Search Results for « Linear dynamical stability »

We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian motion with its Lévy area) as rough path noise the solution coincides with the standard variational solution of the SPDE.

As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently by Baird--Danielo and Lott: nongradient homogeneous expanding Ricci solitons on nilpotent or solvable Lie groups. For all explicitly known nonproduct examples, we demonstrate linear stability of the flow at these fixed points and prove that the linearizations generate strongly continuous semigroups.

We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner-King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense, and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing, and minimal self joinings. The last two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.

Recent experiments in hypersonic vehicle technology have had some successful outcomes (ex. NASA X-43, Boeing X-51) but also have seen some unexpected failures (DARPA Hypersonic Test Vehicle 2). One important failure mode is the problem of engine unstart. Morgan, et al (2012) has discussed and analyzed a Scramjet isolator using two unit problems, viz. that of an oblique shock impinging on a turbulent boundary layer and that of a normal shock train in a constant-area duct. In order to analyze unstart, we use the numerical data obtained from the LES calculations of the normal shock train in a constant area duct and perform temporal linear stability analysis on it under the assumptions of inviscid semi-parallel flow. This paper describes the methodology and the details of this linear stability analysis. The results obtained from the investigations have been documented and interpreted with their relevance to the physical problem. The report concludes that temporal linear stability analysis is insufficient to analyze the problem of unstart. Finally, the need for a new approach (namely, dynamic mode decomposition (DMD)) has been discussed.