Lectures on fractal geometry and dynamical systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc. [u.a.]
2009
|
| Schriftenreihe: | Student mathematical library
52 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XVI, 314 S. Ill., graph. Darst. |
| ISBN: | 9780821848890 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV035882657 | ||
| 003 | DE-604 | ||
| 005 | 20190718 | ||
| 007 | t | ||
| 008 | 091214s2009 xxuad|| |||| 00||| eng d | ||
| 010 | |a 2009028324 | ||
| 020 | |a 9780821848890 |9 978-0-8218-4889-0 | ||
| 035 | |a (OCoLC)600228716 | ||
| 035 | |a (DE-599)BVBBV035882657 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 044 | |a xxu |c US | ||
| 049 | |a DE-83 |a DE-634 |a DE-739 |a DE-188 | ||
| 050 | 0 | |a QA614.86 | |
| 082 | 0 | |a 514/.742 | |
| 084 | |a SK 810 |0 (DE-625)143257: |2 rvk | ||
| 084 | |a 37C45 |2 msc | ||
| 100 | 1 | |a Pesin, Yakov B. |d 1946- |e Verfasser |0 (DE-588)129370533 |4 aut | |
| 245 | 1 | 0 | |a Lectures on fractal geometry and dynamical systems |c Yakov Pesin ; Vaughn Climenhaga |
| 264 | 1 | |a Providence, RI |b American Math. Soc. [u.a.] |c 2009 | |
| 300 | |a XVI, 314 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Student mathematical library |v 52 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Fractals | |
| 650 | 4 | |a Dynamics | |
| 700 | 1 | |a Climenhaga, Vaughn |d 1982- |e Sonstige |0 (DE-588)139069801 |4 oth | |
| 830 | 0 | |a Student mathematical library |v 52 |w (DE-604)BV013184751 |9 52 | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018740200&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-018740200 | ||
Datensatz im Suchindex
| _version_ | 1804140863253643264 |
|---|---|
| adam_text | Contents
Foreword: MASS and
REU
at
Penn
State University
xi
Preface
xiii
Chapter
1.
Basic Concepts and Examples
1
Lecture
1 1
a. A threefold cord: fractals, dynamics, and chaos
1
b. Fractals: intricate geometry and self-similarity
2
с
Dynamics: things that move (or don t)
7
Lecture
2 10
a. Dynamical systems: terminology and notation
10
b. Population models and the logistic map
13
Lecture
3 19
a. A linear map with chaotic behaviour and the
middle-third Cantor set
19
b. The Cantor set and symbolic dynamics
24
Lecture
4 28
a. Some point-set topology
28
b. Metric spaces
31
с
Lebesgue measure
34
Lecture
5 37
a. The
topologica!
structure of symbolic space and the
Cantor set
37
b.
What the coding map doesn t do
41
с
Geometry of Cantor sets
42
Lecture
6 46
a. More general constructions
46
b. Making sense of it all
50
Chapter
2.
Fundamentals of Dimension Theory
53
Lecture
7 53
a. Definition of Hausdorff dimension
53
b. Hausdorff dimension of the middle-third Cantor set
58
с
Alternative definitions of Hausdorff dimension
60
Lecture
8 62
a. Properties of Hausdorff dimension
62
b. Topological dimension
66
Lecture
9 68
a. Comparison of Hausdorff and topological dimension
68
b. Metrics and topologies
71
с
Topology and dimension
74
Lecture
10 .75
a. Hausdorff dimension of Cantor sets
75
b. Moran s Theorem
76
с
Moran
constructions
80
d.
Dynamical constructions and iterated function
systems
82
Lecture
11 85
a. Box dimension: another way of measuring dimension
85
b. Properties of box dimension
88
Lecture
12 90
a. Relationships between the various dimensions
90
b. A
counterexample
94
с
Stability and subadditivity
98
Chapter
3.
Measures: Definitions and Examples
101
Lecture
13 101
a. A little bit of measure theory
101
b. Lebesgue measure and outer measures
105
c.
