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Session 1
Nonlinear Sciences
Chairs:
Qi Ouyang (Peking University, China)
Wei Wang (Nanjing University, China)
Titles and Abstracts
Dwight Barkley (University of Warwick, UK)
Title: Having fun with Adjoints and Spiral Waves
Abstract: I will describe some problems in the dynamics of PDEs for which
numerically solving adjoint problems provides valuable insight. I will focus on the
dynamics of spiral and scroll waves in excitable media. I will show how the
computation of Response Functions (adjoints to Goldstone modes) allows one to
predict and understand a variety of interesting dynamics of these waves, and provides
a basis for their control.
Andrew Birrell (University of Queensland, Australia)
Title: Solvability of BEC/BCS crossover Hamiltonians
Abstract: In the last several decades a prolific field of research has flourished,
dedicated to constructing and analysing exactly solvable quantum models. The
understanding of quantum integrability that made such analysis tractable finds its
origin in the melding of the Quantum Inverse Scattering Method and Bethe ansatz
techniques. This approach can be complicated and difficult to implement. In some
instances an ability to extend the exact solutions to the most general Hamiltonians to
which they are applicable is outside the scope of such techniques. In this presentation,
I will highlight results of a recent paper [1] demonstrating an alternative method, that
enables such a determination. Remarkably this determination does not rely on any
prior knowledge of integrability through the existence of a set of conserved operators.
In particular, we derive the solvability conditions for a general family of Hamiltonians
describing the crossover between the low-temperature phenomena of
superconductivity, in the Bardeen–Cooper-Schrieffer theory, and Bose-Einstein
condensation. We then determine the manifolds in the coupling parameter space for
which these Hamiltonians can be solved exactly.
[1] A Birrell, J Links, and P S Isaac, A variational approach for the quantum inverse
scattering method, Inverse Problems 28 035008, 2012.
Chong-Qing Cheng (Nanjing University, China)
Title: Arnold diffusion and Dynamical Instability of Hamiltonian systems
Abstract: Since the time of Henri Poincare, it has been a center problem in the field
of dynamical systems whether Hamitonian systems are dynamically stable or not.
Closely related to it, there is a conjecture of V.I. Arnold, raised almost half century
ago, i.e. there exist orbits along which the slow (action) variable undergoes substantial
variation. This phenomenon is now called Arnold diffusion. In this talk, I shall present
a survey, from mathematical point of view, on recent progress about this subject.
Steve Gross (University of California Irvine, USA)
Title:
Abstract:
Gang Hu (Beijing Normal University, China)
Title: Chaotic motifs in genomic regulatory networks
Abstract: Chaos should occur often in gene regulatory networks (GRNs) which have
been widely described by nonlinear coupled ordinary differential equations, if their
dimensions are no less than 3. It is therefore puzzling that chaos has never been
reported in GRNs in nature and is also extremely rare in models of GRNs. On the
other hand, the topic of motifs has attracted great attention in studying biological
networks, and network motifs are suggested to be elementary building blocks that
carry out some key functions in the network. In this talk, chaotic motifs (subnetworks
with chaos) in GRNs are discussed. The conclusion is that: (i) chaos can only appear
through competitions between different oscillatory modes with rivaling intensities.
Conditions required for chaotic GRNs are found to be very strict, which make chaotic
GRNs extremely rare. (ii) Chaotic motifs are explored as the simplest few-node
structures capable of producing chaos, and serve as the intrinsic source of chaos of
random fewnode GRNs. Several optimal motifs causing chaos with atypically high
probability are figured out. (iii) Moreover, we discovered that a number of special
oscillators can never produce chaos. These structures bring some advantages on
rhythmic functions and may help us understand the robustness of diverse biological
rhythms. (iv) The methods of dominant phase-advanced driving (DPAD) and DPAD
time fraction are proposed to quantitatively identify chaotic motifs and to explain the
origin of chaotic behaviors in GRNs.
