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Fiedler B. Global Bifurcation of Periodic Solutions with Symmetry
Fiedler B.  Global Bifurcation of Periodic Solutions with Symmetry









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: Global Bifurcation of Periodic Solutions with Symmetry

: Fiedler B.

: en

: /

:

ed2k: ed2k stats

: 1988

: 144

: 14.08.2008

: | | ID
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1986      19 96
ALCON      109
Analytic semigroup      9 12 25f 31 84 91 97f 116
Answers      3 11 96101 106f 115
Approximation      38 84
Approximation, generic      3 10 13 23 33f
Approximation, genericand equivariant      3 13 23 8591 108111
Arms, number of      99f 105
Arms, reduction      99
Arnold Bogdanov Takens singularity      see B-point
Arrhenius kinetics      103
Artin conjecture      19 97
Assumptions, analyticity      22 34 46 87f
Assumptions, boundedness      22 25 8590
Assumptions, equivariance      2 25
Assumptions, genericity      20 49 59 68 84 121f 129f
Assumptions, nondegeneracy      9 22 46 8590 98 112115
Assumptions, regularity      16 130f
auto      109
Axisymmetry      98f
B-point      109 113f
Baire space      13 26
Belousov Zhabotinskii reaction      36 92 99 104f
Benard convection      104
BIFPACK      109
Bifurcation      see also B-point Flip Flip-flop Flop Freezing Hopf E. Jug-handle Ljapunov Period Periodic Saddle-node Torus Stationary Symmetry Type
Bifurcation, diagram      3f 13 31f 68 86 89 109114
Bifurcation, multiparameter      3 9 109 110 114f
Bifurcation, with two parameters      3 103 110 112114
Binary orbit      11f 1824 67 79 85f 94 98 101 108
Biological clock      102
Brouwer degree      4 23 2729 110
Brusselator      92101
Cartan decomposition      99
Catalysis      103
Catastrophe, blue sky      113 115
Catastrophe, theory      9 109
Center      16 93 95
Center and symmetry      48f
Center and virtual periods      42f 46f 85
Center index      17
Center index and orbit index      33 7783 91 104 E.
Center manifold      9
Center, continuum of      34 46f 87
Center, generic      16
Center, generic, genericity of      48f 121f
Center, H-center      16
Change of stability      1f 9 12 17 Hopf E.
Chaos      103 114
Characteristic equation      93f
China      17
Compactness assumption for continua      84f
Compactness assumption for groups      2 39 45 74
Compactness assumption for semiflows      25f 91 122 127f
Complexification      17 23 42 119f
Concentric wave      see Wave
Continuum of periodic solutions      see also Center
Continuum of periodic solutions, excluding period, global      12f 21 24 34 85f 95f 101 110f 113f
Continuum of periodic solutions, including period, global      9 31 96 100 110f
Continuum of periodic solutions, limiting      33 84f
Continuum of periodic solutions, local      107f
Continuum of periodic solutions, unbounded      21 24 27f 33f 88f 99f
Coupled oscillators      2f 8 10 9297 102f 105
Coupled oscillators with electric coupling      103
Crossing number      1 17 78 111 Hopf E. Transverse)
Crossing number, net      23 94f
Crossing number, odd      9 27 96 100
Curvature condition      16 28 30 53 57
Curvature condition, genericity of      103 130f 134
Cusp      109
Cyclic group      6
Degenerate stationary solutions      98 114f
DERPER      109
Dihedral group      3 9297 104 112
Discrete wave      see Wave
Equivariant      see Assumptions Hopf E. Hopf H. Orbit Snake Stationary
Fermat's last theorem      19
Field Koros Noyes model      92
Fixed point subspace      5 10 12 15 45 131
Fixed point subspace and orbit index      81
Fixed point subspace, dimension of      101 102 104 108
Flip      30 57
Flip, doubling      30 55 5961
Flip, pitchfork      5961
Flip-flop      57
Flip-flop, doubling      5961 65f
Flip-flop, pitchfork      5961
FLOP      54
Flop, doubling      54 5961 65f
Floquet multipliers      29 91 109 129 131
Floquet multipliers and Floquet exponents      69
Floquet multipliers and orbit index      7082
Floquet multipliers and virtual periods      43
Floquet multipliers of rotating waves      134
