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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 2

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Peter E. Kloeden , Alexander M. Krasnosel'skii , Pavel Krejčí and  Dmitrii I. Rachinskii
2013, 18(2): i-iii doi: 10.3934/dcdsb.2013.18.2i +[Abstract](37) +[PDF](1512.2KB)
Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62. For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

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Modelling of wheat-flour dough mixing as an open-loop hysteretic process
Robert S. Anderssen and  Martin Kružík
2013, 18(2): 283-293 doi: 10.3934/dcdsb.2013.18.283 +[Abstract](24) +[PDF](332.3KB)
Motivated by the fact that various experimental results yield strong confirmatory support for the hypothesis that the mixing of a wheat-flour dough is essentially a rate-independent process'', this paper examines how the mixing can be modelled using the rigorous mathematical framework developed to model an incremental time evolving deformation of an elasto-plastic material. Initially, for the time evolution of a rate-independent elastic process, the concept is introduced of an "energetic solution'' [24] as the characterization for the rate-independent deformations occurring. The framework in which it is defined is formulated in terms of a polyconvex stored energy density and a multiplicative decomposition of large deformations into elastic and nonelastic (plastic or viscous) components. The mixing of a dough to peak dough development is then modelled as a sequence of incremental elasto-nonelastic deformations. For such incremental processes, the existence of Sobolev solutions is guaranteed. Finally, the limit passage to vanishing time increment leads to the existence of an energetic solution to our problem.
Galerkin finite element methods for semilinear elliptic differential inclusions
Wolf-Jüergen Beyn and  Janosch Rieger
2013, 18(2): 295-312 doi: 10.3934/dcdsb.2013.18.295 +[Abstract](42) +[PDF](413.2KB)
In this paper we consider Galerkin finite element discretizations of semilinear elliptic differential inclusions that satisfy a relaxed one-sided Lipschitz condition. The properties of the set-valued Nemytskii operators are discussed, and it is shown that the solution sets of both, the continuous and the discrete system, are nonempty, closed, bounded, and connected sets in $H^1$-norm. Moreover, the solution sets of the Galerkin inclusion converge with respect to the Hausdorff distance measured in $L^p$-spaces.
Emergence of collective behavior in dynamical networks
Michael Blank
2013, 18(2): 313-329 doi: 10.3934/dcdsb.2013.18.313 +[Abstract](33) +[PDF](425.4KB)
One of the main paradigms of the theory of weakly interacting chaotic systems is the absence of phase transitions in generic situation. We propose a new type of multicomponent systems demonstrating in the weak interaction limit both collective and independent behavior of local components depending on fine properties of the interaction. The model under consideration is related to dynamical networks and sheds a new light to the problem of synchronization under weak interactions.
Optimal control of ODE systems involving a rate independent variational inequality
Martin Brokate and  Pavel Krejčí
2013, 18(2): 331-348 doi: 10.3934/dcdsb.2013.18.331 +[Abstract](53) +[PDF](424.1KB)
This paper is concerned with an optimal control problem for a system of ordinary differential equations with rate independent hysteresis modelled as a rate independent evolution variational inequality with a closed convex constraint $Z\subset \mathbb{R}^m$. We prove existence of optimal solutions as well as necessary optimality conditions of first order. In particular, under certain regularity assumptions we completely characterize the jump behaviour of the adjoint.
Double exponential instability of triangular arbitrage systems
Rod Cross and  Victor Kozyakin
2013, 18(2): 349-376 doi: 10.3934/dcdsb.2013.18.349 +[Abstract](56) +[PDF](450.1KB)
If financial markets displayed the informational efficiency postulated in the efficient markets hypothesis (EMH), arbitrage operations would be self-extinguishing. The present paper considers arbitrage sequences in foreign exchange (FX) markets, in which trading platforms and information are fragmented. In [18,9] it was shown that sequences of triangular arbitrage operations in FX markets containing $4$ currencies and trader-arbitrageurs tend to display periodicity or grow exponentially rather than being self-extinguishing. This paper extends the analysis to $5$ or higher-order currency worlds. The key findings are that in a $5$-currency world arbitrage sequences may also follow an exponential law as well as display periodicity, but that in higher-order currency worlds a double exponential law may additionally apply. There is an ``inheritance of instability'' in the higher-order currency worlds. Profitable arbitrage operations are thus endemic rather that displaying the self-extinguishing properties implied by the EMH.
