Seminars and Colloquia by Series

Equivalence of SRB and physical measures for stochastic dynamical systems

Series
CDSNS Colloquium
Time
Wednesday, April 3, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Blumenthal – Universith of Maryland

It is anticipated that the invariant statistics of many of smooth dynamical systems with a `chaotic’ asymptotic character are given by invariant measures with the SRB property- a geometric property of invariant measures which, roughly, means that the invariant measure is smooth along unstable directions. However, actually verifying the existence of SRB measures for concrete systems is extremely challenging: indeed, SRB measures need not exist, even for systems exhibiting asymptotic hyperbolicity (e.g., the figure eight attractor).

The study of asymptotic properties for dynamical systems in the presence of noise is considerably simpler. One manifestation of this principle is the theorem of Ledrappier and Young ’89, where it was proved that under very mild conditions, stationary measures for a random dynamical system with a positive Lyapunov exponent are automatically random SRB measures (that is, satisfy the random analogue of the SRB property). I will talk today about a new proof of this result in a joint work with Lai-Sang Young. This new proof has the benefit of being (1) conceptually lucid and to-the-point (the original proof is somewhat indirect) and (2) potentially easily adapted to more general settings, e.g., to appropriate infinite-dimensional random dynamics, such as time-t solutions to certain classes SPDE (this generalization is an ongoing work, joint with LSY).

Specialization Models of Network Growth

Series
CDSNS Colloquium
Time
Monday, April 1, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ben Webb – BYU –

One of the characteristics observed in real networks is that, as a network's topology evolves so does the network's ability to perform various complex tasks. To explain this, it has also been observed that as a network grows certain subnetworks begin to specialize the function(s) they perform. We introduce a model of network growth based on this notion of specialization and show that as a network is specialized its topology becomes increasingly modular, hierarchical, and sparser, each of which are properties observed in real networks. This model is also highly flexible in that a network can be specialized over any subset of its components. By selecting these components in various ways we find that a network's topology acquires some of the most well-known properties of real networks including the small-world property, disassortativity, power-law like degree distributions and clustering coefficients. This growth model also maintains the basic spectral properties of a network, i.e. the eigenvalues and eigenvectors associated with the network's adjacency network. This allows us in turn to show that a network maintains certain dynamic properties as the network's topology becomes increasingly complex due to specialization.

Random perturbations of dynamical systems

Series
CDSNS Colloquium
Time
Monday, February 25, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yun Yang – City Univ. NY
The real world is inherently noisy, and so it is natural to consider the random perturbations of deterministic dynamical systems and seek to understand the corresponding asymptotic behavior, i.e., the phenomena that can be observed under long-term iteration. In this talk, we will study the random perturbations of a family of circle maps $f_a$. We will obtain, a checkable, finite-time criterion on the parameter a for random perturbation of $f_a$ to exhibit a unique, and thus ergodic, stationary measure.

Local rigidity of Lyapunov spectrum for toral automorphisms

Series
CDSNS Colloquium
Time
Monday, February 18, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Boris Kalinin – Penn State

We will discuss the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L^4 is irreducible over rationals. We consider a volume-preserving C^1-small perturbation f of L. We show that if the Lyapunov exponents of f with respect to the volume are the same as the Lyapunov exponents of L, then f is C^1+ conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle. This is joint work with Andrey Gogolev and Victoria Sadovskaya

Periodic approximation of Lyapunov exponents for cocycles over hyperbolic systems.

Series
CDSNS Colloquium
Time
Monday, February 18, 2019 - 10:10 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Victoria Sadovskaya – Penn State
We consider a hyperbolic dynamical system (X,f) and a Holder continuous cocycle A over (X,f) with values in GL(d,R), or more generally in the group of invertible bounded linear operators on a Banach space. We discuss approximation of the Lyapunov exponents of A in terms of its periodic data, i.e. its return values along the periodic orbits of f. For a GL(d,R)-valued cocycle A, its Lyapunov exponents with respect to any ergodic f-invariant measure can be approximated by its Lyapunov exponents at periodic orbits of f. In the infinite-dimensional case, the upper and lower Lyapunov exponents of A can be approximated in terms of the norms of the return values of A at periodic points of f. Similar results are obtained in the non-uniformly hyperbolic setting, i.e. for hyperbolic invariant measures. This is joint work with B. Kalinin.

Fluctuation of ergodic sums over periodic orbits

Series
CDSNS Colloquium
Time
Thursday, January 17, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Manfred Denker – Penn State University –
The fluctuations of ergodic sums by the means of global and local specifications on periodic points will be discussed. Results include a Lindeberg-type central limit theorems in both setups of specification. As an application, it is shown that averaging over randomly chosen periodic orbits converges to the integral with respect to the measure of maximal entropy as the period approaches infinity. The results also suggest to decompose the variances of ergodic sums according to global and local sources.

The Technique to Solve the Variable Coefficients Homological Equations

Series
CDSNS Colloquium
Time
Monday, December 17, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hongyu Cheng – MSRI & Nankai University
In the infinite-dimensional KAM theory, solving the homological equations is the one of the main parts. Generally, the coefficients of the homological equations are constants, by comparing the coefficients of the functions, it is easy to solve these equations. If the coefficients of homological equations depend on the angle variables, we call these equations as the variable coefficients homological equations. In this talk we will talk about how to solve these equations.

Gradient-like dynamics: motion near a manifold of quasi-equilibria

Series
CDSNS Colloquium
Time
Monday, September 24, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Bates – Michigan State University
This concerns general gradient-like dynamical systems in Banach space with the property that there is a manifold along which solutions move slowly compared to attraction in the transverse direction. Conditions are given on the energy (or, more generally, Lyapunov functional) that ensure solutions starting near the manifold stay near for a long time or even forever. Applications are given with the vector Allen-Cahn and Cahn-Morral equations. This is joint work with Giorgio Fusco and Georgia Karali.

Chaotic Transition States on the Monkey Saddle

Series
CDSNS Colloquium
Time
Monday, April 16, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Thomas Bartsch – Loughborough University

Please Note: Transition State Theory describes how a reactive system crosses an energy barrier that is marked by a saddle point of the potential energy. The transition from the reactant to the product side of the barrier is regulated by a system of invariant manifolds that separate trajectories with qualitatively different behaviour. The situation becomes more complex if there are more than two reaction channels, or possible outcomes of the reaction. Indeed, the monkey saddle potential, with three channels, is known to exhibit chaotic dynamics at any energy. We investigate the boundaries between initial conditions with different outcomes in an attempt to obtain a qualitative and quantitative description of the relevant invariant structures.

TBA

Occupation times

Series
CDSNS Colloquium
Time
Monday, April 2, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Manfred Heinz Denker – Penn State University
Consider a $T$-preserving probability measure $m$ on a dynamical system $T:X\to X$. The occupation time of a measurable function is the sequence $\ell_n(A,x)$ ($A\subset \mathbb R, x\in X$) defined as the number of $j\le n$ for which the partial sums $S_jf(x)\in A$. The talk will discuss conditions which ensure that this sequence, properly normed, converges weakly to some limit distribution. It turns out that this distribution is Mittag-Leffler and in particular the result covers the case when $f\circ T^j$ is a fractal Gaussian noise of Hurst parameter $>3/4$.

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