Baowen Li (National University of Singapore, Singapore and Tongji University,
China)
Title: Phononics: manipulating heat flow with electronic analog and beyond
Abstract: The form of energy termed heat that typically derives from lattice
vibrations, i.e. the phonons, is usually considered as waste energy and, moreover,
deleterious to information processing. However, in this talk, we attempt to rebut this
common view: By use of tailored models we demonstrate that phonons can be
manipulated like electrons and photons can, thus enabling controlled heat transport.
Moreover, we explain that phonons can be put to beneficial use to carry and process
information.
In a first part we present ways to control heat transport and how to process
information for physical systems which are driven by a temperature bias. Particularly,
we put forward the toolkit of familiar electronic analogs for exercising phononics; i.e.
phononic devices which act as thermal diodes, thermal transistors, thermal logic gates
and thermal memories, etc.. These concepts are then put to work to transport, control
and rectify heat in physical realistic nanosystems by devising practical designs of
hybrid nanostructures that permit the operation of functional phononic devices and, as
well, report first experimental realizations.
Next, we discuss yet richer possibilities to manipulate heat flow by use of time
varying thermal bath temperatures or various other external fields. These give rise to a
plenty of intriguing phononic nonequilibrium phenomena as for example the directed
shuttling of heat, a geometrical phase induced heat pumping, or the phonon Hall
effect, that all may find its way into operation with electronic analogs.
References:
[1] N. B Li, J. Ren, G Zhang, L Wang, P Hanggi, and B Li, Rev. Mod. Phys. 84, 1045
(2012)
Feng Liu (Nanjing University, China)
Title: The dynamics and functions of the p53 network
Abstract: The tumor suppressor p53 lies at the hub of cellular signaling networks that
are activated by various stress signals. As a nuclear transcription factor, p53 can
regulate expression of a large number of target genes. The levels, subcellular
localization, posttranslational modifications, and cofactors of p53 all mediate its
function and dynamics. We have proposed integrated models of the p53 network and
explored how the p53 pathway responds to DNA damage and energy stress by
numerical simulations. We found that p53 can induce distinct cellular outcomes
including cell cycle arrest, senescence, and apoptosis. We characterized the dynamics
of the p53 network in detail and revealed the underlying molecular mechanisms for
the cellular responses to stresses.
Jie Liu (Institute of Applied Physics and Computational Mathematics, China)
Title: Adiabatic theory for Bose-Einstein Condensate (BEC)
Abstract: We investigate adiabatic evolution of quantum states as governed by the
nonlinear Schro¨dinger equation and provide examples of applications with a
nonlinear tunneling model for Bose-Einstein condensates. Our analysis not only spells
out conditions for adiabatic evolution of eigenstates but also characterizes the motion
of noneigenstates which cannot be obtained from the former in the absence of the
superposition principle. We also investigate the Berry phase acquired by an eigenstate
that experienced a nonlinear adiabatic evolution. The circuit integral of the Berry
connection of the instantaneous eigenstate cannot account for the adiabatic geometric
phase, while the Bogoliubov excitations around the eigenstates are found to be
accumulated during the nonlinear adiabatic evolution and contribute a finite phase of
geometric nature. Some possible applications of our theory are discussed.
References:
1)Jie Liu, Biao Wu, and Qian Niu, "Nonlinear evolution of quantum states in the
adiabatic regime", Physical Review Letters 90, 170404 (2003)
2)B.Wu and Jie Liu, "Commutability between the Semiclassical and Adiabatic
Limits",Phys. Rev. Lett. 96, 020405 (2006)
3)Jie Liu and L.B.Fu, "Berry Phase in Nonlinear Systems", Phys. Rev. A 81, 052112
(2010)
4)Sheng-Chang Li, Li-Bin Fu, and Jie Liu, "Adiabatic geometric phase for a
Bose-Einstein condensate coupled to a cavity" ,Phys. Rev. A 84, 053610 (2011)
Laurette Tuckerman (ESPCI, France)
Title: Hexagonal Faraday waves
Abstract: When a fluid layer is subjected to vertical oscillation, subharmonic
standing waves form at its surface; this is the famous instability described by Faraday
in 1831. The patterns first observed were classic: stripes, squares and hexagons. The
discovery in the 1990s of more complicated spatial structures such as quasicrystalline
patterns, superlattices and oscillons led to a resurgence of theoretical interest.
