815 research outputs found
Research in the general area of non-linear dynamical systems Final report, 8 Jun. 1965 - 8 Jun. 1967
Nonlinear dynamical systems research on systems stability, invariance principles, Liapunov functions, and Volterra and functional integral equation
Some remarks on stability for a phase-field model with memory
The phase field system with memory can be viewed as a phenomenological extension of the classical phase equations in which memory effects have been taken into account in both fields. Such memory effects could be important for example during phase transition in polymer melts in the proximity of the glass transition temperature where configurational degrees of freedom in the polymer melt constitute slowly relaxing "internal modes" which are di±cult to model explicitly. They should be relevant in particular to glass-liquid-glass transitions where re-entrance effects have been recently reported [27]. We note that in numerical studies based on sharp interface equations obtained from (PFM), grains have been seen to rotate as they shrink [35, 36]. While further modelling and numerical efforts are now being undertaken, the present manuscript is devoted to strengthening the analytical underpinnings of the model
On Lyapunov-type inequality for a class of quasilinear systems
In this paper, we establish a new Lyapunov-type inequality for quasilinear systems. Our result in special case reduces to the result of Watanabe et al. [J. Inequal. Appl. 242(2012), 1-8]. As an application, we also obtain lower bounds for the eigenvalues of corresponding systems
The convergence problem for dissipative autonomous systems: classical methods and recent advances
The initial motivation of this text was to provide an up to date translation
of the monograph [45] written in french by the first author, taking account of
more recent developments of infinite dimensional dynamics based on the
{\L}ojasiewicz gradient inequality. In order to keep the present work within
modest size bounds and to make it available to the readers without too much
delay, we decided to make a first volume entirely dedicated to the so-called
convergence problem for autonomous systems of dissipative type. We hope that
this volume will help the interested reader to make the connection between the
rather simple background developed in the french monograph and the rather
technical specialized literature on the convergence problem which grew up
rather fast in the recent years
Existence, Uniqueness and Stability of Solutions of a Class of Nonlinear Partial Differential Equations
In this work we present a unified approach for treating the existence, uniqueness and asymptotic stability of classical solutions for a class of nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems
Stationary Multiple Spots for Reaction-Diffusion Systems
In this paper, we review
analytical methods for a rigorous study of the
existence and stability of stationary, multiple
spots for reaction-diffusion systems. We will
consider two classes of reaction-diffusion
systems: activator-inhibitor systems (such as
the Gierer-Meinhardt system) and
activator-substrate systems (such as the
Gray-Scott system or the Schnakenberg model).
The main ideas are presented in the context of
the Schnakenberg model, and these results are
new to the literature.
We will consider the systems in a
two-dimensional, bounded and smooth domain for small diffusion
constant of the activator.
Existence of multi-spots is proved using tools
from nonlinear functional analysis such as
Liapunov-Schmidt reduction and fixed-point
theorems. The amplitudes and positions of spots
follow from this analysis.
Stability is shown in two parts, for
eigenvalues of order one and eigenvalues
converging to zero, respectively. Eigenvalues
of order one are studied by deriving their
leading-order asymptotic behavior and reducing
the eigenvalue problem to a nonlocal eigenvalue
problem (NLEP). A study of the NLEP reveals a
condition for the maximal number of stable
spots.
Eigenvalues converging to zero are investigated
using a projection similar to Liapunov-Schmidt
reduction and conditions on the positions for
stable spots are derived. The Green's function
of the Laplacian plays a central role in the
analysis.
The results are interpreted in the biological,
chemical and ecological contexts. They are
confirmed by numerical simulations
Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization
Short time existence for a surface diffusion evolution equation with
curvature regularization is proved in the context of epitaxially strained
three-dimensional films. This is achieved by implementing a minimizing movement
scheme, which is hinged on the -gradient flow structure underpinning
the evolution law. Long-time behavior and Liapunov stability in the case of
initial data close to a flat configuration are also addressed.Comment: 44 page
Hopf bifurcations in time-delay systems with band-limited feedback
We investigate the steady-state solution and its bifurcations in time-delay
systems with band-limited feedback. This is a first step in a rigorous study
concerning the effects of AC-coupled components in nonlinear devices with
time-delayed feedback. We show that the steady state is globally stable for
small feedback gain and that local stability is lost, generically, through a
Hopf bifurcation for larger feedback gain. We provide simple criteria that
determine whether the Hopf bifurcation is supercritical or subcritical based on
the knowledge of the first three terms in the Taylor-expansion of the
nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the
steady state is shown, which indicates possible quasiperiodic and chaotic
dynamics in these systems. As a result of this investigation, we find that
AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda
et al., Physica D 29 (1987) 223-235] already at the level of steady-state
bifurcations, e.g. bifurcations exist in which limit cycles are created with
periods other than the fundamental ``period-2'' mode found in Ikeda-type
systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear
Phenomen
- …