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Exponential dichotomy of a strongly continuous cocycle $bFi$ is proved to be equivalent to existence of a Ma~{n}e sequence either for $bFi$ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension.
The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n in mathbb{Z}$ such that $T^n A cap A ot = varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations wi
Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such
We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+varepsilon G$, on the parameter $varepsilon$. In particular, we study the limit a
We study two-dimensional stochastic differential equations (SDEs) of McKean--Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such S
We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter $epsilon$ uncouples the system at $epsilon=0$. Using a normal form for $N=2$ identical systems undergoing Hopf bifurcation, w