Dichotomies in Stability Theory

Controlling the Unsteady Analogue of Saddle Stagnation Points

“…Now, the theory discussed previously applies to this situation with hardly any difference. The hyperbolicity of the unperturbed trajectory would need to be expressed in terms of exponential dichotomies [10,35,32], and once again, the persistence of exponential dichotomies under perturbations [10,53,51] ensures that the system (4.1) possesses a nearby hyperbolic trajectory a ε (t) as long as sufficient smoothness and boundedness of v 1 is present. Thus, the results upon which section 2 are premised [4] extend to this situation as well.…”

“…The new hyperbolic trajectory (a ε (t), t) is a "wobbled" version of (a, t). The retention of stable and unstable manifolds also follows directly from the persistence of exponential dichotomies, since these manifolds are locally represented using the projection matrices guaranteed by exponential dichotomies [10,53]. These manifolds are no longer uniform in t and now form time-varying flow separators.…”

“…The flow-regulating trajectory is a time-varying entity in this moving frame and does not correspond to a location at which the fluid velocity is zero. A theoretical definition for hyperbolic trajectories can be given in terms of exponential dichotomies [10,51,41,32,35,7,5], but such a definition is generally difficult to use to locate hyperbolic trajectories even for a specified unsteady Eulerian velocity field. While this hinders the analysis and numerical determination of hyperbolic trajectories, there is 2.…”

“…When ε = 0, the solution (a, t) of (2.2) perturbs to a nearby solution (a ε (t), t) of (2.1), which turns out to continue to be hyperbolic in the following sense. In nonautonomous systems such as (2.1), hyperbolicity needs to be defined in terms of exponential dichotomies [10,41,32,35] of the associated variational equation…”

Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

“…Instead there is an n-dimensional center space, which makes the study of slow eigenvalues more difficult. For an introduction to exponential dichotomies, see [5,23,26,27]. A variant of exponential dichotomies with exponential rate approaching infinity is used in Lemma 5.2.…”