Hysteresis and post Walrasian economics
Rod Cross , Hugh McNamara , Leonid Kalachev and  Alexei Pokrovskii
2013, 18(2): 377-401 doi: 10.3934/dcdsb.2013.18.377 +[Abstract](44) +[PDF](1557.1KB)
The ``new consensus'' DSGE(dynamic stochastic general equilibrium) macroeconomic model has microfoundations provided by a single representative agent. In this model shocks to the economic environment do not have any lasting effects. In reality adjustments at the micro level are made by heterogeneous agents, and the aggregation problem cannot be assumed away. In this paper we show that the discontinuous adjustments made by heterogeneous agents at the micro level mean that shocks have lasting effects, aggregate variables containing a selective, erasable memory of the shocks experienced. This hysteresis framework provides foundations for the post-Walrasian analysis of macroeconomic systems.
A mesoscopic stock market model with hysteretic agents
Michael Grinfeld , Harbir Lamba and  Rod Cross
2013, 18(2): 403-415 doi: 10.3934/dcdsb.2013.18.403 +[Abstract](48) +[PDF](436.0KB)
Following the approach of [22], we derive a system of Fokker-Planck equations to model a stock-market in which hysteretic agents can take long and short positions. We show numerically that the resulting mesoscopic model has rich behaviour, being hysteretic at the mesoscale and displaying bubbles and volatility clustering in particular.
Reduction and identification of dynamic models. Simple example: Generic receptor model
Heikki Haario , Leonid Kalachev and  Marko Laine
2013, 18(2): 417-435 doi: 10.3934/dcdsb.2013.18.417 +[Abstract](35) +[PDF](1263.7KB)
Identification of biological models is often complicated by the fact that the available experimental data from field measurements is noisy or incomplete. Moreover, models may be complex, and contain a large number of correlated parameters. As a result, the parameters are poorly identified by the data, and the reliability of the model predictions is questionable. We consider a general scheme for reduction and identification of dynamic models using two modern approaches, Markov chain Monte Carlo sampling methods together with asymptotic model reduction techniques. The ideas are illustrated using a simple example related to bio-medical applications: a model of a generic receptor. In this paper we want to point out what the researchers working in biological, medical, etc., fields should look for in order to identify such problematic situations in modelling, and how to overcome these problems.
Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay
Mikhail Kamenskii and  Boris Mikhaylenko
2013, 18(2): 437-452 doi: 10.3934/dcdsb.2013.18.437 +[Abstract](32) +[PDF](377.3KB)
This paper proposes an approach to investigate bifurcation of periodic solutions to functional-differential equations of neutral type with a small delay and a small periodic perturbation from the limit cycle under the assumption that there exists adjoint Floquet solutions to the linearized equation.
Asymptotic behaviour of random tridiagonal Markov chains in biological applications
Peter E. Kloeden and  Victor Kozyakin
2013, 18(2): 453-465 doi: 10.3934/dcdsb.2013.18.453 +[Abstract](52) +[PDF](403.3KB)
Discrete-time discrete-state random Markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses the Hilbert projection metric and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself. The proof does not involve probabilistic properties of the sample path $\omega$ and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transitions probabilities, in which case the attractor is a periodic path.
Periodic canard trajectories with multiple segments following the unstable part of critical manifold
Alexander M. Krasnosel'skii , Edward O'Grady , Alexei Pokrovskii and  Dmitrii I. Rachinskii
2013, 18(2): 467-482 doi: 10.3934/dcdsb.2013.18.467 +[Abstract](42) +[PDF](561.9KB)
We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$ portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.