Fluid dynamics      3 104
Fredholm map      35f 69 91 117f
Fredholm map, index      117f 123 130
Freezing      7 50f 67 8183 114 132f
Frequency doubling      102
Frozen wave      see Wave
Fruit of the Loom      130 133
Fuller index      9 29 110 112
GENERIC      13 (see also Approximation Localization Perturbations)
global      see Continuum Hopf E. Stationary
Graph of oscillators      102f
Haar measure      39
Hamiltonian systems      9 31 102
Heteroclinic orbit      103f
Heterogeneous oscillations      3 96105
Hint      65
Homoclinic orbit      106 113115
homogeneous solution      3 9398
Homogeneous solution, nonhomogeneous solution      3 96105
Homotopy invariance of Brouwer degree      23 29 110
Homotopy invariance of the Fuller index      110
Homotopy invariance of the Hopf index      23 106 112114
Homotopy invariance of the orbit index      13 32f 7077 81f 91
Hopf, E. bifurcation      see also Center
Hopf, E. bifurcation, global      1f 7 9f 110f
Hopf, E. bifurcation, global, equivariant      2f 10f 1526 2934 79f 82f 92104 110 113f
Hopf, E. bifurcation, history      9f
Hopf, E. bifurcation, local      1f 9f 79
Hopf, E. bifurcation, local, equivarint      3 10 45f 48f 100 106f
Hopf, E. bifurcation, local, infinite-dimensional      9
Hopf, E. bifurcation, local, planar      9
Hopf, E. index, global equivariant      2 (see also Center)
Hopf, H. theorem      110
Horseshoe      114
Hyperbolicity of periodic solutions and type 0, $\mathbb{O}_k$ solutions      31 127 129
Hyperbolicity of periodic solutions and virtual period/symmetry      11 96 100
Hyperbolicity of periodic solutions at generic centers      16 123f
Hyperbolicity of periodic solutions, rotating waves      51 133
Hypercycle      102
INDEX      see Homotopy invariance Fredholm Fuller Hopf E. Orbit
Induction over period/symmetry      116 124 127134
Infinitesimal rotation      7 25 49 81 88
Inhomogeneous      see Homogeneous
Instability      see Stability Unstable
Integral equations      12 38 103
Irreducible      see Representation
Isotropy group      4
Isotropy group, maximal      12 95 100f 104 106f
Isotropy group, submaximal      108 (see also Virtual)
Isotypic, action      119f
Isotypic, decomposition      22 74f 95 99
Iterated, maps      9 29 54 56f 62
Iterated, multiplication by two      11 18f
J-homomorphism      110
j-jug-handle      111f
Jug-handle      31f 110f
Kupka Smale theorem      13 31 48
Langmuir Hinshelwood kinetics      103
Laplace operator      97f
Laplace operator, discretized      92f 98
Laser equations      104
Lattices, and virtual symmetry      45
Lattices, crystal      103
Lattices, hexagonal      104
Linked periodic solutions      32
Ljapunov center theorem      102
Ljapunov Schmidf reduction and virtual symmetry      37 53
Ljapunov Schmidf reduction at rotating waves      50 132
Ljapunov Schmidf reduction at secondary bifurcations      66 130f
local      see Global
Localization of genericity      116 122 127f
Loop in parameter space      112f
Loop of periodic solutions      31f 111f
Lorenz equation      114
Manifolds of solutions      114f
Maximal      see Isotropy Torus
Minimal      see Period
Mode interaction      104
Models      92 102f
Multiparameter      see Bifurcation
Net crossing      see Crossing number
Network      see Graph
Neural nets      103
Non-free group action      5 110
Normal form      62 102 104 113f
Normal form, Arnold normal form of matrices      119121
Numerical, algorithms      109
Numerical, analysis      4 102 109
Numerical, simulation      98 100 103
Obstruction      112114
Odd      see Crossing number
Operator setting for periodic solutions      7 10 31 48 66 68
Orbit index, equivariant of discrete waves      70
Orbit index, equivariant of rotating waves      81
Orbit index, nonequivariant      32 (see also Floquet Homotopy Stability)
Orthogonal groups, $O(2) \times SO(2)$      104
Orthogonal groups, O(2)      14 104 108f 112 114
Orthogonal groups, O(3)      92 97101 107
Orthogonal groups, O(N)      2
Orthogonal groups, SO(2)      6 14 34 103 110 115
