Dynamical systems: stability, symbolic dynamics, and chaos
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton ; London ; New York
CRC Press
1999
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Studies in advanced mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | 504 Seiten Illustrationen |
| ISBN: | 0849384958 9780849384950 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV013085992 | ||
| 003 | DE-604 | ||
| 005 | 20190705 | ||
| 007 | t | ||
| 008 | 000403s1999 a||| |||| 00||| eng d | ||
| 020 | |a 0849384958 |c hbk |9 0-8493-8495-8 | ||
| 020 | |a 9780849384950 |c hbk |9 978-0-8493-8495-0 | ||
| 035 | |a (OCoLC)245877428 | ||
| 035 | |a (DE-599)BVBBV013085992 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-29 |a DE-703 |a DE-19 |a DE-91G |a DE-384 |a DE-634 |a DE-20 |a DE-11 |a DE-739 |a DE-83 |a DE-188 | ||
| 050 | 0 | |a QA614.8 | |
| 082 | 0 | |a 515.352 | |
| 084 | |a SK 520 |0 (DE-625)143244: |2 rvk | ||
| 084 | |a SK 960 |0 (DE-625)143275: |2 rvk | ||
| 084 | |a 11 |2 ssgn | ||
| 084 | |a MAT 344f |2 stub | ||
| 084 | |a 58Fxx |2 msc | ||
| 084 | |a PHY 066f |2 stub | ||
| 100 | 1 | |a Robinson, Clark |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Dynamical systems |b stability, symbolic dynamics, and chaos |c Clark Robinson |
| 250 | |a Second edition | ||
| 264 | 1 | |a Boca Raton ; London ; New York |b CRC Press |c 1999 | |
| 300 | |a 504 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Studies in advanced mathematics | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Verzweigung <Mathematik> - Differenzierbares dynamisches System - Chaos - Strukturelle Stabilität | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematische Methode |0 (DE-588)4155620-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differenzierbares dynamisches System |0 (DE-588)4137931-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differenzierbares dynamisches System |0 (DE-588)4137931-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Dynamisches System |0 (DE-588)4013396-5 |D s |
| 689 | 1 | 1 | |a Mathematische Methode |0 (DE-588)4155620-3 |D s |
| 689 | 1 | |8 1\p |5 DE-604 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008914417&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-008914417 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Datensatz im Suchindex
| _version_ | 1804127783061815296 |
|---|---|
| adam_text | Contents
Chapter I. Introduction 1
1.1 Population Growth Models, One Population 2
1.2 Iteration of Real Valued Functions as Dynamical Systems .. 3
1.3 Higher Dimensional Systems 5
1.4 Outline of the Topics of the Chapters 9
Chapter II. One Dimensional Dynamics by Iteration 13
2.1 Calculus Prerequisites 13
*2.2 Periodic Points 15
*2.2.1 Fixed Points for the Quadratic Family 20
*2.3 Limit Sets and Recurrence for Maps 22
*2.4 Invariant Cantor Sets for the Quadratic Family 26
?2.4.1 Middle Cantor Sets 26
*2.4.2 Construction of the Invariant Cantor Set 30
2.4.3 The Invariant Cantor Set for i 4 33
*2.5 Symbolic Dynamics for the Quadratic Map 38
*2.6 Conjugacy and Structural Stability 41
*2.7 Conjugacy and Structural Stability of the Quadratic Map.. 47
2.8 Homeomorphisms of the Circle 50
2.9 Exercises 58
Chapter III. Chaos and Its Measurement 65
3.1 Sharkovskii s Theorem 65
3.1.1 Examples for Sharkovskii s Theorem 72
3.2 Subshifts of Finite Type 74
3.3 Zeta Function 80
3.4 Period Doubling Cascade 82
3.5 Chaos 84
3.6 Liapunov Exponents 88
3.7 Exercises 91
Chapter IV. Linear Systems 95
4.1 Review: Linear Maps and the Real Jordan Canonical Form. 95
*4.2 Linear Differential Equations 97
*4.3 Solutions for Constant Coefficients 99
*4.4 Phase Portraits 104
*4.5 Contracting Linear Differential Equations 108
*4.6 Hyperbolic Linear Differential Equations 113
*4.7 Topologically Conjugate Linear Differential Equations 115
*4.8 Nonhomogeneous Equations 117
*4.9 Linear Maps 118
4.9.1 Perron Frobenius Theorem 125
4.10 Exercises 129
* Core Sections
CONTENTS
Chapter V. Analysis Near Fixed Points and Periodic Orbits 133
*5.1 Review: Differentiation in Higher Dimensions 133
?5.2 Review: The Implicit Function Theorem 136
