We consider random dynamical systems with bounded noise and their associated set-valued mappings. Broadly speaking, the set-valued mapping, associated to a random dynamical system with bounded noise, maps an initial condition to the set of all possible outcomes of the application of the random dynamical system.


In Chapter \ref{ch:ews} we consider the topological bifurcations of some special sets defined for set-valued mappings, called minimal invariant sets. A minimal invariant set is left invariant by the application of a set-valued map and does not contain proper invariant sets. By the results provided by Lamb, Rasmussen, and Rodrigues in \cite{LambRasmussenRodrigues}, a topological bifurcation of a minimal invariant set models a, so called, critical transition, i.e. a drastic change of the state of a physical system.

We will show that the derivative of some special maps, called the extremal maps, describe the bifurcation of a minimal invariant set. Based on the derivative of the extremal maps, we develop a new early warning signal, i.e. a method for predicting a critical transition.

In Chapter \ref{ch:dich}, we will switch the context back to random dynamical systems. We will introduce a dichotomy spectrum for noninvertible linear random dynamical systems and we will prove a new version of the spectral theorem. This will be a direct continuation of the work by Callaway, Son, Lamb and Rasmussen in \cite{callaway2013dichotomy}, where they introduce a notion of spectrum for \emph{invertible} linear random dynamical systems, and prove a spectral theorem.

In Chapter \ref{ch:appl}, we consider a one dimensional random dynamical system with bounded noise exhibiting a pitchfork bifurcation. We will show that our early warning signal, developed in Chapter \ref{ch:ews}, and the spectrum, calculated for the linearization, describe a bifurcation: the derivative of the extremal map at the boundary reaches the value $1$ and the spectrum expands to contain $1$ at the bifurcation point.