Surprisingly, the Faraday instability has been little studied numerically, with the first
2D simulation carried out in 2000 by Chen. The first 3D simulation, carried out by
our group using a front-tracking/immersed-boundary method, reproduced the
experimental hexagonal standing waves; the variation of the interface height over one
oscillation period is illustrated above. These waves are succeeded, however, by
long-time recurrent alternation between new patterns we have called quasi-hexagonal
and beaded stripes, interconnected by new spatio-temporal symmetries.
Bing-Hong Wang (University of Science and Technology of China, China)
Title: Transportation Dynamics on Networks of Mobile Agents Based on the Greedy
Routing
Abstract: Due to the increasing importance of communication networks such as the
Internet and networks of mobile phone users in modern society, the traffic of
information flows in these networks have attracted more and more attention. In
communication networks, information packets are forwarded from sources to
destinations by specific routing protocol. To enhance the transportation capacity of
communication networks, researchers have designed various routing algorithms, including
the shortest path, the integration of static and dynamic information, the local routing,
the efficient routing, the pheromone routing protocol, the greedy algorithm, and so on.
Previous studies about traffic dynamics usually focus on static networks, where nodes
are motionless and links among nodes keep fixed. In our recent paper [1], we studied
the transportation dynamics on networks of mobile agents. We assume that agents
move on a plane and the searching area of an agent is a circle centered at itself. In that
paper, information packets were delivered according to random routing algorithm, that
is, a packet at an agent is forwarded to another randomly chosen agent in its’ searching
area. The random routing can be applied in the case where moving agents cannot
obtain the information about other agents’ positions. However, if an agent can know
other agents’ positions, the random routing is not an effective algorithm for a packet
to quickly reach the destination. Utilizing the information of agents’ positions, we
now propose a greedy routing for networks of mobile agents. In the greedy routing, every
time step a packet at an agent is delivered to another agent whose distance f r om the
destination is shortest among all the agents in the searched neighbor region. We have
found that the greedy routing markedly enhances the transportation capacity of
networks compared with random routing. Based on the greedy routing, We find that the
transportation capacity of t h e network increases as the communication radius
increases. The transportation capacity of networks reaches the highest at the moderate
moving speed. We also find that the average delivering time increases as the moving
speed increases but decreases as the communication radius increases.
[1] Yang H-X, Wang W-X, Xie Y-B, Lai Y-C and Wang B-H, Phys. Rev. E 83, (2011)
016102
Zhigang Zheng (Beijing Normal University, China)
Title: Nonlinear wave dynamics of the Transport Process on Nonlinear Lattices
Abstract: Energy transport on nonlinear lattices may exhibit novel wave behaviors
such as solitary waves, phonons, discrete breathers, and so on. We discuss the
nonlinear wave dynamics and explore their macroscopic consequences in transport
process such as diffusion and heat conductions.
Changsong Zhou (Hong Kong Baptist University, Hong Kong, China)
Title: Avalanche Dynamics in Hierarchical Neural Networks
Abstract: Cerebral cortical brain networks possess a number of prominent features of
structure and dynamics in the systems level. Firstly, they are structured hierarchically,
from large-scale regions of the whole brain, via cortical areas and area
sub-compartments organized as structural and functional maps, to cortical columns,
and finally circuits made up of individual neurons. Due to this organization of
hierarchical modular network, the connectivity of the local neural network can be
quite dense, while the whole network is very sparse. A balance of excitation and
inhibition also features local recurrent neural network. Second, the networks display
self-organized sustained activity, which is persistent in the absence of external stimuli.
At the systems level, such activity is characterized by complex rhythmical oscillations
over a broadband background, while at the cellular level; neuronal discharges have
been observed to display avalanches, indicating that cortical networks are at the state
of self-organized criticality.
In this talk, I will present our recent study about the impact of hierarchical modular
organization in balanced neural network on the generation of sustained critical
avalanches dynamics and the potential function role in information processing.