119 research outputs found
On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions
We study the non-autonomously forced Burgers equation
on the space interval with two sets of the boundary conditions:
the Dirichlet and periodic ones. For both situations we prove that there exists
the unique bounded trajectory of this equation defined for all . Moreover we demonstrate that this trajectory attracts all
trajectories both in pullback and forward sense. We also prove that for the
Dirichlet case this attraction is exponential
Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation
We present an algorithm for the rigorous integration of Delay Differential
Equations (DDEs) of the form . As an application, we
give a computer assisted proof of the existence of two attracting periodic
orbits (before and after the first period-doubling bifurcation) in the
Mackey-Glass equation
Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof
We prove the existence of globally attracting solutions of the viscous
Burgers equation with periodic boundary conditions on the line for some
particular choices of viscosity and non-autonomous forcing. The attract- ing
solution is periodic if the forcing is periodic. The method is general and can
be applied to other similar partial differential equations. The proof is
computer assisted.Comment: 38 pages, 1 figur
Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations
We describe a topological method to study the dynamics of dissipative PDEs on
a torus with rapidly oscillating forcing terms. We show that a dissipative PDE,
which is invariant with respect to Galilean transformations, with a large
average initial velocity can be reduced to a problem with rapidly oscillating
forcing terms. We apply the technique to the Burgers equation, and the
incompressible 2D Navier-Stokes equations with a time-dependent forcing. We
prove that for a large initial average speed the equation admits a bounded
eternal solution, which attracts all other solutions forward in time. For the
incompressible 3D Navier-Stokes equations we establish existence of a locally
attracting solution
Connecting orbits for a singular nonautonomous real Ginzburg-Landau type equation
We propose a method for computation of stable and unstable sets associated to
hyperbolic equilibria of nonautonomous ODEs and for computation of specific
type of connecting orbits in nonautonomous singular ODEs. We apply the method
to a certain a singular nonautonomous real Ginzburg-Landau type equation, which
that arises from the problem of formation of spots in the Swift-Hohenberg
equation.Comment: 36 pages, 6 figure
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