Uniqueness of Kubo-Martin-Schwinger states for classical dynamical systems with infinite-dimensional phase space

1980 ◽  
Vol 44 (2) ◽  
pp. 697-702
Author(s):  
A. A. Arsen'ev
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Infinite Dimensional ◽  
Dimensional Phase Space

DISSIPATIVE SOLITON PULSATIONS WITH PERIODS BEYOND THE LASER CAVITY ROUND TRIP TIME

2005 ◽  
Vol 14 (02) ◽  
pp. 177-194 ◽  
Author(s):  
N. AKHMEDIEV ◽  
J. M. SOTO-CRESPO ◽  
M. GRAPINET ◽  
Ph. GRELU
Keyword(s):  
Phase Space ◽  
Nonlinear System ◽  
Fiber Laser ◽  
Laser Cavity ◽  
Continuous Model ◽  
Round Trip Time ◽  
Round Trip ◽  
Infinite Dimensional ◽  
Trip Time ◽  
Dimensional Phase Space

We review recent results on periodic pulsations of the soliton parameters in a passively mode-locked fiber laser. Solitons change their shape, amplitude, width and velocity periodically in time. These pulsations are limit cycles of a dissipative nonlinear system in an infinite-dimensional phase space. Pulsation periods can vary from a few to hundreds of round trips. We present a continuous model of a laser as well as a model with parameter management. The results of the modeling are supported with experimental results obtained using a fiber laser.

scholarly journals Attracting sets for one class of asymptotically compact systems with pulsed perturbation

2021 ◽  
pp. 140-148
Author(s):  
Oleksiy Kapustyan ◽  
Nataliia Gorban
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Compact Semigroup ◽  
Global Attractors ◽  
Energy Functional ◽  
Weakly Nonlinear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Limit Value ◽  
Fixed Subset

The authors consider the pulsed dynamical systems generated by evolutionary processes. The trajectories of these processes undergo the pulsed perturbation when the energy functional reaches some fixed limit value.  The generalization of the classical theory of global attractors of infinite dimensional dynamical systems in case of systems with impulse actions is carried out.  It is established that for the dissipative pulsed dynamical system generated by the asymptotically compact semigroup, there exists a uniform attractor, i.e., a compact uniformly attracting set, minimal among all such sets in the phase space of the system. The result is applied to the weakly nonlinear wave equation with dissipation, the trajectories of which are subjected to impulsive perturbations upon attainment of a certain fixed subset in the phase space, so called the impulse set.

scholarly journals DYNAMICAL ASPECTS OF LIE-POISSON STRUCTURES

1993 ◽  
Vol 08 (31) ◽  
pp. 2973-2987 ◽  
Author(s):  
F. LIZZI ◽  
G. MARMO ◽  
G. SPARANO ◽  
P. VITALE
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Poisson Bracket ◽  
Lie Groups ◽  
Hamiltonian Systems ◽  
Quantum Groups ◽  
Equations Of Motion ◽  
Poisson Structures ◽  
Dimensional Phase Space ◽  
Deformation Procedure

Quantum groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at SU(2) and SU(1, 1), as submanifolds of a four-dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some Hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure.

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