Periodic solutions of isotone hybrid systems
Thomas I. Seidman and  Olaf Klein
2013, 18(2): 483-493 doi: 10.3934/dcdsb.2013.18.483 +[Abstract](24) +[PDF](383.9KB)
Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work [7] of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.
Canard explosion in chemical and optical systems
Elena Shchepakina and  Olga Korotkova
2013, 18(2): 495-512 doi: 10.3934/dcdsb.2013.18.495 +[Abstract](33) +[PDF](917.1KB)
The paper deals with the study of the relation between the Andronov--Hopf bifurcation, the canard explosion and the critical phenomena for the van der Pol's type system of singularly perturbed differential equations. Sufficient conditions for the limit cycle birth bifurcation in the case of the singularly perturbed systems are investigated. We use the method of integral manifolds and canards techniques to obtain the conditions under which the system possesses the canard cycle. Through the application to some chemical and optical models it is shown that the canard point should be considered as the critical value of the control parameter.
Canard cascades
Vladimir Sobolev
2013, 18(2): 513-521 doi: 10.3934/dcdsb.2013.18.513 +[Abstract](27) +[PDF](858.3KB)
The existence of canard cascades is studied in the paper as a problem of gluing of stable and unstable one-dimensional slow invariant manifolds at turning points. This way of looking is made feasible to establish the existence of canard cascades, that can be considered as a generalization of canards. A further development of this approach, with applications to the van der Pol equation and a problem of population dynamics, is contained in the paper.
On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited
Alexander Veretennikov
2013, 18(2): 523-549 doi: 10.3934/dcdsb.2013.18.523 +[Abstract](31) +[PDF](441.2KB)
We establish a large deviation principle for stochastic differential equations with averaging in the case when all coefficients of the fast component depend on the slow one, including diffusion.
Ohm-Hall conduction in hysteresis-free ferromagnetic processes
Augusto Visintin
2013, 18(2): 551-563 doi: 10.3934/dcdsb.2013.18.551 +[Abstract](32) +[PDF](390.1KB)
Electromagnetic processes in a ferromagnetic conductor (e.g., an electric transformer) are here described by coupling the Maxwell equations with nonlinear constitutive laws of the form $$ \vec B \in \mu_0\vec H + {\mathcal M}(x) \vec H/|\vec H|, \qquad \vec J = \sigma(x) \big( \vec E + \vec E_a(x,t) + h(x)\vec J \!\times\! \vec B \big). $$ Here $\vec E_a$ stands for an applied electromotive force; the saturation ${\mathcal M}(x)$, the conductivity $\sigma(x)$ and the Hall coefficient $h(x)$ are also prescribed. The first relation accounts for hysteresis-free ferromagnetism, the second one for the Ohm law and the Hall effect.
    This model leads to the formulation of an initial-value problem for a doubly-nonlinear parabolic-hyperbolic system in the whole $R^3$. Existence of a weak solution is proved, via approximation by time-discretization, derivation of a priori estimates, and passage to the limit. This final step rests upon a time-dependent extension of the Murat and Tartar div-curl lemma, and on compactness by strict convexity.
Equicontinuous sweeping processes
Alexander Vladimirov
2013, 18(2): 565-573 doi: 10.3934/dcdsb.2013.18.565 +[Abstract](29) +[PDF](309.0KB)
We prove that the sweeping process on a "regular" class of convex sets is equicontinuous. Classes of polyhedral sets with a given finite set of normal vectors are regular, as well as classes of uniformly strictly convex sets. Regularity is invariant to certain operations on classes of convex sets such as intersection, finite union, arithmetic sum and affine transformation.
The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields
Igor G. Vladimirov
2013, 18(2): 575-600 doi: 10.3934/dcdsb.2013.18.575 +[Abstract](42) +[PDF](743.5KB)
We consider an "elastic'' version of the statistical mechanical monomer-dimer problem on the $n$-dimensional integer lattice. Our setting includes the classical "rigid'' formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan'' EDM systems where the dimer potential is a weighted $l_1$-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.

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