Hausdorff measures
109
Lecture
14 110
a. Choosing a good outer measure
110
b. Bernoulli measures on symbolic space 111
с
Measures on Cantor sets
113
d. Markov measures
114
Lecture
15 117
a. The support of a measure
117
b. Subshifts of finite type: One-dimensional Markov
maps
120
Chapter
4.
Measures and Dimensions
123
Lecture
16 123
a. The Uniform Mass Distribution Principle: Using
measures to determine dimension
123
b. Pointwise dimension and the Non-uniform Mass
Distribution Principle
125
Lecture
17 128
a. Variable pointwise dimension
128
b. Hausdorff dimension of exact dimensional measures
135
с
Pointwise dimension of Hausdorff measures
137
Lecture
18 138
a. Local entropy
138
b. Kolmogorov-Sinai entropy
142
с
Topologica!
entropy
143
Lecture
19 146
a. Entropy of Markov measures
146
b. Hausdorff dimension of Markov constructions
149
Lecture
20 151
a. Lyapunov exponents
151
b. Fractals within fractals
155
Chapter
5.
Discrete-Time Systems: The FitzHugh-Nagumo
Model
159
Lecture
21 159
a. The FitzHugh-Nagumo model for neurons
159
b.
Numerical investigations: From continuous to discrete
164
Lecture
22 167
a. Studying the local map
167
b. Stability of fixed points for general maps
169
Lecture
23 175
a. Stability of fixed points for the FitzHugh-Nagumo
model
175
b. Periodic points
177
Lecture
24 181
a. Beyond period-doubling: Down the rabbit hole
181
b. Becoming one-dimensional
185
Chapter
6.
The Bifurcation Diagram for the Logistic Map
191
Lecture
25 191
a. Bifurcations of the logistic map
191
b. Classifying bifurcations
194
Lecture
26 198
a. The period-doubling cascade
198
b. Chaos at the end of the bifurcation diagram
199
с
The centre cannot hold: escape to infinity
202
Lecture
27 204
a. Finding the relevant part of phase space:
ω
-limit
sets
204
b. Windows of stability in the bifurcation diagram
206
с
Chaos outside the windows of stability
207
Chapter
7.
Chaotic Attractors and Persistent Chaos
209
Lecture
28 209
a. Trapping regions
209
b. Attractors
212
Lecture
29 215
a. The
Smale-
Williams solenoid
215
b. Uniform hyperbolicity
218
c. Symbolic dynamics
221
Lecture
30 224
a. Dimension of direct products
224
b. Quantifying the attractor
228
c. Lyapunov
exponents in multiple dimensions
229
d. The non-conformal case
231
e. The attractor for the FitzHugh-Nagumo map
232
Chapter
8.
Horseshoes and Intermittent Chaos
233
Lecture
31 233
a. The
Smale
horseshoe: A trapping region that isn t
233
b. Hausdorff dimension of the horseshoe
237
с
Symbolic dynamics on the
Smale
horseshoe
239
Lecture
32 241
a. Variations on a theme: Other horseshoes
241
b. Intermittent chaos vs. persistent chaos
245
с
Homoclinic orbits and horseshoes
247
Chapter
9.
