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Bridges Th.J., Furter J.E. — Singularity Theory and Equivariant Symplectic Maps
Bridges Th.J., Furter J.E. — Singularity Theory and Equivariant Symplectic Maps

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Название: Singularity Theory and Equivariant Symplectic Maps

Авторы: Bridges Th.J., Furter J.E.

Аннотация:

The monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighborhood of a fixed point. The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the correspondence between orbits of a symplectic map and critical points of an action functional. New results on singularity theory for equivariant gradient bifurcation problems are obtained and then used to classify singularities of bifurcating period-q points. Of particular interest is that a general framework for analyzing group-theoretic aspects and singularities of symplectic maps (particularly period-q points) is presented. Topics include: bifurcations when the symplectic map has spatial symmetry and a theory for the collision of multipliers near rational points with and without spatial symmetry. The monograph also includes 11 self-contained appendices each with a basic result on symplectic maps. The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant bifurcation theory.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 226

Добавлена в каталог: 03.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$Z_n$ (cyclic group)      1 14 63 85
$\mathbb{D}_n$ (dihedral group)      18 69 89 133
action      11.
Action, action functional      10
Action, Lagrangian action      10.
Action-angle variables      8
Admissable variations      11
Algebraic transversality      53
Area-preserving map      85 101 146 174
Arnold, V.      9 21 47 151
Aubry, S.      2.
Averaging over the group      49 53 59 211.
Bartsch, T.      5 127—129
Bernstein, D.      2
Bhowal, A.      164.
Bifurcation, degenerate      77 101.
Bifurcation, diagrams      74 77 98—99 105—106 108 113 115 159 163—165 183
Bifurcation, multiparameter      9 85
Bifurcation, reduced bifurcation equation      19 20 137—138 155
Block diagonalization      16 121 125.
Bridges, T.      6 7 144—145 149 151 164 173—174 193
Bruce, J.      48.
Casimir function      146 147 173
Catastrophe map      45
Category, Lusternik — Schnirelmann      5 129.
Category, WII-category      5 129
Circulant matrix      14
Clapp, M.      5 128.
Classification theorems for paths      47.
Classification theorems, $\mathbb{D}_4$ problems      90.
Classification theorems, $\mathbb{D}_q$, problems      89.
Classification theorems, $\mathbb{Z}_$q problems      70 73 74
Classification theorems, $\mathcal{A}^{\mathbb{Z}_q}$ for potentials      65.
Codimension, $K_{\Delta}$      47—48.
Codimension, gradient      35 38 39 50
Codimension, infinite      34.
Codimension, topological      70 73 74 89
Collision of multipliers at third root of unity      117.
Collision of multipliers at ±i      160.
Collision of multipliers, collision singularity      23 189
Collision of multipliers, irrational      149 171.
Collision of multipliers, rational      149
Collision of multipliers, rational, nonlinear      150.
Collision of multipliers, secondary      164 170 171.
Configuration space      11.
Configuration space, linear stability in      190—192.
Configuration space, signature in      185—190
Contact group $(\mathcal{K}^{\Gamma}_{\lambda})$      36
Critical dimension      127
Curl operator      49
Curve selection lemma      58
Cushman, R.      7 142 144—145 149 173—174 193.
Cyclic group      see $\mathbb{Z}_n$
Damon, J.      45 46 52 53 58 212
