Featured minisymposia at Snowbird

By Peter van Heijster
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The new thing at Snowbird this year: featured minisymposia. What are they about and why should you attend them? An overview:

Title: Advances in the theory and computation of Hamiltonian systems
By: Marian Gidea and Tere M. Seara
When: Sunday, May 19, 2:45pm – 5:00pm
Where: Ballroom I
Intro: The featured presentations of this minisymposium will outline some recent theoretical advances in Hamiltonian instability, celestial mechanics, and PDE, as well as applications.

One of the main objectives in studying dynamical systems is to acquire information on the qualitative aspects of theirs solutions. A special interest is presented by invariant sets (e.g., periodic and quasi-periodic solutions), and by solutions whose past or future asymptotic behavior is related to such invariant sets. These invariant objects can be used to approximate the behavior of nearby solutions.

A fundamental question is to decide which solutions are stable, i.e., their motion remains localized, and which are unstable, i.e., their motion experiences large changes in phase space, or behaves in an unpredictable manner.

To study stability and instability, it is important to detect the geometric/topological invariant objects that organize the dynamics (local phenomena) and the connections between them (global phenomena).

Some of the talks in this minisymposium deal with local phenomena in Hamiltonian systems and Hamiltonian PDE: special solutions which are localized in the phase space, breathers in some Klein-Gordon equation, bifurcations in simplified Hamiltonian models of the nonlinear Schrödinger equation (NLS) , binary and triple collisions in the planar three-body and four-body problem.

Some other talks deal with the study of global phenomena in Hamiltonian systems and with the problem of instability. For integrable Hamiltonian systems it is well know that all motions are stable, as they are confined to invariant tori; the instability problem asserts that typical, arbitrarily small perturbations of integrable Hamiltonian systems always possess unstable solutions. The recent years witnessed a spectacular progress on this problem. This minisimposium will showcase two research directions that proved to be successful in detecting instability in classical Hamiltonian systems: variational methods, which have been used to prove generic type of results, and geometric methods, which seem suitable to deal with concrete systems. In terms of applications, Hamiltonian instability can be exploited to design low energy spacecraft trajectories, and for Earth’s satellite repositioning.

An unstable trajectory in the spatial circular restricted three-body problem (Credit: A. Delshams, M. Gidea, P. Roldan).


Title: Celestial Dynamics
By: Josep Masdemont and Gerard Gomez
When: Monday, May 20, 2:30pm – 4:45pm
Where: Maybird
Intro: In this section we will briefly survey the salient works of the past decades, describe some of the most interesting applications in recent missions, and conclude with the most recent advances and applications of Dynamical Systems to Celestial Mechanics and Astrodynamics problems.

Mission design and other recent work on celestial mechanics are featured in minisymposium MS41.
Picture taken from Why do math?


Title: Delayed Oscillators
By: Gábor Stépán
When: Wednesday, May 22, 2:30pm – 4:45pm
Where: Ballroom III
Intro: Apart of the first delayed models of population dynamics, the delayed mechanical oscillators have served as a driving force for the development of the theory of functional differential equations since the middle of the 20th century. The lectures provide a brief review of the actual delayed oscillator models of engineering and present the recent results of their analyses from discrete delays through distributed, stochastic and state-dependent ones together with the corresponding applied numerical techniques.

A delayed Mathieu equation.
Picture taken from Gábor Stépán's web page


Title: Dynamics and Control of Neurons and Networks,Parts I and II
By: Victoria Booth, Jeff Moehlis and Michal Zochowski
When: Part I: Sunday, May 19, 2:45pm – 5:00pm and
When: Part II: Monday, May 20, 2:30pm – 4:45pm and
Where: Ballroom II
Intro: Neurons are the cells that form the basic structural unit of the nervous system. There are roughly one hundred billion neurons in the human brain, and these have a variety of shapes, sizes, and electrophysiological properties. Neurons receive inputs from other neurons via chemical and electrical synapses, and can respond by generating an action potential, or a spike, which is then communicated to other neurons. The spike activity patterns among networks of coupled neurons ultimately allows each of us to do many things, including 1) receiving sensory information from inside or outside our body, 2) interpreting, storing and retrieving this information, and 3) initiating motor responses.

