322 research outputs found
Homoclinic Orbits In Slowly Varying Oscillators
We obtain existence and bifurcation theorems for homoclinic orbits in three-dimensional flows that are perturbations of families of planar Hamiltonian systems. The perturbations may or may not depend explicitly on time. We show how the results on periodic orbits of the preceding paper are related to the present homoclinic results, and apply them to a periodically forced Duffing equation with weak
feedback
Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit
We apply the dynamical approach to the study of the second order semi-linear
elliptic boundary value problem in a cylindrical domain with a small parameter
at the second derivative with respect to the "time" variable corresponding to
the axis of the cylinder.
We prove that, under natural assumptions on the nonlinear interaction
function and the external forces, this problem possesses the uniform attractors
and that these attractors tend to the attractor of the limit parabolic
equation. Moreover, in case where the limit attractor is regular, we give the
detailed description of the structure of these uniform attractors when the
perturbation parameter is small enough, and estimate the symmetric distance
between the perturbed and non-perturbed attractors
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
Asymptotic behavior of periodic solutions in one-parameter families of Li\'{e}nard equations
In this paper, we consider one--parameter () families of Li\'enard
differential equations. We are concerned with the study on the asymptotic
behavior of periodic solutions for small and large values of . To
prove our main result we use the relaxation oscillation theory and a
topological version of the averaging theory. More specifically, the first one
is appropriate for studying the periodic solutions for large values of
and the second one for small values of . In particular, our
hypotheses allow us to establish a link between these two theories
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
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
Quasistatic dynamical systems
We introduce the notion of a quasistatic dynamical system, which generalizes
that of an ordinary dynamical system. Quasistatic dynamical systems are
inspired by the namesake processes in thermodynamics, which are idealized
processes where the observed system transforms (infinitesimally) slowly due to
external influence, tracing out a continuous path of thermodynamic equilibria
over an (infinitely) long time span. Time-evolution of states under a
quasistatic dynamical system is entirely deterministic, but choosing the
initial state randomly renders the process a stochastic one. In the
prototypical setting where the time-evolution is specified by strongly chaotic
maps on the circle, we obtain a description of the statistical behaviour as a
stochastic diffusion process, under surprisingly mild conditions on the initial
distribution, by solving a well-posed martingale problem. We also consider
various admissible ways of centering the process, with the curious conclusion
that the "obvious" centering suggested by the initial distribution sometimes
fails to yield the expected diffusion.Comment: 40 page
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