Continuous-Time Systems: The
Lorenz
Model
253
Lecture
33 253
a. Continuous-time systems: Basic concepts
253
b. Fixed points of continuous-time systems
256
Lecture
34 259
a. The pendulum
259
b. Two-dimensional systems
263
Lecture
35 267
a. The
Lorenz
equations
267
b. Beyond the linear mindset
269
с
Examining the
Lorenz
system
269
Lecture
36 274
a. Passing to
a Poincaré
map
274
b. Horseshoes in the
Lorenz
system
277
Lecture
37 281
a. The
Lorenz
attractor
281
b. The geometric
Lorenz
attractor
281
с
Dimension of the geometric
Lorenz
attractor
285
d. Back to the
Lorenz
attractor, and beyond
286
Appendix
289
Hints to selected exercises
295
Suggested Reading
299
Bibliography
305
Index
309
|
| any_adam_object | 1 |
| author | Pesin, Yakov B. 1946- |
| author_GND | (DE-588)129370533 (DE-588)139069801 |
| author_facet | Pesin, Yakov B. 1946- |
| author_role | aut |
| author_sort | Pesin, Yakov B. 1946- |
| author_variant | y b p yb ybp |
| building | Verbundindex |
| bvnumber | BV035882657 |
| callnumber-first | Q - Science |
| callnumber-label | QA614 |
| callnumber-raw | QA614.86 |
| callnumber-search | QA614.86 |
| callnumber-sort | QA 3614.86 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 810 |
| ctrlnum | (OCoLC)600228716 (DE-599)BVBBV035882657 |
| dewey-full | 514/.742 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514/.742 |
| dewey-search | 514/.742 |
| dewey-sort | 3514 3742 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01555nam a2200409 cb4500</leader><controlfield tag="001">BV035882657</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20190718 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">091214s2009 xxuad|| |||| 00||| eng d</controlfield><datafield tag="010" ind1=" " ind2=" "><subfield code="a">2009028324</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780821848890</subfield><subfield code="9">978-0-8218-4889-0</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)600228716</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV035882657</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">xxu</subfield><subfield code="c">US</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-83</subfield><subfield code="a">DE-634</subfield><subfield code="a">DE-739</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA614.86</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">514/.742</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 810</subfield><subfield code="0">(DE-625)143257:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">37C45</subfield><subfield code="2">msc</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Pesin, Yakov B.</subfield><subfield code="d">1946-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)129370533</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Lectures on fractal geometry and dynamical systems</subfield><subfield code="c">Yakov Pesin ; Vaughn Climenhaga</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Providence, RI</subfield><subfield code="b">American Math. Soc. [u.a.]</subfield><subfield code="c">2009</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XVI, 314 S.</subfield><subfield code="b">Ill., graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Student mathematical library</subfield><subfield code="v">52</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Includes bibliographical references and index</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Fractals</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Dynamics</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Climenhaga, Vaughn</subfield><subfield code="d">1982-</subfield><subfield code="e">Sonstige</subfield><subfield code="0">(DE-588)139069801</subfield><subfield code="4">oth</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Student mathematical library</subfield><subfield code="v">52</subfield><subfield code="w">(DE-604)BV013184751</subfield><subfield code="9">52</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Passau</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018740200&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-018740200</subfield></datafield></record></collection> |
| id | DE-604.BV035882657 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:06:44Z |
| institution | BVB |
| isbn | 9780821848890 |
| language | English |
| lccn | 2009028324 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018740200 |
| oclc_num | 600228716 |
| open_access_boolean | |
| owner | DE-83 DE-634 DE-739 DE-188 |
| owner_facet | DE-83 DE-634 DE-739 DE-188 |
| physical | XVI, 314 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | American Math. Soc. [u.a.] |
| record_format | marc |
| series | Student mathematical library |
| series2 | Student mathematical library |
| spelling | Pesin, Yakov B. 1946- Verfasser (DE-588)129370533 aut Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga Providence, RI American Math. Soc. [u.a.] 2009 XVI, 314 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Student mathematical library 52 Includes bibliographical references and index Fractals Dynamics Climenhaga, Vaughn 1982- Sonstige (DE-588)139069801 oth Student mathematical library 52 (DE-604)BV013184751 52 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018740200&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pesin, Yakov B. 1946- Lectures on fractal geometry and dynamical systems Student mathematical library Fractals Dynamics |
| title | Lectures on fractal geometry and dynamical systems |
| title_auth | Lectures on fractal geometry and dynamical systems |
| title_exact_search | Lectures on fractal geometry and dynamical systems |
| title_full | Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga |
| title_fullStr | Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga |
| title_full_unstemmed | Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga |
| title_short | Lectures on fractal geometry and dynamical systems |
| title_sort | lectures on fractal geometry and dynamical systems |
| topic | Fractals Dynamics |
| topic_facet | Fractals Dynamics |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018740200&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013184751 |
| work_keys_str_mv | AT pesinyakovb lecturesonfractalgeometryanddynamicalsystems AT climenhagavaughn lecturesonfractalgeometryanddynamicalsystems |