Symmetry and Perturbation Theory in Nonlinear Dynamics

By: Giampaolo Cicogna, Guiseppe Gaeta

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Hardcover | 1 November 1999

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228 Pages

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24.13 x 16.51 x 1.91

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has been in the of a Symmetry major ingredient development quantum perturba tion and it is a basic of the of theory, ingredient theory integrable (Hamiltonian and of the the use in context of non Hamiltonian) systems; yet, symmetry gen eral is rather recent. From the of view of nonlinear perturbation theory point the use of has become dynamics, widespread only through equivariant symmetry bifurcation in this attention has been confined to linear even theory; case, mostly symmetries. in recent the and of methods for dif Also, theory practice symmetry years ferential has become and has been to a equations increasingly popular applied of the of the book Olver This by variety problems (following appearance [2621). with is and deals of nature theory deeply geometrical symmetries general (pro vided that described i.e. in this context there is are vector no they by fields), to limit attention to linear reason symmetries. In this look the basic tools of i.e. normal book we at perturbation theory, introduced Poincar6 about and their inter a forms (first by century ago) study action with with no limitation to linear ones. We focus on the most symmetries, basic fixed the and i.e. a setting, systems having point (at origin) perturbative around thus is local.

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Differential Calculus & Equations Science Physics Applied Physics Applied Mathematics Nonlinear Science Quantum Physics & Quantum Mechanics & Quantum Field Theory Statistical Physics

Mathematical Physics Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

Introduction	p. 1
Definitions and Properties	p. 5
Notations	p. 5
The Restricted Problem: Equations	p. 5
Periodic Solutions and Periodic Orbits	p. 6
The Period-in-Family	p. 8
Structure of Families	p. 9
Family Segments	p. 12
Symmetry	p. 12
Stability	p. 13
Generating Orbits and Keplerian Orbits	p. 14
Species	p. 16
Generating Orbits of the First Species	p. 21
Kinds	p. 21
First Kind	p. 22
The Case n = 1	p. 22
The Case n &neq; 1	p. 23
Second Kind	p. 24
Symmetric Orbits	p. 26
Asymmetric Orbits	p. 31
Summary	p. 33
Generating Orbits of the Second Species	p. 35
Arcs	p. 35
Supporting Ellipses and Types	p. 38
Type 1	p. 41
S-Arcs	p. 42
The (A, Z) Plane	p. 48
S-Arc Families	p. 52
T-Arcs and T-Arc Families	p. 55
Overview	p. 59
Types 2, 3, 4	p. 59
Ends of S- and T-Arc Families	p. 62
Extremums of C: Arc Family Segments	p. 68
Orbits	p. 73
Second Species Families	p. 75
Ends of Family Segments	p. 76
Generating Orbits of the Third Species	p. 79
¿-Generating Orbits	p. 79
Continuation from ¿ = 0 to ¿ > 0	p. 82
First Species	p. 82
Second Species	p. 82
¿-Generating Orbits for 0 < ¿ < 1/3	p. 83
First Species	p. 84
Second Species	p. 85
Third Species	p. 88
¿-Generating Orbits for ¿ = 1/3	p. 88
Continuation from ¿ < 1/3 to ¿ = 1/3	p. 91
Family f	p. 91
Family a	p. 91
Family c	p. 92
Family g'	p. 92
Family g	p. 92
Recapitulation	p. 92
Other Families	p. 93
¿-Generating Orbits for ¿ > 1/3	p. 93
Continuation from ¿ = 1/3 to ¿ > 1/3	p. 94
Conclusions	p. 94
Bifurcation Orbits	p. 95
First Species and First Species	p. 96
First Kind and First Kind	p. 96
First Kind and Second Kind, Symmetric	p. 96
First Kind and Second Kind, Asymmetric	p. 97
Second Kind, Symmetric and Symmetric	p. 97
Second Kind, Symmetric and Asymmetric	p. 97
Second Kind, Asymmetric and Asymmetric	p. 97
Second Species and First or Second Species	p. 98
Total Bifurcation	p. 99
Partial Bifurcation	p. 115
Third Species and First or Second Species	p. 123
Third Species and Third Species	p. 123
Recapitulation	p. 123
Junctions: Symmetry	p. 125
First Species Bifurcations	p. 125
Third Species Bifurcations	p. 126
Second Species Bifurcations	p. 126
Partial Bifurcation	p. 127
Total Bifurcation	p. 128
Conclusions	p. 134
Junctions: Broucke's Principle	p. 137
Definition	p. 137
Side of Passage for a Node	p. 139
Type 1	p. 140
Type 2	p. 143
Type 3	p. 145
Side of Passage for an Antinode	p. 147
Type 1	p. 147
Type 2	p. 150
Type 3	p. 151
Results: Partial Bifurcation	p. 152
Type 1	p. 152
Type 2	p. 160
Type 3	p. 162
Results: Total Bifurcation	p. 162
Type 1	p. 162
Type 2	p. 167
Type 3	p. 167
Recapitulation	p. 167
Fragments	p. 171
Introduction	p. 171
Accidents	p. 172
Explanation of the Fragment Tables	p. 173
First Species Family Segments	p. 174
First Kind: Retrograde Orbits	p. 174
First Kind: Direct Orbits	p. 174
Second Kind	p. 175
Second Species Family Segments	p. 179
Detection of Bifurcation Orbits	p. 179
Data on Arc Family Segments	p. 183
Fragments	p. 190
Third Species Family Segments	p. 196
Generating Families	p. 203
Algorithm	p. 203
Explanation of the Tables	p. 205
Natural Families	p. 206
Family a	p. 207
Family b	p. 208
Family c	p. 209
Family f	p. 209
Family g	p. 210
Family h	p. 211
Family i	p. 213
Family l	p. 214
Family m	p. 214
Summary	p. 214
Other Families	p. 214
Comparison with Computed Families	p. 220
Family a	p. 221
Family b	p. 223
Family c	p. 224
Family f	p. 225
Family g	p. 225
Family h	p. 225
Family i	p. 225
Families l and m	p. 229
Final Comments	p. 232
Correspondence Between Old and New Notations	p. 235
Arcs	p. 235
Arc Families	p. 237
Critical Arcs	p. 240
The Domain D₂	p. 243
The Curved Boundary ¿	p. 243
S-Arc Families in Domain D₂	p. 245
Number of Branches	p. 251
Partial Bifurcation	p. 251
Type 1	p. 251
Type 2	p. 253
Type 3	p. 254
Total Bifurcation	p. 255
Type 1	p. 256
Type 2	p. 258
Type 3	p. 260
Conclusions	p. 264
Index of Definitions	p. 265
Index of Notations	p. 269
References	p. 275
Table of Contents provided by Publisher. All Rights Reserved.

Symmetry and Perturbation Theory in Nonlinear Dynamics

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