural
group,
85
structurally
stable,
190,
245
Sub-bundle,
90
Subalgebra,
78
Subcovering
of
X,
36
Subgroup,
25
Submanifold,
51,
125
Subspace,
29
Surgery,
63
452
Suspension,
149,
150
Symbol
of
the
differential
operator
P,
132,
138
Symplectic
form,
97
Symplectic
manifold,
97
Tangent
bundle
of
M,
84,
88,
125
Tangent
vector
at
p,
49
Tensor
bundle,
89
Tensor
field,
89
The
analytic
index
of
P,
142
The
differential
of
f,
51
The
restriction
of
f
to
X,
34
The
restriction
of
the
flow,
156
The
tangent
vector
to
the
curve
a
at
t,
54
Time map,
146
Topological
conjugacy,
153,
194
Topological
dynamics,
145
Topological
equivalence,
154
Topological
group,
74
Topological
manifold,
45
Topological
property
of
X,
35
Topological
space,
27
Topological
vector
space,
123
Topologically
conjugate,
153,
155
Topologically
equivalent,
155
Topologically
transitive,
261
Topology
induced,
29
Totally
stable,
226
Trajectory,
55
Transitive,
74
Transverse
to
the
orbit,
193
Trivial,
79,
85
Trivial
bundle,
86
Tubular
neighborhood
of
M,
107
Tychonoff
Theorem,
37
Typical
fiber,
83
453