Abstract: This is an overview of the work titled âInvariant densities for dynamical systems with random switchingâ by Yuri Bakhtin and Tobias Hurth published in Nonlinearity in 2012. In the paper they provide sufficient conditions for the existence of a unique ACIP for the random dynamical system resulting from an nonautonomous ODE with RHS that randomly switches among a finite family of smooth vector fields.
Abstract: ASIPs tell us that trajectories of dynamical systems are almost like Brownian motion trajectories with an error small compared to the size of the trajectories. In this talk we present how these extremely useful results can be obtained via the Nagaev Guivarc'h method.
Abstract: I will talk about Wenxiang Sun, Xueting Tian and Edson Vagasâs paper âNon-uniformly hyperbolic flows and shadowing.â Consider a hyperbolic ergodic measure of a C^1 flow on a compact manifold with a dominated splitting on the linear Poincare flow, we have some weak shadowing property. As an application of this result, we prove that the measure can be approached on the weak* topology by measures supported on periodic orbits.
Abstract: This talk concerns the stability of a nonlinear parabolic potential model for a swarm. We will discuss the global behavior of a simplified decoupled system, and the local behavior of systems with more agents. We have generalized the model to curved surfaces, and will present numerical results displaying their behavior.