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Series: CDSNS Colloquium

We will consider the nonlinear elliptic PDEs driven by the fractional Laplacian with superlinear or asymptotically linear terms or combined nonlinearities. An L^infinity regularity result is given using the De Giorgi-Stampacchia iteration method. By the Mountain Pass Theorem and other nonlinear analysis methods, the local and global existence and multiplicity of non-trivial solutions for these equations are established. This is joint work with Yuanhong Wei.

Series: CDSNS Colloquium

Some relevant Hamiltonian systems in Celestial Mechanics have first integrals in involution. A classic technique to study such systems, known as symplectic reduction, is based in reducing the number of degrees of freedom by using the first integrals. In this talk we present two a posteriori KAM theorems for Hamiltonian systems with first integrals in involution, including the isoenergetic case, without using symplectic reduction. The approach leads to efficient numerical methods and validating techniques.This is a joint work with Alejandro Luque.

Series: CDSNS Colloquium

Over recent years, a great deal of analytical studies and modeling simulations have been brought together to identify the key signatures that allow dynamically similar nonlinear systems from diverse origins to be united into a single class. Among these key structures are bifurcations of homoclinic and heteroclinic connections of saddle equilibria and periodic orbits. Such homoclinic structures are the primary cause for high sensitivity and instability of deterministic chaos in various systems. Development of effective, intelligent and yet simple algorithms and tools is an imperative task for studies of complex dynamics in generic nonlinear systems. The core of our approach is the reduction of the time evolution of a characteristic observable in a system to its symbolic representation to conjugate or differentiate between similar behaviors. Of our particular consideration are the Lorenz-like systems and systems with spiral chaos due to the Shilnikov saddle-focus. The proposed approach and tools will let one detect homoclinic and heteroclinic orbits, and carry out state of the art studies homoclinic bifurcations in parameterized systems of diverse origins.

Series: CDSNS Colloquium

A special class of dynamical systems that we will focus on are substitutions. This class of systems provides a variety of ergodic theoretic behavior and is connected to self-similar interval exchange transformations. During this talk we will explore rigidity sequences for these systems. A sequence $\left( n_m \right)$ is a rigidity sequence for the dynamical system $(X,T,\mu)$ if $\mu(T^{n_m}A\cap A)\rightarrow \mu(A)$ for all positive measure sets $A$. We will discuss the structure of rigidity sequences for substitutions that are rank-one and substitutions that have constant length. This is joint work with Jon Fickenscher.

Series: CDSNS Colloquium

Several modern footbridges around the world have experienced large lateral vibrations during crowd loading events. The onset of large-amplitude bridge wobbling has generally been attributed to crowd synchrony; although, its role in the initiation of wobbling has been challenged. In this talk, we will discuss (i) the contribution of a single pedestrian into overall, possibly unsynchronized, crowd dynamics, and (ii) detailed, yet analytically tractable, models of crowd phase-locking. The pedestrian models can be used as "crash test dummies" when numerically probing a specific bridge design. This is particularly important because the U.S. code for designing pedestrian bridges does not contain explicit guidelines that account for the collective pedestrian behavior. This talk is based on two recent papers: Belykh et al., Science Advances, 3, e1701512 (2017) and Belykh et al., Chaos, 26, 116314 (2016).

Series: CDSNS Colloquium

We present a mean field model of electroencephalographic activity in the brain, which is composed of a system of coupled ODEs and PDEs. We show the existence and uniqueness of weak and strong solutions of this model and investigate the regularity of the solutions. We establish biophysically plausible semidynamical system frameworks and show that the semigroups of weak and strong solution operators possess bounded absorbing sets. We show that there exist parameter values for which the semidynamical systems do not possess a global attractor due to the lack of the compactness property. In this case, the internal dynamics of the ODE components of the solutions can create asymptotic spatial discontinuities in the solutions, regardless of the smoothness of the initial values and forcing terms.

Series: CDSNS Colloquium

If h is a homeomorphism on a compact manifold which is chain-recurrent, we will try to understand when the lift of h to an abelian cover is also chain-recurrent. This has consequences on closed geodesics in manifold of negative curvature.

Series: CDSNS Colloquium

For the tipping elements in the Earth’s climate system, the most important issue to address is how stable is the desirable state against random perturbations. Extreme biotic and climatic events pose severe hazards to tropical rainforests. Their local effects are extremely stochastic and difficult to measure. Moreover, the direction and intensity of the response of forest trees to such perturbations are unknown, especially given the lack of efficient dynamical vegetation models to evaluate forest tree cover changes over time. In this study, we consider randomness in the mathematical modelling of forest trees by incorporating uncertainty through a stochastic differential equation. According to field-based evidence, the interactions between fires and droughts are a more direct mechanism that may describe sudden forest degradation in the south-eastern Amazon. In modeling the Amazonian vegetation system, we include symmetric α-stable Lévy perturbations. We report results of stability analysis of the metastable fertile forest state. We conclude that even a very slight threat to the forest state stability represents L´evy noise with large jumps of low intensity, that can be interpreted as a fire occurring in a non-drought year. During years of severe drought, high-intensity fires significantly accelerate the transition between a forest and savanna state.

Series: CDSNS Colloquium

I will consider the isotropic XY quantum chain with a transverse magnetic field acting
on a single site and analyze the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. It has been shown in the early 70’s
that, in the thermodynamic limit, the state of such system obeys a linear time-dependent
Schrodinger equation with a memory term.
I will consider two different regimes, namely when the perturbation has non-zero or
zero average, and I will show that if the magnitute of the potential is small enough then
for large enough frequencies the state approaches a periodic orbit synchronized with the
potential. Moreover I will provide the explicit rate of convergence to the asymptotics.
This is a joint work with G. Genovese.

Series: CDSNS Colloquium

When perturbed with a small periodic forcing, two (or more) coupledconservative oscillators can exhibit instabilities: trajectories thatbecome unstable while accumulating ``unbounded'' energy from thesource. This is known as Arnold diffusion, and has been traditionallyapplied to celestial mechanics, for example to study the stability ofthe solar system or to explain the Kirkwood gaps in the asteroid belt.However, such phenomenon could be extremely useful in energyharvesting systems as well, whose aim is precisely to capture as muchenergy as possible from a source.In this talk we will show a first step towards the application ofArnold diffusion theory in energy harvesting systems. We will consideran energy harvesting system based on two piezoelectric oscillators.When forced to oscillate, for instance when driven by a small periodicvibration, such oscillators create an electrical current which chargesan accumulator (a capacitor or a battery). Unfortunately, suchoscillators are not conservative, as they are not perfectly elastic(they exhibit damping).We will discuss the persistence of normally hyperbolic invariantmanifolds, which play a crucial role in the diffusing mechanisms. Bymeans of the parameterization method, we will compute such manifoldsand their associated stable and unstable manifolds. We will alsodiscuss the Melnikov method to obtain sufficient conditions for theexistence of homoclinic intersections.