295 research outputs found
Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations
The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization
dynamics in the presence of an applied field and a spin polarized current. In
the case of axial symmetry and with focus on one space dimension, we
investigate the emergence of space-time patterns in the form of wavetrains and
coherent structures, whose local wavenumber varies in space. A major part of
this study concerns existence and stability of wavetrains and of front- and
domain wall-type coherent structures whose profiles asymptote to wavetrains or
the constant up-/down-magnetizations. For certain polarization the Slonczewski
term can be removed which allows for a more complete charaterization, including
soliton-type solutions. Decisive for the solution structure is the polarization
parameter as well as size of anisotropy compared with the difference of field
intensity and current intensity normalized by the damping
Control of integrable Hamiltonian systems and degenerate bifurcations
In this dissertation, we study the control of near-integrable systems. A near-integrable system is one whose phase space has a similar structure to an integrable system during short time periods and for some parameter regime. We begin by studying the control of integrable Hamiltonian systems. The controller targets an exact solution to the integrable system using dissipative and conservative terms. We find that a Takens-Bogdanov bifurcation occurs in the limit of no dissipative control. The presence of a Takens-Bogdanov bifurcation implies that the control is highly susceptible to noise. We illustrate our results using a two- and four-dimensional integrable systems generated by low order truncations of solutions to the nonlinear Schrodinger equation (NLS). We then extend our results to a near-integrable system related to the NLS; the Ginzburg-Landau equation. We attempt to control the Ginzburg-Landau equation to a plane wave solution of the NLS. We show that for a certain parameter regime; a Takens-Bogdanov bifurcation occurs in the limit of no dissipative control. Through this example, we show that solutions of integrable systems can be viable control targets for related near-integrable systems
Hysteresis in Adiabatic Dynamical Systems: an Introduction
We give a nontechnical description of the behaviour of dynamical systems
governed by two distinct time scales. We discuss in particular memory effects,
such as bifurcation delay and hysteresis, and comment the scaling behaviour of
hysteresis cycles. These properties are illustrated on a few simple examples.Comment: 28 pages, 10 ps figures, AMS-LaTeX. This is the introduction of my
Ph.D. dissertation, available at
http://dpwww.epfl.ch/instituts/ipt/berglund/these.htm
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
Complex Patterns in Extended Oscillatory Systems
Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenüber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und Stabilitätsanalyse werden Instabilitäten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der Grenzübergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklärt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der Kalziumsignalübertragung in Zellen identifiziert
Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the
simplest microscopic description of the thermodynamic transition from a fluid
state to a crystalline state. The resulting phase field crystal model describes
a variety of spatially localized structures, in addition to different spatially
extended periodic structures. The location of these structures in the
temperature versus mean order parameter plane is determined using a combination
of numerical continuation in one dimension and direct numerical simulation in
two and three dimensions. Localized states are found in the region of
thermodynamic coexistence between the homogeneous and structured phases, and
may lie outside of the binodal for these states. The results are related to the
phenomenon of slanted snaking but take the form of standard homoclinic snaking
when the mean order parameter is plotted as a function of the chemical
potential, and are expected to carry over to related models with a conserved
order parameter.Comment: 40 pages, 13 figure
Critical behavior of a Ginzburg-Landau model with additive quenched noise
We address a mean-field zero-temperature Ginzburg-Landau, or \phi^4, model
subjected to quenched additive noise, which has been used recently as a
framework for analyzing collective effects induced by diversity. We first make
use of a self-consistent theory to calculate the phase diagram of the system,
predicting the onset of an order-disorder critical transition at a critical
value {\sigma}c of the quenched noise intensity \sigma, with critical exponents
that follow Landau theory of thermal phase transitions. We subsequently perform
a numerical integration of the system's dynamical variables in order to compare
the analytical results (valid in the thermodynamic limit and associated to the
ground state of the global Lyapunov potential) with the stationary state of the
(finite size) system. In the region of the parameter space where metastability
is absent (and therefore the stationary state coincide with the ground state of
the Lyapunov potential), a finite-size scaling analysis of the order parameter
fluctuations suggests that the magnetic susceptibility diverges quadratically
in the vicinity of the transition, what constitutes a violation of the
fluctuation-dissipation relation. We derive an effective Hamiltonian and
accordingly argue that its functional form does not allow to straightforwardly
relate the order parameter fluctuations to the linear response of the system,
at odds with equilibrium theory. In the region of the parameter space where the
system is susceptible to have a large number of metastable states (and
therefore the stationary state does not necessarily correspond to the ground
state of the global Lyapunov potential), we numerically find a phase diagram
that strongly depends on the initial conditions of the dynamical variables.Comment: 8 figure
Self-Organization, Coherence and Turbulence in Laser Optics
In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubic-quintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent “internal modes” that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure, namely, a dissipative Raman soliton
- …