Workshop on Theory and Applications of
Random/Non-autonomous Dynamical Systems
8-15 September, 2014
Dynamical Systems Group, Department of Mathematics, Imperial College London
13:00-14:00 Christian Rodrigues (Max Planck Institute, Leipzig, Germany)
Stochastic stability and the representation of Markov chains by random maps
Amongst the main concerns of Dynamics one wants to decide whether
asymptotic states are robust under random perturbations. For practical
applications, the randomly perturbed dynamics is given by a Markov chain
model. Nevertheless, in order to derive conclusions from the mathematical
perspective about stability, asymptotic states, invariance, etc.,
one is always enforced to use a model based on iteration of maps chosen
at random with a given probability. The two approaches however are in general not equivalent. In this talk we systematically investigate the problem of
representing Markov chains by families of random maps, and which
regularity of these maps can be achieved depending on the properties
of the probability measures. Our key idea is to use techniques from
optimal transport to select optimal such maps. From this scheme, we
not only deduce the representation by measurable and continuous random
maps, but also obtain conditions for the to construct random
diffeomorphisms from a given Markov chain. This is a joint work with
Jost, and Kell.
15:00-16:00 Rainer Klages (Queen Mary University of London)
Diffusion in randomly perturbed dissipative dynamics
Dynamical systems having many coexisting attractors present
interesting properties from both fundamental theoretical and
modelling points of view. When such dynamics is under bounded random
perturbations, the basins of attraction are no longer invariant and
there is the possibility of transport among them. Here we introduce
a basic theoretical setting which enables us to study this hopping
process from the perspective of anomalous transport using the
concept of a random dynamical system with holes. We apply it to a
simple model by investigating the role of hyperbolicity for the
transport among basins. We show numerically that our system exhibits
non-Gaussian position distributions, power-law escape times, and
subdiffusion. Our simulation results are reproduced consistently
from stochastic Continuous Time Random Walk theory.
13:00-14:30 Hiroki Sumi (Osaka University, Japan)
Negativity of Lyapunov exponents of generic random dynamical systems of complex polynomials
In this talk, we consider random dynamical systems of complex polynomial
maps on the Riemann sphere.
It is well-known that for each rational map $f$ on the Riemann sphere
with $\deg (f)\geq 2$, the Hausdorff dimension of the set of points $z$ in the Riemann sphere for which the Lyapunov exponent of the dynamics of $f$ is positive, is positive. However, we show that for generic i.i.d. random dynamical systems of complex polynomials, the following holds. For all but countably many points $x$ in the Riemann sphere, for almost every sequence $\gamma =(\gamma _{1}, \gamma _{2}, \gamma _{3},\cdots )$ of polynomials, the Lyapunov exponent along $\gamma $ starting with $x$ is negative. Note that the above statement cannot hold in the usual iteration dynamics of a single rational map $f$ with $\deg (f)\geq 2.$ Therefore the picture of the random complex dynamics is completely different from that of the usual complex dynamics. We remark that even if the chaos of the random system disappears in the $C^{0}$ sense, the chaos of the system may remain in the $C^{1}$ sense, and we have to consider the ``gradation between chaos and order''.
15:00-16:00 Masayuki Asaoka (Kyoto University, Japan)
Arbitrary growth of the number of periodic points and universal dynamics in one-dimensional semigroups
Theory of random dynamics studies properties of almost every
path of iterated function systems. In this talk, we discuss
how bad the dynamics can be for a special path and how does
it effect the average of dynamical quantity of iterated function
systems. More precisely, we prove the following theorem:
There exists an open subset of the set of smooth iterated
function systems on the unit interval with three generators
such that
(a) For all most path, the n-th iteration is uniform contraction,
and hence it has unique fixer point for any large n,
(b) For any generic element of U, the limsup of the average of
the number of fixed points of n-th iterations grows
superexpoentially.
The result illustrates that the behavior of the average in finite
time is different from that of a.e. paths.
We also proves that for generic element in U there exists a path
along which `any' dynamics is realized.
Free discussion
13:00-14:00 Hiroshi Teramoto (Hokkaido University, Japan)
Extracting coherent molecular vibrational modes by nonlinear time series analysis
Nonlinear resonance is one of the most efficient mechanisms for vibrational energy transfers among vibrational modes. In this talk, we propose a method to extract vibrational modes from a given time series to understand the underlying mechanism behind the energy transfers in terms of nonlinear resonances.
14:10-15:10 Yuzuru Sato (Hokkaido University, Japan/Imperial College London/London Mathematical Laboratoy)
Random basin in dice roll
Dynamics of dice roll on heterogeneous environments is studied in terms of random basin strucuture. Uncertainty exponents are numerically estimated in a random dynamical system and final state sensitivity under the presence of noise is investigated. Extracting random maps from experimental data of dice roll is also briefly discussed.
13:00-14:00 Tsuyoshi Chawanya (Osaka University, Japan)
On bifurcation phenomena in the quasi-periodically driven logistic map
system
Quasi-periodically forced logistic map system is one of the
representative systems where strange non-chaotic attractors(SNAs) are
observed. In this talk, we show that the numerical analysis on the
bifurcation phenomena in QPLM can be carried out efficiently using a
map describing the evolution of "line segment" in the logistic map
system, and exhibit some of the obtained results including phase
diagram with fine resolution indicating the existence of variety of
bifurcation senario from a stable torus to a chaotic attractor with or
without SNA in the middle.
13:00-14:00 Jun-nosuke Teramae (Osaka University, Japan)
Stochastic dynamics and computation in network models of cortical spiking neurons
Neurons in the brain sustain spontaneous ongoing activity with highly irregular spike trains. While the irregular activity has been regarded as ignorable background noise, recent experiments reveal that the spontaneous activity is significant player of computation in the brain. In this talk, introducing a recently proposed deterministic model of cortical network of spiking neurons that stably generates and maintains spontaneous ongoing activity with highly irregular spike trains, we study stochastic dynamics of the ongoing activity and discuss possible roles of the stochastic spiking for neural computation.
14:10-15:10 Tiago Pereira (Imperial College London)
Dynamics in Heterogeneous Networks: Emergence at Various Scales
Recent results reveal that typical real-world networks have various
levels of connectivity. These networks exhibit emergent behaviour at
various levels. Striking examples are found in the brain, where
synchronisation between highly connected neurons coordinate and shape
the network development. These phenomena remain a major challenge. I
will discuss a probabilistic dimension reduction principle to describe
the network dynamics. I show that, at large levels of connectivity,
the high-dimensional network dynamics can be reduced to a few
macroscopic equations. The strategy is to describe ensembles of random
networks, and the dynamics almost every initial state. This reduction
provides the opportunity to explore the coherent properties at various
network connectivity scales. This is a joint work with Sebastian van Strien and Jeroen Lamb.
Organisers
Yuzuru Sato (Hokkaido University/Imperial College London/London Mathematical Laboratory)
Martin Rasmussen (Imperial College London)
Dimitry Turaev (Imperial College London)
Jeroen Lamb (Imperial College London)
Contacts
Yuzuru Sato
Room 532, Huxley Building, Department of Mathematics, Imperial College London
South Kensington Campus, London SW7 2AZ
Phone: +44 (0)79 6355 4120 (mobile)
Email: yuzuru.sato"@"imperial.ac.uk
and
RIES Complex Systems Research Group / Department of Mathematics
Hokkaido University, Kita 12 Nishi 6, Kita-ku, Sapporo, Hokkaido 060-0812, Japan
Phone: +81-(0)11-706-2889
Email: ysato"@"math.sci.hokudai.ac.jp