*5.2.1 Higher Dimensional Implicit Function Theorem .. 138
*5.2.2 The Inverse Function Theorem 139
*5.2.3 Contraction Mapping Theorem 140
*5.3 Existence of Solutions for Differential Equations 142
*5.4 Limit Sets and Recurrence for Flows 148
*5.5 Fixed Points for Nonlinear Differential Equations 151
?5.5.1 Nonlinear Sinks 153
*5.5.2 Nonlinear Hyperbolic Fixed Points 155
*5.5.3 Liapunov Functions Near a Fixed Point 156
*5.6 Stability of Periodic Points for Nonlinear Maps 158
5.7 Proof of the Hartman Grobman Theorem 160
?5.7.1 Proof of the Local Theorem 166
5.7.2 Proof of the Hartman Grobman Theorem for Flows 167
?5.8 Periodic Orbits for Flows 168
5.8.1 The Suspension of a Map 173
5.8.2 An Attracting Periodic Orbit for the
Van der Pol Equations 174
5.8.3 Poincare Map for Differential Equations
in the Plane 179
?5.9 Poincare Bendixson Theorem 181
?5.10 Stable Manifold Theorem for a Fixed Point of a Map 183
5.10.1 Proof of the Stable Manifold Theorem 187
5.10.2 Center Manifold 200
?5.10.3 Stable Manifold Theorem for Flows 202
?5.11 The Inclination Lemma 203
5.12 Exercises 204
Chapter VI. Hamiltonian Systems 215
6.1 Hamiltonian Differential Equations 215
6.2 Linear Hamiltonian Systems 220
6.3 Symplectic Diffeomorphisms 223
6.4 Normal Form at Fixed Point 227
6.5 KAM Theorem 231
6.6 Exercises 233
Chapter VII. Bifurcation of Periodic Points 237
7.1 Saddle Node Bifurcation 237
7.2 Saddle Node Bifurcation in Higher Dimensions 239
7.3 Period Doubling Bifurcation 244
7.4 Andronov Hopf Bifurcation for Differential Equations 249
7.5 Andronov Hopf Bifurcation for Diffeomorphisms 256
7.6 Exercises 259
Chapter VIII. Examples of Hyperbolic Sets and Attractors 263
?8.1 Definition of a Manifold 263
?8.1.1 Topology on Space of Differentiable Functions 265
?8.1.2 Tangent Space 266
?8.1.3 Hyperbolic Invariant Sets 269
* Core Sections
CONTENTS
*8.2 Transitivity Theorems 273
*8.3 Two Sided Shift Spaces 275
8.3.1 Subshifts for Nonnegative Matrices 275
*8.4 Geometric Horseshoe 277
8.4.1 Horseshoe for the Henon Map 283
*8.4.2 Horseshoe from a Homoclinic Point 287
8.4.3 Nontransverse Homoclinic Point 297
*8.4.4 Homoclinic Points and Horseshoes for Flows 299
8.4.5 Melnikov Method for Homoclinic Points 302
8.4.6 Fractal Basin Boundaries 307
*8.5 Hyperbolic Toral Automorphisms 308
8.5.1 Markov Partitions for Hyperbolic Toral
Automorphisms 313
8.5.2 Ergodicity of Hyperbolic Toral Automorphisms... 320
8.5.3 The Zeta Function for Hyperbolic Toral Automor¬
phisms 322
*8.6 Attractors 326
*8.7 The Solenoid Attractor 328
8.7.1 Conjugacy of the Solenoid to an Inverse Limit 333
8.8 The DA Attractor 334
8.8.1 The Branched Manifold 338
*8.9 Plykin Attractors in the Plane 338
8.10 Attractor for the Henon Map 341
8.11 Lorenz Attractor 344
8.11.1 Geometric Model for the Lorenz Equations 347
8.11.2 Homoclinic Bifurcation to a Lorenz Attractor — 353
*8.12 Morse Smale Systems 353
8.13 Exercises 361
Chapter IX. Measurement of Chaos in Higher Dimensions 369
9.1 Topological Entropy 369
9.1.1 Proof of Two Theorems on Topological Entropy.. 379
9.1.2 Entropy of Higher Dimensional Examples 386
9.2 Liapunov Exponents 387
9.4 Sinai Ruelle Bowen Measure for an Attractor 392
9.4 Fractal Dimension 393
9.5 Exercises 398
Chapter X. Global Theory of Hyperbolic Systems 403
10.1 Fundamental Theorem of Dynamical Systems 403
10.1.1 Fundamental Theorem for a Homeomorphism 410
10.2 Stable Manifold Theorem for a Hyperbolic Invariant Set.... 411
10.3 Shadowing and Expansiveness 414
10.4 Anosov Closing Lemma 418
10.5 Decomposition of Hyperbolic Recurrent Points 419
10.6 Markov Partitions for a Hyperbolic Invariant Set 425
10.7 Local Stability and Stability of Anosov Diffeomorphisms ... 435
10.8 Stability of Anosov Flows 438
10.9 Global Stability Theorems 441
10.10 Exercises 444
* Core Sections
CONTENTS
Chapter XI. Generic Properties 449
11.1 Kupka Smale Theorem 449
11.2 Transversality 453
11.3 Proof of the Kupka Smale Theorem 455
11.4 Necessary Conditions for Structural Stability 461
11.5 Nondensity of Structural Stability 464
11.6 Exercises 466
Chapter XII. Smoothness of Stable Manifolds and
