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1531-3492

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## Discrete & Continuous Dynamical Systems - B

September 2010 , Volume 14 , Issue 2

Special issue

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

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2010, 14(2): i-iii
doi: 10.3934/dcdsb.2010.14.2i

*+*[Abstract](1201)*+*[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.

2010, 14(2): 307-326
doi: 10.3934/dcdsb.2010.14.307

*+*[Abstract](1948)*+*[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.

2010, 14(2): 327-351
doi: 10.3934/dcdsb.2010.14.327

*+*[Abstract](1797)*+*[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.

2010, 14(2): 353-365
doi: 10.3934/dcdsb.2010.14.353

*+*[Abstract](1599)*+*[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.

2010, 14(2): 367-387
doi: 10.3934/dcdsb.2010.14.367

*+*[Abstract](1515)*+*[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.

2010, 14(2): 389-407
doi: 10.3934/dcdsb.2010.14.389

*+*[Abstract](1492)*+*[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.

2010, 14(2): 409-428
doi: 10.3934/dcdsb.2010.14.409

*+*[Abstract](1609)*+*[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.

2010, 14(2): 429-438
doi: 10.3934/dcdsb.2010.14.429

*+*[Abstract](1234)*+*[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.

2010, 14(2): 439-455
doi: 10.3934/dcdsb.2010.14.439

*+*[Abstract](1584)*+*[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.

2010, 14(2): 457-472
doi: 10.3934/dcdsb.2010.14.457

*+*[Abstract](1705)*+*[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.

2010, 14(2): 473-493
doi: 10.3934/dcdsb.2010.14.473

*+*[Abstract](2579)*+*[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.

2010, 14(2): 495-513
doi: 10.3934/dcdsb.2010.14.495

*+*[Abstract](1802)*+*[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.

2010, 14(2): 515-557
doi: 10.3934/dcdsb.2010.14.515

*+*[Abstract](2485)*+*[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.

2010, 14(2): 559-586
doi: 10.3934/dcdsb.2010.14.559

*+*[Abstract](1585)*+*[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.

2010, 14(2): 587-602
doi: 10.3934/dcdsb.2010.14.587

*+*[Abstract](1433)*+*[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.

2010, 14(2): 603-627
doi: 10.3934/dcdsb.2010.14.603

*+*[Abstract](1310)*+*[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.

2010, 14(2): 629-654
doi: 10.3934/dcdsb.2010.14.629

*+*[Abstract](1584)*+*[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 }.

2010, 14(2): 655-673
doi: 10.3934/dcdsb.2010.14.655

*+*[Abstract](1389)*+*[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.

2010, 14(2): 675-697
doi: 10.3934/dcdsb.2010.14.675

*+*[Abstract](1304)*+*[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.

2010, 14(2): 699-717
doi: 10.3934/dcdsb.2010.14.699

*+*[Abstract](1468)*+*[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.

2010, 14(2): 719-732
doi: 10.3934/dcdsb.2010.14.719

*+*[Abstract](1416)*+*[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.

2010, 14(2): 733-737
doi: 10.3934/dcdsb.2010.14.733

*+*[Abstract](1370)*+*[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.

2010, 14(2): 739-776
doi: 10.3934/dcdsb.2010.14.739

*+*[Abstract](1576)*+*[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.

2010, 14(2): 777-792
doi: 10.3934/dcdsb.2010.14.777

*+*[Abstract](1505)*+*[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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