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Discrete & Continuous Dynamical Systems - B

2010 , Volume 14 , Issue 2

Special issue
dedicated to Peter E. Kloeden on the occasion of his 60th birthday

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Preface
Tomás Caraballo, José Real and Russell Johnson
2010, 14(2): i-iii doi: 10.3934/dcdsb.2010.14.2i +[Abstract](205) +[PDF](528.6KB)
Abstract:
The purpose of this special volume of the DCDS is to honor Peter Kloeden on the occasion of his 60th birthday, and to recognize his scientific accomplishments. We know that he will continue to provide enlightening and fascinating contributions to Mathematics and to allied fields.
   It is not possible in the space available to give a detailed description of his work. Therefore, we will only give a brief summary, highlighting some of the themes he has taken up and the results he has obtained.

For more information please click the “Full Text” above.
Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions
María Anguiano, Tomás Caraballo, José Real and José Valero
2010, 14(2): 307-326 doi: 10.3934/dcdsb.2010.14.307 +[Abstract](329) +[PDF](268.4KB)
Abstract:
The existence of a pullback attractor for a reaction-diffusion equations in an unbounded domain containing a non-autonomous forcing term taking values in the space $H^{-1}$, and with a continuous nonlinearity which does not ensure uniqueness of solutions, is proved in this paper. The theory of set-valued non-autonomous dynamical systems is applied to the problem.
Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation
José M. Arrieta and Simone M. Bruschi
2010, 14(2): 327-351 doi: 10.3934/dcdsb.2010.14.327 +[Abstract](282) +[PDF](323.7KB)
Abstract:
We continue the analysis started in [3] and announced in [2], studying the behavior of solutions of nonlinear elliptic equations $\Delta u+f(x,u)=0 $ in $\Omega$ ε with nonlinear boundary conditions of type $\frac{\partial u}{\partial n}+g(x,u)=0$, when the boundary of the domain varies very rapidly. We show that if the oscillations are very rapid, in the sense that, roughly speaking, its period is much smaller than its amplitude and the function $g$ is of a dissipative type, that is, it satisfies $g(x,u)u\geq b|u|^{d+1}$, then the boundary condition in the limit problem is $u=0$, that is, we obtain a homogeneus Dirichlet boundary condition. We show the convergence of solutions in $H^1$ and $C^0$ norms and the convergence of the eigenvalues and eigenfunctions of the linearizations around the solutions. Moreover, if a solution of the limit problem is hyperbolic (non degenerate) and some extra conditions in $g$ are satisfied, then we show that there exists one and only one solution of the perturbed problem nearby.
Averaging of ordinary differential equations with slowly varying averages
Zvi Artstein
2010, 14(2): 353-365 doi: 10.3934/dcdsb.2010.14.353 +[Abstract](263) +[PDF](190.9KB)
Abstract:
The averaging method asserts that a good approximation to the solution of a time varying ordinary differential equation with small amplitude is the solution of the averaged equation, and that the error is maintained small on a long time interval. We establish a similar result allowing the averaged equation to vary in time, thus allowing slowly varying averages of the original equation. Both the modeling issue and the estimation of the resulting errors are addressed.
Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems
Flaviano Battelli and Ken Palmer
2010, 14(2): 367-387 doi: 10.3934/dcdsb.2010.14.367 +[Abstract](247) +[PDF](236.4KB)
Abstract:
We consider a singularly perturbed system with a normally hyperbolic centre manifold. Assuming the existence of a fast homoclinic orbit to a point of the centre manifold belonging to a hyperbolic periodic solution for the slow system, we prove an old and a new result concerning the existence of solutions of the singularly perturbed system that are homoclinic to a periodic solution of the system on the centre manifold. We also give examples in dimensions greater than three of Sil'nikov [16] periodic-to-periodic homoclinic orbits.
Two-sided error estimates for the stochastic theta method
Wolf-Jürgen Beyn and Raphael Kruse
2010, 14(2): 389-407 doi: 10.3934/dcdsb.2010.14.389 +[Abstract](240) +[PDF](249.0KB)
Abstract:
Two-sided error estimates are derived for the strong error of convergence of the stochastic theta method. The main result is based on two ingredients. The first one shows how the theory of convergence can be embedded into standard concepts of consistency, stability and convergence by an appropriate choice of norms and function spaces. The second one is a suitable stochastic generalization of Spijker's norm (1968) that is known to lead to two-sided error estimates for deterministic one-step methods. We show that the stochastic theta method is bistable with respect to this norm and that well-known results on the optimal $\mathcal{O}(\sqrt{h})$ order of convergence follow from this property in a natural way.
