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Discrete & Continuous Dynamical Systems - B

2011 , Volume 16 , Issue 1

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The Euler-Maruyama approximations for the CEV model
Vyacheslav M. Abramov , Fima C. Klebaner and  Robert Sh. Lipster
2011, 16(1): 1-14 doi: 10.3934/dcdsb.2011.16.1 +[Abstract](38) +[PDF](426.8KB)
The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_s ds+\int_0^t\sigma (X^+_s)^p dW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0 \le t \le T$, in the Skorokhod metric, by giving a new approximation by continuous processes. We calculate ruin probabilities as an example of such approximation. The ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the limiting distribution is discontinuous at zero. To approximate the size of the jump at zero we use the Levy metric, and also confirm the convergence numerically.
Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay
Matthieu Alfaro and  Arnaud Ducrot
2011, 16(1): 15-29 doi: 10.3934/dcdsb.2011.16.15 +[Abstract](38) +[PDF](406.3KB)
We investigate the singular limit, as $\varepsilon \to 0$, of the Fisher equation $\partial _t u=\varepsilon \Delta u + \varepsilon ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with slow exponential decay, therefore completing the analysis presented in [1]. We prove that the sharp interface limit moves with a constant speed depending dramatically on the tails of the initial data. We make a thorough analysis of both the generation and motion of interface, thus providing a new estimate of the thickness of the transition layers.
Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains
Dimitra Antonopoulou and  Georgia Karali
2011, 16(1): 31-55 doi: 10.3934/dcdsb.2011.16.31 +[Abstract](50) +[PDF](552.0KB)
We consider a generalized Stochastic Cahn-Hilliard equation with multiplicative white noise posed on bounded convex domains in $R^d$, $d=1,2,3$, with piece-wise smooth boundary, and introduce an additive time dependent white noise term in the chemical potential. Since the Green's function of the problem is induced by a convolution semigroup, we present the equation in a weak stochastic integral formulation and prove existence of solution when $d\leq 2$ for general domains, and for $d=3$ for domains with minimum eigenfunction growth, without making use of any explicit expression of the spectrum and the eigenfunctions. The analysis is based on stochastic integral calculus, Galerkin approximations and the asymptotic spectral properties of the Neumann Laplacian operator. Existence is also derived for some non-convex cases when the boundary is smooth.
Finite to infinite steady state solutions, bifurcations of an integro-differential equation
Samir K. Bhowmik , Dugald B. Duncan , Michael Grinfeld and  Gabriel J. Lord
2011, 16(1): 57-71 doi: 10.3934/dcdsb.2011.16.57 +[Abstract](49) +[PDF](482.1KB)
We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is increased to examine the transition from an uncountably infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.
The dynamics of a low-order model for the Atlantic multidecadal oscillation
Henk Broer , Henk Dijkstra , Carles Simó , Alef Sterk and  Renato Vitolo
2011, 16(1): 73-107 doi: 10.3934/dcdsb.2011.16.73 +[Abstract](32) +[PDF](5210.4KB)
Observational and model based studies provide ample evidence for the presence of multidecadal variability in the North Atlantic sea-surface temperature known as the Atlantic Multidecadal Oscillation (AMO). This variability is characterised by a multidecadal time scale, a westward propagation of temperature anomalies, and a phase difference between the anomalous meridional and zonal overturning circulations.
    We study the AMO in a low-order model obtained by projecting a model for thermally driven ocean flows onto a 27-dimensional function space. We study bifurcations of attractors by varying the equator-to-pole temperature gradient ($\Delta T$) and a damping parameter ($\gamma$).
    For $\Delta T = 20^\circ$C and $\gamma = 0$ the low-order model has a stable equilibrium corresponding to a steady ocean flow. By increasing $\gamma$ to 1 a supercritical Hopf bifurcation gives birth to a periodic attractor with the spatio-temporal signature of the AMO. Through a period doubling cascade this periodic orbit gives birth to Hénon-like strange attractors. Finally, we study the effects of annual modulation by introducing a time-periodic forcing. Then the AMO appears through a Hopf-Neĭmark-Sacker bifurcation. For $\Delta T = 24^\circ$C we detected at least 11 quasi-periodic doublings of the invariant torus. After these doublings we find quasi-periodic Hénon-like strange attractors.
Multi-group asset flow equations and stability
Gunduz Caginalp and  Mark DeSantis
2011, 16(1): 109-150 doi: 10.3934/dcdsb.2011.16.109 +[Abstract](46) +[PDF](625.3KB)
We consider a two-group asset flow model of a financial instrument with one group focused on price trend, the other on value. We prove the existence of both stable and unstable regions for the system of differential equations and show that a strong motivation based on (particularly recent) price trend is associated with instability. Numerical computations using a set of typical parameters describe precise regions of stability and instability. A precise limiting connection between the discrete and differential equations is also established.
Periodic and quasi--periodic orbits of the dissipative standard map
Alessandra Celletti and  Sara Di Ruzza
2011, 16(1): 151-171 doi: 10.3934/dcdsb.2011.16.151 +[Abstract](34) +[PDF](332.8KB)
We present analytical and numerical investigations of the dynamics of the dissipative standard map. We first study the existence of periodic orbits by using a constructive version of the implicit function theorem; then, we introduce a parametric representation, which provides the interval of the drift parameter ensuring the existence of a periodic orbit with a given period. The determination of quasi--periodic attractors is efficiently obtained using the parametric representation combined with a Newton's procedure, aimed to reduce the error of the approximate solution provided by the parametric representation. These methods allow us to relate the drift parameter of the periodic orbits to that of the invariant attractors, as well as to constrain the drift of a periodic orbit within Arnold's tongues in the parameter space.
