Classification of soliton solutions in an infinite-dimensional phase space on the basis of the theory of dynamical systems

1985 ◽  
Vol 65 (3) ◽  
pp. 1234-1239 ◽  
Author(s):  
V. M. Eleonskii ◽  
N. E. Kulagin ◽  
N. S. Novozhilova
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Soliton Solutions ◽  
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.

Discontinuity-induced bifurcations of piecewise smooth dynamical systems

2010 ◽  
Vol 368 (1930) ◽  
pp. 4915-4935 ◽  
Author(s):  
M. di Bernardo ◽  
S. J. Hogan
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Limit Cycles ◽  
State Of The Art ◽  
Piecewise Smooth ◽  
Hybrid Dynamical Systems ◽  
Current State ◽  
Relevant Class

This paper presents an overview of the current state of the art in the analysis of discontinuity-induced bifurcations (DIBs) of piecewise smooth dynamical systems, a particularly relevant class of hybrid dynamical systems. Firstly, we present a classification of the most common types of DIBs involving non-trivial interactions of fixed points and equilibria of maps and flows with the manifolds in phase space where the system is non-smooth. We then analyse the case of limit cycles interacting with such manifolds, presenting grazing and sliding bifurcations. A description of possible classification strategies to predict and analyse the scenarios following such bifurcations is also discussed, with particular attention to those methodologies that can be applied to generic n -dimensional systems.

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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