Neurons are also dynamical systems: there is a set of variables that describe the state of a neuron, and a rule that describes how these variables evolve and respond to external inputs. There is by now a long, fruitful history of using dynamical systems techniques to understand the dynamics of individual neurons and the dynamics of coupled networks of neurons. Dynamical systems techniques have helped to develop quantitative understanding of the interactions among neuron response properties, connection topology of a network and dynamics of the synaptic coupling between neurons that combine to generate spike activity patterns in a network. However, we are still very far from completely understanding how the very complex dynamical system that is the brain actually works.

More recently, there has been growing interest in controlling spike activity in neuronal networks, motivated, for example, by treatment of neurological diseases such as Parkinson's disease and epilepsy. Talks in this Featured Minisymposium will address different dynamical systems approaches for understanding the dynamics and control of neurons and networks.

Title: Dynamics of Marine Ecosystems
By: Drew LaMar and Leah Shaw
When: Wednesday, May 22, 2:30pm – 4:45pm
Where: Ballroom II
Intro: There exist many challenges, both biological and modeling, inherent in marine ecosystems. The talks in this mini-symposium cover a wide-range of topics, from shellfish to phytoplankton, as well as many overarching questions, such as the effects of nutrient availability, environmental variability and climate change on population dynamics.

Title: Dynamics of Networks of Networks
By: Jürgen Kurths
When: Wednesday, May 22, 2:30pm – 4:45pm
Where: Ballroom I
Intro: Our world is more connected than ever. Worldwide individual network systems are becoming increasingly interconnected and interdependent yielding a network of networks. The network of networks presents a new scientific paradigm of significant challenge. They behave significantly different from isolated networks studied so far and are prone to richer and unexpected phenomena. Here theoretical results as well as applications to the Earth System, cargo ship movement and bioinvasion as well as chemical oscillators are presented.

Title: Dynamics of Planet Earth
By: Hans Kaper
When: Monday, May 20, 2:30pm – 4:45pm
Where: Ballroom I
Intro: More than 100 scientific societies, universities, research institutes, and organizations all over the world have joined forces to dedicate 2013 as a special year for the Mathematics of Planet Earth (MPE2013). The Web site http://mpe2013.org lists a full schedule of activities organized under the umbrella of MPE2013, including workshops, symposia, and special lectures. A daily blog provides a forum for informal discussion of the many themes covered by MPE2013, and a series of public lectures (supported by the Simons Foundation) by well-known mathematicians highlights Mathematics of Planet Earth for a worldwide audience.

Important topics covered by MPE2013 include the Earth's climate system and issues of sustainability. Many problems involve dynamical systems, and this year's Snowbird meeting will host a Featured Minisymposium on "Dynamics of Planet Earth" as part of MPE2013. The minisymposium will feature an overview talk by the organizer and four talks on specific applications of dynamical systems and bifurcation theory to the Earth's climate system. Chris Danforth (U Vermont) will demonstrate a novel method for improving forecasts during integration of a weather model. Mary Silber (Northwestern U) will discuss tipping points in the context of bifurcation theory, using case studies of possible tipping points in models of Arctic sea-ice retreat and desertification. Marty Anderies (Arizona State U), who is interested in land use and the carbon cycle, will explore the relationship between nonlinear dynamics and planetary boundaries. Mary Lou Zeeman (Bowdoin College and Cornell U) will focus on issues of sustainability and will explore how a decision-support viewpoint may inspire new questions for dynamical systems.



Title: Lagrangian Dynamics in Geophysical Flows
By: M. Josefina Olascoaga
When: Sunday, May 19, 2:45pm – 5:00pm
Where: Maybird
Intro: The need of understanding and forecasting Lagrangian transport in complex geophysical flows has become quite pressing by the occurrence of several environmental disasters in recent years (oil spills, volcanic ash releases, radioactive contamination leaks). This mini-symposium is devoted to discuss advances in dynamical systems that enable detection of structures around which Lagrangian transport is organized, thereby facilitating its understanding and providing means for its forecasting.

