2,657 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
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
On the stationarity of linearly forced turbulence in finite domains
A simple scheme of forcing turbulence away from decay was introduced by
Lundgren some time ago, the `linear forcing', which amounts to a force term
linear in the velocity field with a constant coefficient. The evolution of
linearly forced turbulence towards a stationary final state, as indicated by
direct numerical simulations (DNS), is examined from a theoretical point of
view based on symmetry arguments. In order to follow closely the DNS the flow
is assumed to live in a cubic domain with periodic boundary conditions. The
simplicity of the linear forcing scheme allows one to re-write the problem as
one of decaying turbulence with a decreasing viscosity. Scaling symmetry
considerations suggest that the system evolves to a stationary state, evolution
that may be understood as the gradual breaking of a larger approximate symmetry
to a smaller exact symmetry. The same arguments show that the finiteness of the
domain is intimately related to the evolution of the system to a stationary
state at late times, as well as the consistency of this state with a high
degree of isotropy imposed by the symmetries of the domain itself. The
fluctuations observed in the DNS for all quantities in the stationary state can
be associated with deviations from isotropy. Indeed, self-preserving isotropic
turbulence models are used to study evolution from a direct dynamical point of
view, emphasizing the naturalness of the Taylor microscale as a self-similarity
scale in this system. In this context the stationary state emerges as a stable
fixed point. Self-preservation seems to be the reason behind a noted similarity
of the third order structure function between the linearly forced and freely
decaying turbulence, where again the finiteness of the domain plays an
significant role.Comment: 15 pages, 7 figures, changes in the discussion at the end of section
VI, formula (60) correcte
- …