The implicit Euler scheme for one-sided Lipschitz differential inclusions
Wolf-Jüergen Beyn and Janosch Rieger
2010, 14(2): 409-428 doi: 10.3934/dcdsb.2010.14.409 +[Abstract](298) +[PDF](270.6KB)
Abstract:
We propose a set-valued version of the implicit Euler scheme for relaxed one-sided Lipschitz differential inclusions and prove that the defining implicit inclusions have a well-defined solution. Furthermore, we give a convergence analysis based on stability theorems, which shows that the set-valued implicit Euler method inherits all favourable stability properties from the single-valued scheme. The impact of spatial discretization is discussed, a fully discretized version of the scheme is analyzed, and a numerical example is given.
Carrying an inverted pendulum on a bumpy road
Mari Paz Calvo and Jesus M. Sanz-Serna
2010, 14(2): 429-438 doi: 10.3934/dcdsb.2010.14.429 +[Abstract](233) +[PDF](185.0KB)
Abstract:
We study the stabilization by means of random impulses of an unstable linear oscillator. Almost sure exponential stability is proved for some combinations of the parameter values.
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions
Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss and José Valero
2010, 14(2): 439-455 doi: 10.3934/dcdsb.2010.14.439 +[Abstract](330) +[PDF](225.4KB)
Abstract:
In this work we present the existence and uniqueness of pullback and random attractors for stochastic evolution equations with infinite delays when the uniqueness of solutions for these equations is not required. Our results are obtained by means of the theory of set-valued random dynamical systems and their conjugation properties.
Escape rates and Perron-Frobenius operators: Open and closed dynamical systems
Gary Froyland and Ognjen Stancevic
2010, 14(2): 457-472 doi: 10.3934/dcdsb.2010.14.457 +[Abstract](312) +[PDF](255.7KB)
Abstract:
We study the Perron-Frobenius operator $\mathcal{P}$ of closed dynamical systems and certain open dynamical systems. We prove that the presence of a large positive eigenvalue $\rho$ of $\mathcal{P}$ guarantees the existence of a 2-partition of the phase space for which the escape rates of the open systems defined on the two partition sets are both slower than $-\log\rho$. The open systems with slow escape rates are easily identified from the Perron-Frobenius operators of the closed systems. Numerical results are presented for expanding maps of the unit interval. We also apply our technique to shifts of finite type to show that if the adjacency matrix for the shift has a large positive second eigenvalue, then the shift may be decomposed into two disjoint subshifts, both of which have high topological entropies.
Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion
María J. Garrido–Atienza, Kening Lu and Björn Schmalfuss
2010, 14(2): 473-493 doi: 10.3934/dcdsb.2010.14.473 +[Abstract](473) +[PDF](254.5KB)
Abstract:
In this paper we study nonlinear stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than $1/2$. We show that these SPDEs generate random dynamical systems (or stochastic flows) by using the stochastic calculus for an fBm where the stochastic integrals are defined by integrands given by fractional derivatives. In particular, we emphasize that the coefficients in front of the fractional noise are non-trivial.
Zero, one and two-switch models of gene regulation
Somkid Intep and Desmond J. Higham
2010, 14(2): 495-513 doi: 10.3934/dcdsb.2010.14.495 +[Abstract](325) +[PDF](250.2KB)
Abstract:
We compare a hierarchy of three stochastic models in gene regulation. In each case, genes produce mRNA molecules which in turn produce protein. The simplest model, as described by Thattai and Van Oudenaarden (Proc. Nat. Acad. Sci., 2001), assumes that a gene is always active, and uses a first-order chemical kinetics framework in the continuous-time, discrete-space Markov jump (Gillespie) setting. The second model, proposed by Raser and O'Shea (Science, 2004), generalizes the first by allowing the gene to switch randomly between active and inactive states. Our third model accounts for other effects, such as the binding/unbinding of a transcription factor, by using two independent on/off switches operating in AND mode. We focus first on the noise strength, which has been defined in the biological literature as the ratio of the variance to the mean at steady state. We show that the steady state variance in the mRNA and protein for the three models can either increase or decrease when switches are incorporated, depending on the rate constants and initial conditions. Despite this, we also find that the overall noise strength is always greater when switches are added, in the sense that one or two switches are always noisier than none. On the other hand, moving from one to two switches may either increase or decrease the noise strength. Moreover, the steady state values may not reflect the relative noise levels in the transient phase. We then look at a hybrid version of the two-switch model that uses stochastic differential equations to describe the evolution of mRNA and protein. This is a simple example of a multiscale modelling approach that allows for cheaper numerical simulations. Although the underlying chemical kinetics appears to be second order, we show that it is possible to analyse the first and second moments of the mRNA and protein levels by applying a generalized version of Ito's lemma. We find that the hybrid model matches the moments of underlying Markov jump model for all time. By contrast, further simplifying the model by removing the diffusion in order to obtain an ordinary differential equation driven by a switch causes the mRNA and protein variances to be underestimated.