Applications 469
12.1 Differentiate Invariant Sections for Fiber Contractions 469
12.2 Differentiability of Invariant Splitting 477
12.4 Differentiability of the Center Manifold 480
12.4 Persistence of Normally Contracting Manifolds 480
12.5 Exercises 484
References 487
Index 501
|
| any_adam_object | 1 |
| author | Robinson, Clark |
| author_facet | Robinson, Clark |
| author_role | aut |
| author_sort | Robinson, Clark |
| author_variant | c r cr |
| building | Verbundindex |
| bvnumber | BV013085992 |
| callnumber-first | Q - Science |
| callnumber-label | QA614 |
| callnumber-raw | QA614.8 |
| callnumber-search | QA614.8 |
| callnumber-sort | QA 3614.8 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 520 SK 960 |
| classification_tum | MAT 344f PHY 066f |
| ctrlnum | (OCoLC)245877428 (DE-599)BVBBV013085992 |
| dewey-full | 515.352 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.352 |
| dewey-search | 515.352 |
| dewey-sort | 3515.352 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | Second edition |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02248nam a2200541 c 4500</leader><controlfield tag="001">BV013085992</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20190705 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">000403s1999 a||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0849384958</subfield><subfield code="c">hbk</subfield><subfield code="9">0-8493-8495-8</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780849384950</subfield><subfield code="c">hbk</subfield><subfield code="9">978-0-8493-8495-0</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)245877428</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV013085992</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-29</subfield><subfield code="a">DE-703</subfield><subfield code="a">DE-19</subfield><subfield code="a">DE-91G</subfield><subfield code="a">DE-384</subfield><subfield code="a">DE-634</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-739</subfield><subfield code="a">DE-83</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA614.8</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">515.352</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 520</subfield><subfield code="0">(DE-625)143244:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 960</subfield><subfield code="0">(DE-625)143275:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">11</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 344f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">58Fxx</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">PHY 066f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Robinson, Clark</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Dynamical systems</subfield><subfield code="b">stability, symbolic dynamics, and chaos</subfield><subfield code="c">Clark Robinson</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">Second edition</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Boca Raton ; London ; New York</subfield><subfield code="b">CRC Press</subfield><subfield code="c">1999</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">504 Seiten</subfield><subfield code="b">Illustrationen</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="0" ind2=" "><subfield code="a">Studies in advanced mathematics</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Hier auch später erschienene, unveränderte Nachdrucke</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Verzweigung <Mathematik> - Differenzierbares dynamisches System - Chaos - Strukturelle Stabilität</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Differentiable dynamical systems</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Dynamisches System</subfield><subfield code="0">(DE-588)4013396-5</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Mathematische Methode</subfield><subfield code="0">(DE-588)4155620-3</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Differenzierbares dynamisches System</subfield><subfield code="0">(DE-588)4137931-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Differenzierbares