Darboux theorem      122 202
Davis, P.      14 15 88
Dellnitz, M.      175 202
Dewar, R.      199 200.
Discriminant variety      45
Distinguished parameter      3 89.
du-Plessis, A.      48.
Elliptic fixed point      9
Equivalence, $\mathcal{A}^{\mathbb{D_q}}$      89 91.
Equivalence, $\mathcal{A}^{\mathbb{Z}_q}$      63—65.
Equivalence, $\mathcal{A}^{\mathbb{Z}_q}$ (for gradients)      68—69
Equivalence, $\mathcal{K}^{\mathbb{D}_q}$      89.
Equivalence, dynamical      199
Equivalence, equivariant contact $(\mathcal{K}^{\Gamma})$      33 35 36
Equivalence, equivariant right $(\mathcal{R}^{\Gamma})$      3 33
Equivalence, left-right $(\mathcal{A})$      34
Equivalence, path      2 34 40—41
Equivalence, topological      70
Equivariant, preparation theorem      55 56 62
Equivariant, singularity theory      33
Equivariant, splitting lemma      3 20 124—125 185
Equivariant, symplectic map      119.
Field, M.      58.
Fixed-point subspace      55 126 214.
Floquet theory (discrete)      24
Floquet, multipliers, exponents      24 31 167.
Flowcharts, $\mathbb{Z}_3$-invariant potentials      68.
Flowcharts, $\mathbb{Z}_4$-invariant potentials      78.
Flowcharts, $\mathbb{Z}_q$q-invariant potentials      80
Fourier matrix      18 125.
Fundamental lemma      46 61
Fundamental Lemma, proof      56
Fundamental theorem      47.
Fundamental Theorem, proof      58
Furter, J.      69 89—90 104 108 118 152.
G-orbit      60.
Gel'fand, I.      136
Generating function      11
Gibson, C      209.
Golubitsky, M.      3 20 34 63 69 89—91 93 214.
Gradient, bifurcation problem      33.
Gradient, higher-order terms      40
Group action      19.
Group orbit      135
Group, compact Lie      38 121
Group, Lie      60
Group, spatial      18.
GSS-II      35—37 40 48 51 56—57 61—62 75 121 126 132—133 213—214
Hamiltonian systems      3 120 130
Hessian matrix      15 21 23 154 166 212
Higher-order terms      39 48
Hilbert basis      20 137 144.
Howard, J.      28 186
Hummel, A.      3 34.
Ideal, instrinsic      48
Invariant circle      146.
Invariant function      137
Invariant function, $\mathbb{D}_{[p,q]}$      206
Invariant function, $\mathbb{Z}_q$      63 67
Involution      18.
Involution, anti-symplectic      101 197 209
Isotypic decomposition      121.
Katok, A.      2
Kielhoefer, H.      128
Kook, H.      2 24 26
Kozlov, V.      21.
Krawcewicz, W.      129.
Krein locus      171.
Lagrangian (continuous time)      141 142
Lagrangian generating function      1—2
Lahiri, A.      7 164
Lidskii, V.      136
Liftable vectorfields      53.
Looijenga, E.      45 58.
MacKay, R.      2 7 10 12 24 28 144 149 173—174 186 190
Magnus, R.      185
Manifold      59 60.
Manifold, analytic      44.
Manifold, solid torus      136.
Manifold, submanifold      60
Marsden, J.      175
Marzantowicz, W.      129
Mather's Lemma      60—61
Mather, J.      3 60
Mawhin, J.      210 212
Meiss, J.      2 10 12 24 26 190 199 200
Melbourne, I.      175 202
Meyer, K.      1 2 6 10 20—21 87—88 103
Modal parameters      34
Mode interactions      10 205
Momentum map      194
Mond, D.      3 35 53 58
Montaldi, J.      3 5 35 40 120 128 143 202.
Morse index      19 127.
Moser, J.      10 120 130
Nakayama's lemma      73
Neishtadt, A.      21
Noether's theorem      5 194.
Noether's theorem, conserved quantities      5 136 141 194—196.
Non-degeneracy conditions      67—68 70 74 79 82
Normal form for bifurcation function      20 23 36 74 155 181.
Normal form, unipotent normal form      173.
Normal form, Williamson normal form      192
Normal space, extended      37.
O(2)      5.
O(2), equivariant syniplectic maps      132 143