Taylor expansions of solutions of stochastic partial differential equations
Arnulf Jentzen
2010, 14(2): 515-557 doi: 10.3934/dcdsb.2010.14.515 +[Abstract](450) +[PDF](454.5KB)
Abstract:
The solution of a stochastic partial differential equation (SPDE) of evolutionary type is with respect to a reasonable state space in general not a semimartingale anymore and does therefore in general not satisfy an Itô formula like the solution of a finite dimensional stochastic ordinary differential equation. Consequently, stochastic Taylor expansions of the solution of a SPDE can not be derived by an iterated application of Itô's formula. Recently, in [Jentzen and Kloeden, Ann. Probab. 38 (2010), no. 2, 532-569] in the case of SPDEs with additive noise an alternative approach for deriving Taylor expansions has been introduced by using the mild formulation of the SPDE and by an appropriate recursion technique. This method is used in this article to derive Taylor expansions of arbitrarily high order of the solution of a SPDE with non-additive noise.
Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials
Russell Johnson and Luca Zampogni
2010, 14(2): 559-586 doi: 10.3934/dcdsb.2010.14.559 +[Abstract](306) +[PDF](345.8KB)
Abstract:
The class of generalized reflectionless Schrödinger potentials was introduced by Marchenko-Lundina and was analyzed by Kotani. We state and prove various results concerning those stationary ergodic processes of Schrödinger potentials which are contained in this class.
Polynomial reformulation of the Kuo criteria for v- sufficiency of map-germs
Victor Kozyakin
2010, 14(2): 587-602 doi: 10.3934/dcdsb.2010.14.587 +[Abstract](217) +[PDF](260.2KB)
Abstract:
In the paper a set of necessary and sufficient conditions for v-sufficiency (equiv. sv-sufficiency) of jets of map-germs $f:(\mathbb{R}^{n},0)\to (\mathbb{R}^{m},0)$ is proved which generalize both the Kuiper-Kuo and the Thom conditions in the function case ($m=1$) so as the Kuo conditions in the general map case ($m>1$). Contrary to the Kuo conditions the conditions proved in the paper do not require to verify any inequalities in a so-called horn-neighborhood of the (a'priori unknown) set $f^{-1}(0)$. Instead, the proposed conditions reduce the problem on v-sufficiency of jets to evaluating the local Łojasiewicz exponents for some constructively built polynomial functions.
On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models
Adam Larios and E. S. Titi
2010, 14(2): 603-627 doi: 10.3934/dcdsb.2010.14.603 +[Abstract](236) +[PDF](356.0KB)
Abstract:
We prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to the Euler-Voigt inviscid $\alpha$-regularization of the three-dimensional Euler equations of ideal incompressible fluids. Moreover, we establish the convergence of strong solutions of the Euler-Voigt model to the corresponding solution of the three-dimensional Euler equations for inviscid flow on the interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of the Euler equations based on this inviscid regularization. The coupling of a magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid irresistive magneto-hydrodynamic (MHD) system. Global regularity of the regularized MHD system is also established.
Mutational inclusions: Differential inclusions in metric spaces
Thomas Lorenz
2010, 14(2): 629-654 doi: 10.3934/dcdsb.2010.14.629 +[Abstract](262) +[PDF](350.8KB)
Abstract:
The focus of interest is how to extend ordinary differential inclusions beyond the traditional border of vector spaces. We aim at an existence theorem for solutions whose values are in a given metric space.
   In the nineties, Aubin suggested how to formulate ordinary differential equations and their main existence theorems in metric spaces: mutational equations (which are quite similar to the quasidifferential equations of Panasyuk). Now the well-known Antosiewicz-Cellina Theorem is extended to so-called mutational inclusions. It provides new results about nonlocal set evolutions in R N .
