570 research outputs found
Dissipative periodic and chaotic patterns to the KdV--Burgers and Gardner equations
We investigate the KdV-Burgers and Gardner equations with dissipation and
external perturbation terms by the approach of dynamical systems and
Shil'nikov's analysis. The stability of the equilibrium point is considered,
and Hopf bifurcations are investigated after a certain scaling that reduces the
parameter space of a three-mode dynamical system which now depends only on two
parameters. The Hopf curve divides the two-dimensional space into two regions.
On the left region the equilibrium point is stable leading to dissapative
periodic orbits. While changing the bifurcation parameter given by the velocity
of the traveling waves, the equilibrium point becomes unstable and a unique
stable limit cycle bifurcates from the origin. This limit cycle is the result
of a supercritical Hopf bifurcation which is proved using the Lyapunov
coefficient together with the Routh-Hurwitz criterion. On the right side of the
Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using
Shil'nikov's theorem which requires the construction of a homoclinic orbit,
while for the Gardner equation the supercritical Hopf bifurcation leads only to
a stable periodic orbit.Comment: 13 pages, 12 figure
Thermosolutal and binary fluid convection as a 2 x 2 matrix problem
We describe an interpretation of convection in binary fluid mixtures as a
superposition of thermal and solutal problems, with coupling due to advection
and proportional to the separation parameter S. Many properties of binary fluid
convection are then consequences of generic properties of 2 x 2 matrices. The
eigenvalues of 2 x 2 matrices varying continuously with a parameter r undergo
either avoided crossing or complex coalescence, depending on the sign of the
coupling (product of off-diagonal terms). We first consider the matrix
governing the stability of the conductive state. When the thermal and solutal
gradients act in concert (S>0, avoided crossing), the growth rates of
perturbations remain real and of either thermal or solutal type. In contrast,
when the thermal and solutal gradients are of opposite signs (S<0, complex
coalescence), the growth rates become complex and are of mixed type.
Surprisingly, the kinetic energy of nonlinear steady states is governed by an
eigenvalue problem very similar to that governing the growth rates. There is a
quantitative analogy between the growth rates of the linear stability problem
for infinite Prandtl number and the amplitudes of steady states of the minimal
five-variable Veronis model for arbitrary Prandtl number. For positive S,
avoided crossing leads to a distinction between low-amplitude solutal and
high-amplitude thermal regimes. For negative S, the transition between real and
complex eigenvalues leads to the creation of branches of finite amplitude, i.e.
to saddle-node bifurcations. The codimension-two point at which the saddle-node
bifurcations disappear, separating subcritical from supercritical pitchfork
bifurcations, is exactly analogous to the Bogdanov codimension-two point at
which the Hopf bifurcations disappear in the linear problem
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
pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual
pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path
for elliptic systems of PDEs over bounded 2D domains, based on the Matlab
pdetoolbox. The new features include a more efficient use of FEM, easier
switching between different single parameter continuations, genuine
multi-parameter continuation (e.g., fold continuation), more efficient
implementation of nonlinear boundary conditions, cylinder and torus geometries
(i.e., periodic boundary conditions), and a general interface for adding
auxiliary equations like mass conservation or phase equations for continuation
of traveling waves. The package (library, demos, manuals) can be downloaded at
www.staff.uni-oldenburg.de/hannes.uecker/pde2pat
Bifurcation from relative periodic solutions
Published versio
Nonintegrable Schrodinger Discrete Breathers
In an extensive numerical investigation of nonintegrable translational motion
of discrete breathers in nonlinear Schrodinger lattices, we have used a
regularized Newton algorithm to continue these solutions from the limit of the
integrable Ablowitz-Ladik lattice. These solutions are shown to be a
superposition of a localized moving core and an excited extended state
(background) to which the localized moving pulse is spatially asymptotic. The
background is a linear combination of small amplitude nonlinear resonant plane
waves and it plays an essential role in the energy balance governing the
translational motion of the localized core. Perturbative collective variable
theory predictions are critically analyzed in the light of the numerical
results.Comment: 42 pages, 28 figures. to be published in CHAOS (December 2004
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