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Ramussen M. — Attractivity and Bifurcation for Nonautonomous Dynamical Systems
Ramussen M. — Attractivity and Bifurcation for Nonautonomous Dynamical Systems

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Название: Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Автор: Ramussen M.

Аннотация:

Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this book, two different approaches are developed which are based on special definitions of local attractivity and repulsivity. It is shown that these notions lead to nonautonomous Morse decompositions, which are useful to describe the global asymptotic behavior of systems on compact phase spaces. Furthermore, methods from the qualitative theory for linear and nonlinear systems are derived, and nonautonomous counterparts of the classical one-dimensional autonomous bifurcation patterns are developed.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 212

Добавлена в каталог: 13.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
(p, T)-attractive, definition      21
(p, T)-attractive, linearized attractivity      128
(p, T)-attractor, definition      19
(p, T)-attractor, example      19 20
(p, T)-bifurcation, adiabatic bifurcation theory      133
(p, T)-bifurcation, definition      44
(p, T)-bifurcation, example      46 47 141 150
(p, T)-dichotomy spectrum, definition      98
(p, T)-dichotomy spectrum, of autonomous systems      109
(p, T)-dichotomy, definition      83
(p, T)-dichotomy, nonhyperbolic      84
(p, T)-dichotomy, Roughness Theorem      113
(p, T)-repeller, definition      19 22
(p, T)-repeller, example      19 20
(p, T)-repulsive, definition      21
(p, T)-repulsive, linearized repulsivity      128
(p, T)-transition      44
Adiabatic solution, definition      131
Adiabatic solution, existence      131
Adiabatic system      130
All-time attractive, definition      21
All-time attractive, linearized attractivity      126
All-time attractor, definition      18
All-time attractor, example      18 19
All-time bifurcation, definition      43
All-time bifurcation, dichotomy spectrum of linearization      125
All-time bifurcation, example      46 47 137 144
All-time dichotomy spectrum, corresponding Morse decomposition      106
All-time dichotomy spectrum, definition      98
All-time dichotomy spectrum, example      102
All-time dichotomy spectrum, spectral manifolds      105
All-time exponential dichotomy, definition      83
All-time exponential dichotomy, nonhyperbolic      84
All-time exponential dichotomy, Roughness Theorem      112
All-time exponential dichotomy, uniqueness of the invariant projector      86
All-time repeller, definition      18 22
All-time repeller, example      18 19
All-time repulsive, definition      21
All-time repulsive, linearized repulsivity      126
All-time transition, definition      43
All-time transition, example      144
Attractor      40
Center manifold reduction      48 179
Cocycle      9
D-bifurcation      49
Domain of all-time attraction      26
Domain of all-time repulsion      26
Domain of future attraction      25
Domain of future repulsion      26
Domain of past attraction      25
Domain of past repulsion      25
Duffing-van der Pol oscillator      190
Dynamical system      9
Euclidean norm      8
Euclidean scalar product      8
Exponential dichotomy      83
Forward attractor      41
Future asymptotically autonomous      153
Future attractive, definition      21
Future attractive, linearized attractivity      126
Future attractor, definition      16
Future attractor, example      18
Future attractor, existence      36
Future attractor, nonuniqueness      32
Future attractor-repeller pair, definition      53
Future attractor-repeller pair, forward convergence      55
Future attractor-repeller pair, Lyapunov function      62
Future attractor-repeller pair, nonuniqueness      56
Future attractor-repeller pair, pushforward convergence      55
Future bifurcation, definition      43
Future bifurcation, dichotomy spectrum of linearization      125
Future bifurcation, example      45 47 137 144 177 181 189
Future dichotomy spectrum, corresponding Morse decomposition      106
Future dichotomy spectrum, definition      98
Future dichotomy spectrum, example      102
Future dichotomy spectrum, Lyapunov spectrum      107
Future dichotomy spectrum, spectral manifolds      105
Future exponential dichotomy, definition      83
Future exponential dichotomy, dynamical properties      86
Future exponential dichotomy, nonhyperbolic      84
Future exponential dichotomy, nonuniqueness of the invariant projector      86 97
Future exponential dichotomy, Roughness Theorem      112
Future Morse decomposition definition      57
Future Morse decomposition finer      58
Future Morse decomposition finest      78
Future Morse decomposition forward convergence      61 65 75