Three dimensional system of globally modified Navier-Stokes equations with infinite delays
Pedro Marín-Rubio, Antonio M. Márquez-Durán and José Real
2010, 14(2): 655-673 doi: 10.3934/dcdsb.2010.14.655 +[Abstract](217) +[PDF](240.8KB)
Abstract:
Existence and uniqueness of solution for a globally modified version of Navier-Stokes equations containing infinite delay terms are established. Moreover, we also analyze the stationary problem and, under suitable additional conditions, we obtain global exponential decay of the solutions of the evolutionary problem to the stationary solution.
On a doubly nonlinear Cahn-Hilliard-Gurtin system
Alain Miranville and Giulio Schimperna
2010, 14(2): 675-697 doi: 10.3934/dcdsb.2010.14.675 +[Abstract](225) +[PDF](350.2KB)
Abstract:
Our aim in this paper is to study a doubly nonlinear Cahn-Hilliard-type system. In particular, we prove existence and uniqueness results and the existence of global attractors.
Dynamics of the fuzzy logistic family
Juan J. Nieto, M. Victoria Otero-Espinar and Rosana Rodríguez-López
2010, 14(2): 699-717 doi: 10.3934/dcdsb.2010.14.699 +[Abstract](226) +[PDF](337.0KB)
Abstract:
In this work, we study the global dynamics of the fuzzy quadratic family $F_a(x)=G_a(x,x)$, where $a \in\mathbb{R}$, $G_a(x,y)=ax(1-y)$, and $x, y \in E^1$ are elements of the set of fuzzy real numbers. We analyze the set of fixed points of $F_a$ and the behavior of each fuzzy number $x \in E^1$ under iteration by $F_a$, with $a>1$. For $0 < a \leq 1$, we study some stability properties for the fixed points of $F_a$ in $[\chi_{\{0\}}, \chi_{\{1\}}]$. We observe different types of attractors, including chaos. We show that our formulation includes and extends classical results for the real quadratic family, since the set of crisp fuzzy numbers is invariant. Finally, we present some applications and physical considerations in relation with the logistic family.
Global bifurcations from the center of mass in the Sitnikov problem
Rafael Ortega and Andrés Rivera
2010, 14(2): 719-732 doi: 10.3934/dcdsb.2010.14.719 +[Abstract](239) +[PDF](210.0KB)
Abstract:
The Sitnikov problem is a restricted three body problem where the eccentricity of the primaries acts as a parameter. We find families of symmetric periodic solutions bifurcating from the equilibrium at the center of mass. These families admit a global continuation up to excentricity $e=1$. The same techniques are applicable to the families obtained by continuation from the circular problem ($e=0$). They lead to a refinement of a result obtained by J. Llibre and R. Ortega.
Variational shadowing
Sergei Yu. Pilyugin
2010, 14(2): 733-737 doi: 10.3934/dcdsb.2010.14.733 +[Abstract](258) +[PDF](103.2KB)
Abstract:
We introduce a variational shadowing property of diffeomorphisms and show that this property is equivalent to structural stability. Bibliography: 8 titles.
Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach
Christian Pötzsche
2010, 14(2): 739-776 doi: 10.3934/dcdsb.2010.14.739 +[Abstract](326) +[PDF](643.0KB)
Abstract:
We investigate local bifurcation properties for nonautonomous difference and ordinary differential equations. Extending a well-established autonomous theory, due to our arbitrary time dependence, equilibria or periodic solutions typically do not exist and are replaced by bounded complete solutions as possible bifurcating objects.
   Under this premise, appropriate exponential dichotomies in the variational equation along a nonhyperbolic solution on both time axes provide the necessary Fredholm theory in order to employ a Lyapunov-Schmidt reduction. Among other results, this yields nonautonomous versions of the classical fold, transcritical and pitchfork bifurcation patterns.
A criterion for asymptotic straightness of force fields
Jürgen Scheurle and Stephan Schmitz
2010, 14(2): 777-792 doi: 10.3934/dcdsb.2010.14.777 +[Abstract](292) +[PDF](210.7KB)
Abstract:
We consider the equations of motion arising from the classical scattering problem for potentials decreasing sufficiently fast at infinity. It is common to impose some conditions on the potential which guarantee that the paths of particles moving to infinity have straight lines as asymptotes. In this paper a new criterion is given by which one can decide whether or not a given potential has this special property called asymptotic straightness.

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