The theory of nonautonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this book. The focus is on dissipative systems and nonautonomous attractors, in particular the recently introduced concept of pullback attractors. Linearization theory, invariant manifolds, Lyapunov functions, Morse decompositions and bifurcations for nonautonomous systems and set-valued generalizations are also considered as well as applications to numerical approximations, switching systems and synchronization. Parallels with corresponding theories of control and random dynamical systems are briefly sketched. With its clear and systematic exposition, many examples and exercises, as well as its interesting applications, this book can serve as a text at the beginning graduate level. It is also useful for those who wish to begin their own independent research in this rapidly developing area.

Nonautonomous Dynamical Systems

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in Charrier et al. (SIAM J Numer Anal, 2013), to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Frechet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen---Loeve expansion of the random coefficient. Numerical results complete the paper.

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Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

Oscillation of third-order neutral differential equations

On the oscillation of higher-order half-linear delay differential equations

Deterioration of European apartment buildings

On the oscillation of certain third order nonlinear functional differential equations

We consider the two-step bifurcation scenario which has been studied by L. Arnold and his co-workers. We formulate a "continuous case" and a "measurable case" of the scenario, and present results and conjectures regarding sufficient conditions that it take place.

Two-step transition in nonautonomous bifurcations: an explanation

A method to compute the volume of a molecule

We give sufficient conditions for local solutions to some fourth order semilinear ordinary differential equations to blow up in finite time with wide oscillations, a phenomenon not visible for lower order equations. The result is then applied to several classes of semilinear partial differential equations in order to characterize the blow up of solutions including, in particular, its applications to a suspension bridge model. We also give numerical results which describe this oscillating blow up and allow us to suggest several open problems and to formulate some related conjectures.

Wide Oscillation Finite Time Blow Up for Solutions to Nonlinear Fourth Order Differential Equations

We adopt the multilevel Monte Carlo method introduced by M. Giles (Multilevel Monte Carlo path simulation, Oper. Res. 56(3):607–617, 2008) to SDEs with additive fractional noise of Hurst parameter H>1/2. For the approximation of a Lipschitz functional of the terminal state of the SDE we construct a multilevel estimator based on the Euler scheme. This estimator achieves a prescribed root mean square error of order e with a computational effort of order e−2.

Raffaella Pavani

Papers

On the oscillation of certain third order nonlinear functional differential equations

Two-step transition in nonautonomous bifurcations: an explanation

A method to compute the volume of a molecule

Wide Oscillation Finite Time Blow Up for Solutions to Nonlinear Fourth Order Differential Equations

Multilevel Monte Carlo for stochastic differential equations with additive fractional noise