Future Morse decomposition Lyapunov function      63
Future Morse decomposition nonuniqueness      66 77
Future Morse decomposition pushforward convergence      59 65 76
Future rejecting      30
Future repeller, definition      16 22
Future repeller, example      18
Future repeller, existence      31 36
Future repeller, uniqueness      31
Future repulsive, definition      21
Future repulsive, linearized repulsivity      126
Future transition, definition      43
Future transition, example      144 177 189
General solution      194
Hausdorff distance      8
Hausdorff semi-distance      8
Hopf bifurcation, autonomous      181
Hopf bifurcation, nonautonomous      188 189
Invariant projector, definition      82
Invariant projector, null space      82
Invariant projector, range      82
lim inf      11
lim sup      11
Lorenz equation      178
Lyapunov exponent      106
Lyapunov spectrum      106
Lyapunov stability      39
Lyapunov — Schmidt reduction      48
Morse set      57
Nonautonomous dynamical system, definition      9
Nonautonomous dynamical system, examples      9
Nonautonomous dynamical system, linear      81
Nonautonomous invariant manifold, all-time      117
Nonautonomous invariant manifold, center      124
Nonautonomous invariant manifold, center-stable      124
Nonautonomous invariant manifold, center-unstable      124
Nonautonomous invariant manifold, definition      116
Nonautonomous invariant manifold, future      123
Nonautonomous invariant manifold, linear      68 81
Nonautonomous invariant manifold, past      122
Nonautonomous invariant manifold, stable      124
Nonautonomous invariant manifold, strongly stable      124
Nonautonomous invariant manifold, strongly unstable      124
Nonautonomous invariant manifold, uniqueness      124
Nonautonomous invariant manifold, unstable      124
Nonautonomous set      11
Orbit      11
Ordinary differential equation      193
P-bifurcation      49
Past absorbing      30
Past asymptotically autonomous      153
Past attractive, definition      21
Past attractive, linearized attractivity      126
Past attractor, definition      13
Past attractor, example      15
Past attractor, existence      31 36
Past attractor, uniqueness      31
Past attractor-repeller pair, backward convergence      55
Past attractor-repeller pair, definition      52
Past attractor-repeller pair, Lyapunov function      62
Past attractor-repeller pair, nonuniqueness      56
Past attractor-repeller pair, pullback convergence      55
Past bifurcation, definition      42
Past bifurcation, dichotomy spectrum of linearization      125
Past bifurcation, example      45 47 137 144 175 180 188
Past dichotomy spectrum, corresponding Morse decomposition      106
Past dichotomy spectrum, definition      98
Past dichotomy spectrum, example      102
Past dichotomy spectrum, Lyapunov spectrum      107
Past dichotomy spectrum, spectral manifolds      105
Past exponential dichotomy, definition      82
Past exponential dichotomy, dynamical properties      85
Past exponential dichotomy, nonhyperbolic      84
Past exponential dichotomy, nonuniqueness of the invariant projector      86 97
Past exponential dichotomy, Roughness Theorem      112
Past Morse decomposition, backward convergence      60 64 75
Past Morse decomposition, definition      57
Past Morse decomposition, finer      58
Past Morse decomposition, finest      78
Past Morse decomposition, Lyapunov function      63
Past Morse decomposition, nonuniqueness      65 77
Past Morse decomposition, pullback convergence      59 64 75
Past repeller, definition      13 21
Past repeller, example      15
Past repeller, existence      36
Past repeller, nonuniqueness      32
Past repulsive, definition      21
Past repulsive, linearized repulsivity      126
Past transition, definition      42
Past transition, example      144 175 188
Pitchfork bifurcation, autonomous      149 175
Pitchfork bifurcation, nonautonomous      45 144 150 175 177
Projective space      68 196
Pullback attractor      41
Quasibounded      8
Radius of (p, T)-attraction      23
Radius of (p, T)-repulsion      23
Radius of all-time attraction      23
Radius of all-time repulsion      23
Radius of future attraction      22
Radius of future repulsion      22
Radius of past attraction      22
Radius of past repulsion      22
Repeller      40
Resolvent set      99
Saddle node bifurcation, autonomous      180
Saddle node bifurcation, nonautonomous      180 181
Singularly-perturbed system      130
Solution      21
Spectral theorem      101
Theorem of Selgrade      78
Time reversal      28
Transcritical bifurcation, autonomous      140 180
Transcritical bifurcation, nonautonomous      46 137 141 180 181
Transition operator      194
Uniform attractor      41
Variation of constants      194
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