O(2), invariant generating function      132
Orbit space      144.
Palais, R.      126
Path formulation      3 34 40—41
Percival, I.      2
Period-q point      13
Pfenniger, D.      7
Planar pendulum      142 143
Poisson bracket      145
Potential function      3 33—34
Potential function in $\mathcal{E}^{\Gamma}_o$      47.
Pull-back mapping      49.
q-resonant subspace      119 123
Quaternionic algebra      123.
Rannou map      85.
Rannou, F.      85
Rational Collision Theorem      155 161
Recognition problem      40
Representation, absolutely irreducible      130
Residue      25 166—167
Resonance      10
Reversibility      196.
Reversibility, CS-reversibility      18 101 196 199.
Reversibility, k-reversibility      101.
Reversibility, R-reversibility      101 197 199
Reversible maps      18 22 196 209.
Reversible period-q points      22
Rimmer, R.      1 22.
Roberts, R.      5 40 42 63 69 89—91 93 120 128 143 202
Rotation number      6 9.
Roy, T.      164.
Scalar product, $(\cdot, \cdot)$ on $\mathbb{R}^n$      11
Scalar product, $< \cdot, \cdot> $on $\mathbb{R}^n$      38 204
Scalar product, $<\cdot, \cdot>$ on $X^n_q$      14
Scalar product, $[ \cdot, \cdot]$ on $X^n_q$      11
Schaeffer, D.      3 20 34
Schmidt, D.      6
Schoenberg, I.      88
Schwartz's Theorem      20
Sevryuk, M.      7
Signature      127 193.
Signature for equivariant maps      127 188.
Signature on configuration space      185—190
Signature, Krein's definition      186
Slodowy, P.      44—45.
SO(2)-symmetric rational collision      181
Sokol'skij, A.      6.
Spectrum, second variation of action      15 154
Spectrum, symplectic matrix      28.
Spherical pendulum      140—143
Stability for equivariant maps      146.
Stability, Instability Lemma      24 163 212.
Stability, linear, on configuration space      190—192
Stability, reduced      23.
Stability, Stability Lemma I      23 26.
Stability, Stability Lemma II      171
Standard map, generalized      85 101
Stark, J.      2 10 12.
Starzhinskii, V.      186.
Stewart, I      5 20 40 120 128 143 202
Stratification, Whitney      58
Subgroup, isotropy      134 207 213
Subgroup, isotropy, lattice      207—209.
Subgroup, normalizer      126.
Subgroup, r-spatial      126.
Subgroup, r-spatio-temporal      126.
Subgroup, twisted      126 213—214.
Subgroup, Weyl      126 181
Subharmonic bifurcation      121 130
Symmetry group (discrete)      18
Symmetry, of solutions      129 131.
Symmetry, spatial or temporal      18 126 128.
Symmetry, symmetric criticality      126.
Symplectic form      7 202.
Symplectic group $Sp(2n, \mathbb{R})$      13 151 192—193
Symplectic Lie algebra $sp(2n, \mathbb{R})$      13 151
Symplectic operator      7 196 202
Symplectomorphism      3
Symplectomorphism, q-regular      123.
Tangent map      24 168
Tangent space      36 64.
Tangent, extended tangent space      50 65 73
Tangent, extended tangent space, gradient      38.
Tangent, restricted tangent space      37 50
Tensor product      14
Terao, H.      53.
Torus      143
Twist      199
Twist mapping      199.
Unfolding      37 39 44 52
Unfolding, universal      37 40 48 65.
Unfolding, universal, (gradient) $\mathcal{G}^{\Gamma}$      35 39
Unfolding, versal      37 47 150.
Universality Theorem      39
Universality Theorem, proof      51.
van der Meer, J.      6 175 182
Vanderbauwhede, A.      45.
Variational Principle      11.
Wall, C      42 48
Weinstein — Moser Theory      10 120 130.
Weinstein, A.      10 120 130
Willem, M.      210 212
Williamson, J.      193.
Yakubovich, V.      186
Yau, S.      42.
Zuppa, C.      34.
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