dynamisches System</subfield><subfield code="0">(DE-588)4137931-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Dynamisches System</subfield><subfield code="0">(DE-588)4013396-5</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2="1"><subfield code="a">Mathematische Methode</subfield><subfield code="0">(DE-588)4155620-3</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2=" "><subfield code="8">1\p</subfield><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</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=008914417&sequence=000002&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:BVB01-008914417</subfield></datafield><datafield tag="883" ind1="1" ind2=" "><subfield code="8">1\p</subfield><subfield code="a">cgwrk</subfield><subfield code="d">20201028</subfield><subfield code="q">DE-101</subfield><subfield code="u">https://d-nb.info/provenance/plan#cgwrk</subfield></datafield></record></collection> |
| id | DE-604.BV013085992 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:38:50Z |
| institution | BVB |
| isbn | 0849384958 9780849384950 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008914417 |
| oclc_num | 245877428 |
| open_access_boolean | |
| owner | DE-29 DE-703 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-384 DE-634 DE-20 DE-11 DE-739 DE-83 DE-188 |
| owner_facet | DE-29 DE-703 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-384 DE-634 DE-20 DE-11 DE-739 DE-83 DE-188 |
| physical | 504 Seiten Illustrationen |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Studies in advanced mathematics |
| spelling | Robinson, Clark Verfasser aut Dynamical systems stability, symbolic dynamics, and chaos Clark Robinson Second edition Boca Raton ; London ; New York CRC Press 1999 504 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Studies in advanced mathematics Hier auch später erschienene, unveränderte Nachdrucke Verzweigung <Mathematik> - Differenzierbares dynamisches System - Chaos - Strukturelle Stabilität Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd rswk-swf Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Differenzierbares dynamisches System (DE-588)4137931-7 gnd rswk-swf Differenzierbares dynamisches System (DE-588)4137931-7 s DE-604 Dynamisches System (DE-588)4013396-5 s Mathematische Methode (DE-588)4155620-3 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008914417&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Robinson, Clark Dynamical systems stability, symbolic dynamics, and chaos Verzweigung <Mathematik> - Differenzierbares dynamisches System - Chaos - Strukturelle Stabilität Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd Mathematische Methode (DE-588)4155620-3 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4155620-3 (DE-588)4137931-7 |
| title | Dynamical systems stability, symbolic dynamics, and chaos |
| title_auth | Dynamical systems stability, symbolic dynamics, and chaos |
| title_exact_search | Dynamical systems stability, symbolic dynamics, and chaos |
| title_full | Dynamical systems stability, symbolic dynamics, and chaos Clark Robinson |
| title_fullStr | Dynamical systems stability, symbolic dynamics, and chaos Clark Robinson |
| title_full_unstemmed | Dynamical systems stability, symbolic dynamics, and chaos Clark Robinson |
| title_short | Dynamical systems |
| title_sort | dynamical systems stability symbolic dynamics and chaos |
| title_sub | stability, symbolic dynamics, and chaos |
| topic | Verzweigung <Mathematik> - Differenzierbares dynamisches System - Chaos - Strukturelle Stabilität Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd Mathematische Methode (DE-588)4155620-3 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd |
| topic_facet | Verzweigung <Mathematik> - Differenzierbares dynamisches System - Chaos - Strukturelle Stabilität Differentiable dynamical systems Dynamisches System Mathematische Methode Differenzierbares dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008914417&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT robinsonclark dynamicalsystemsstabilitysymbolicdynamicsandchaos |