Lagrangian transport in the Gulf of Mexico.
Taken from Josefina Olascoaga's web page.


Title: Localized Pattern Formation
By: Alastair Rucklidge
When: Sunday, May 19, 2:45pm – 5:00pm
Where: Ballroom III
Intro: Localized Pattern Formation has been a topic of intense study over the last 10-15 years or so. In the classic scenario, a subcritical pattern-forming instability can lead to bistability between the featureless state and the patterned state, as well as a family of localized patches of pattern between extended featureless regions. This minisymposium will start with an overview of the subject, and go on to present recent work on methods of analysis and on extensions of this scenario, to localized turbulent patches, to two-dimensional pattern formation, to parametrically forced problems such as the Faraday wave problem, and to localized patterns of one wavenumber in a background of pattern of another wavenumber.

This minisymposium is arranged in memory of Thomas Wagenknecht, a mathematician at the University of Leeds and a keen Snowbird supporter, who died last year at the age of 37.

A localized patch of a short-wavelength pattern in the middle of a background of a long-wavelength pattern. (Credit: David Bentley, Thomas Wagenknecht and Alastair Rucklidge.).


Title: Recent Developments in KAM Theory
By: Rafael de la Llave and Alex Haro
When: Wednesday, May 22, 2:30pm – 4:45pm
Where: Maybird
Intro: In the late 50's and early 60's Kolmogorov, Arnold and Moser developed a systematic theory to study persistence of quasi-periodic solutions, introducing a set of technics that is nowadays known under the name of KAM theory.

In contrast with the theory of persistence of periodic solutions,which can be studied with soft methods, the theory of persistence of quasi-periodic solutions requires sophisticated analysis and -- somewhat surprisingly -- delicate questions of number theory. The geometry and regularity of the system matter as well as quantitative estimates.

The KAM theory is not only important conceptually, but also practically. Geometrically, quasi-periodic solutions correspond to invariant tori, and these objects are barriers for transport or landmarks that organize the long term behavior of a large region of the phase space. Moreover, these invariant tori contain, in some sense, the regular dynamics of the system. It is then important to develop efficient numerical algorithms of computation of invarian tori.

In recent years there have been several important developments in the mathematical aspects, the applications and the numerical treatment. Remarkably, all these developments feed onto each other and lead to progress. We hope to present in the special session an introduction of these developments at the same time, with the participation of researches coming from different fields of research.

The issues of the session are, then, quite varied, but with the common link being the study of existence and persistence of invariant tori, in the three aspects mentioned above. On the mathematical side: studies of degenerate systems, different geometries and contexts, improved estimates, infinite dimensional systems. On the applications side: quasi-periodic solutions in oceanography and in atmospheric research, applications in fluid dynamics and celestial mechanics. On the numerical side: efficient and reliable algorithms which can compute invariant tori up to the boundary of existence and reveal a very tantalizing picture of breakdown.

A KAM torus about to break down. (From Jour. Stat. Pys 150, 6 (2013)).


Title: Vortex Dynamics
By: Silas Alben
When: Monday, May 20, 2:30pm – 4:45pm
Where: Ballroom III
Intro: The topic of vortex dynamics has been a major research area in dynamical systems for several decades. This session highlights some newer applications of vortex dynamics to problems related to locomotion and stability of rigid and flexible bodies in fluids.
The minisymposium showcases recent work in which the dynamics of vortices (and related flow structures) play important roles. The studies are both fundamental and applied, with applications to (and inspiration from) problems in engineering and biology. The topics considered include the dynamics of singular and regularized distributions of vorticity, the formation of vorticity at solid boundaries, and interactions among ensembles of vortical structures and solid bodies and boundaries.

A flexible filament interacting with a point vortex.
Taken from Silas Alben's web page.

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