Parameters      see Bifurcation
Pattern      3f 6 97 99 104
Period      see also Virtual
Period, blowup      21 24 50 68 110114
Period, doubling      13 18 30f 47 5162 7276 81 110f 127 130
Period, doubling, cascade      114
Period, jump      13 29 59f 73 86 111f 127
Period, lower bound on      90 127 129
Period, minimal      5 12 31f 49f 73 110f
Period, minimal and transversality      31 116 124f
Period, upper bound on      132 134
Periodic solutions      see Bifurcation Hopf E. Hyperbolicity Linked Operator Poincare Symmetry
Perturbations      see also Generic Singular Transversality
Perturbations, breaking equivariance      13f 104 109
Perturbations, of centers      49 86 122f
Perturbations, of matrices      118f
Perturbations, of Poincare maps      62 66 124f
Perturbations, of rotating waves      7 132f
Perturbations, of stationary solutions      2831
Petri dish      6
Pitchfork      30 53f 57 59f 71f 109 112 127
PITCON      109
PL methods      109
Poincare map      29f 50 5568 118 124f 131 134
Poincare section      29 50 55 115 124 130
Poincare time      55 62f 125
Poincare Andronov Hopf bifurcation      see Hopf E.
Proper      117 123 130
Quaternions      114f
Questions      see also Answers
Questions, open      68 112 115
Questions, principal      2
Reaction diffusion systems      3 10 97101 109
Regularity      see Assumptions
Relative, equilibrium      115
Relative, periodic solution      115
Representation, cyclic      17 22 118f
Representation, irreducible      17 22 74f 109 110 118f
Representation, subspace      2 12 22f 75 81 95 99 120
Rescaling      7 35 7077 80
Resolvent      25f 69 91
Resonance      102 108 114
Roots of unity      30f 43 119 127 129 134
Rotating wave      see Wave
Rotation number      see Torus
Saddle-node bifurcation      31
Sard theorem      13 28 31 118
Sard theorem, Smale Sard theorem      118
Scylla & Charybdis      109
Silnikov homoclinic theorem      114
Singular perturbations      96 99 101
Singularity      4 13f 48 104 109 112f
Singularity, Arnold Bogdanov Takens      see B-point
Singularity, theory      9 104 106 109
Sink      80 83
Smale      see Horseshoe Sard
Snake      21
Snake, equivariant      79
Snake, global      33 79 82
Solution      see Manifold Periodic Stationary
Source      80 83
Speed      see Rotating wave
Spherical harmonics      98f 107
Spherical pendulum      102
Spiral cells      104
spirals      3f 99f 104f
Stability      104 108f
Stability and orbit index      104
Stability, exchange of      33 73 76 7779 104
Stationary solutions, bifurcation of      see also Bifurcation Degenerate Flop Saddle-node Turn
Stationary solutions, bifurcation of global      9 27f 34
Stationary solutions, bifurcation of global equivariant      10 110
Stationary solutions, bifurcation of local      9
Stationary solutions, bifurcation of local equivariant      10
Stratification      116
Stroboscope      66
Strongly monotone systems      109
Swallow-tail      109
Symmetry      see also Virtual
Symmetry, breaking      10 31 52 57 59f 66 102 112 127 131 134
Symmetry, of periodic solutions      6
Symmetry, of periodic solutions, control over      913 21 24 96f 110f
Syntheses      34 45 112
Target pattern      see Pattern
Taylor Couette flow      104
Temptation      64
Topological, approach      9f 13 29 68 103 106 109114
Topological, restrictions      4
Topological, results      3 9f 109f
Topological, charge      99f
Topology, of sets      84
Topology, uniform operator      25 73 77
Topology, weak      13 26 130
Torus, group      45 104
Torus, invariant      103f 114f
Torus, invariant, periodically foliated      59 104 115
Torus, invariant, primary bifurcation of      106
Torus, invariant, rotation number on      115
Torus, invariant, secondary bifurcation of      104f 114f
Torus, lattice      45
Torus, maximal      99f
Transversality      116134
Transversality, assumption      117 124 130f
Transversality, theorem      117 122 127 130134
Transverse crossing      16 28 30 53 57 71f 123 130f
Turing ring      2 7 9297 102105
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