Dynamic phase transition for binary systems in cylindrical geometry
I-Liang Chern and  Chun-Hsiung Hsia
2011, 16(1): 173-188 doi: 10.3934/dcdsb.2011.16.173 +[Abstract](54) +[PDF](354.7KB)
In this article, we present a dynamic phase transition and stability analysis for the Cahn-Hilliard equations in cylindrical geometry. Two types of phase transitions (the continuous type and the jump type) are determined explicitly in terms of relevant physical and geometric parameters.
Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction
Sheng-Chen Fu
2011, 16(1): 189-196 doi: 10.3934/dcdsb.2011.16.189 +[Abstract](28) +[PDF](312.4KB)
In this paper we consider a reaction-diffusion system which describes the acidic nitrate-ferroin reaction. We first show that there exists a minimum speed travelling wave solution. Then some estimates of the minimum speed(s) are derived. Finally, we find that the set of admissible wave speed is $[c_{m i n},\infty)$ under certain condition.
Existence theorem for a model of dryland vegetation
Yukie Goto , Danielle Hilhorst , Ehud Meron and  Roger Temam
2011, 16(1): 197-224 doi: 10.3934/dcdsb.2011.16.197 +[Abstract](32) +[PDF](467.6KB)
In this article, we consider the dryland vegetation model proposed by Gilad et al [6, 7]. This model consists of three nonlinear parabolic partial differential equations, one of which is degenerate parabolic of the family of the porous media equation [3, 7], and we prove the existence of its weak solutions. Our approach based on the classical Galerkin methods combines and makes use of techniques, parabolic regularization, truncation, maximum principle, compactness. We observe in this way various properties and regularity results of the solutions.
On a class of three dimensional Navier-Stokes equations with bounded delay
Sandro M. Guzzo and  Gabriela Planas
2011, 16(1): 225-238 doi: 10.3934/dcdsb.2011.16.225 +[Abstract](43) +[PDF](359.6KB)
In this paper we consider a three dimensional Navier-Stokes type equations with delay terms. We discuss the existence of weak and strong solutions and we study the asymptotic behavior of the strong solutions. Moreover, under suitable assumptions, we show the exponential stability of stationary solutions.
Stabilization via symmetry switching in hybrid dynamical systems
Sebastian Hage-Packhäuser and  Michael Dellnitz
2011, 16(1): 239-263 doi: 10.3934/dcdsb.2011.16.239 +[Abstract](53) +[PDF](1947.5KB)
With a view to stabilization issues of hybrid systems exhibiting a regular structure in terms of symmetry, we introduce the concept of symmetry switching and relate symmetry-induced switching strategies to the asymptotic stability of switched linear systems. To this end, a general notion of hybrid symmetries for switched systems is established whereupon orbital switching is treated which builds on the existence of hybrid symmetries. In the main part, we formulate and prove sufficient conditions for asymptotic stability under slow symmetry switching. As an example of both theoretical and practical interest, we examine time-varying networks of dynamical systems and perform stabilization by means of orbital switching. Behind all that, this work is meant to provide the groundwork for the treatment of equivariant bifurcation phenomena of hybrid systems.
A rigorous derivation of hemitropy in nonlinearly elastic rods
Timothy J. Healey
2011, 16(1): 265-282 doi: 10.3934/dcdsb.2011.16.265 +[Abstract](34) +[PDF](435.4KB)
We consider a class of nonlinearly hyperelastic rods with helical symmetry, cf. [7]. Such a rod is mechanically invariant under the symmetries of a circular-cylindrical helix. Examples include idealized DNA molecules, wire ropes and cables. We examine the limit as the pitch of the helix characterizing the symmetry approaches zero and show that the resulting model is a hemitropic rod. The former is mechanically invariant under all proper rotations about its centerline and generally possesses chirality or handedness in its mechanical response, cf. [7]. An isotropic rod is also rotationally invariant but, in addition, enjoys certain reflection symmetries, which rule out chirality. Isotropy implies hemitropy, but the converse is not generally true. We employ both averaging methods and methods of gamma convergence to obtain the effective or homogenized (hemitropic) problem, the latter not corresponding to a naïve average.
Feedback stabilization methods for the numerical solution of ordinary differential equations
Iasson Karafyllis and  Lars Grüne
2011, 16(1): 283-317 doi: 10.3934/dcdsb.2011.16.283 +[Abstract](45) +[PDF](1577.8KB)
In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations. We apply tools from nonlinear control theory, specifically Lyapunov function and small-gain based feedback stabilization methods for systems with a globally asymptotically stable equilibrium point. Proceeding this way, we derive conditions under which the step size selection problem is solvable (including a nonlinear generalization of the well-known A-stability property for the implicit Euler scheme) as well as step size selection strategies for several applications.
Vanishing singularity in hard impacting systems
Soumya Kundu , Soumitro Banerjee and  Damian Giaouris
2011, 16(1): 319-332 doi: 10.3934/dcdsb.2011.16.319 +[Abstract](33) +[PDF](654.4KB)
It is known that the Jacobian of the discrete-time map of an impact oscillator in the neighborhood of a grazing orbit depends on the square-root of the distance the mass would have gone beyond the position of the wall if the wall were not there. This results in an infinite stretching of the phase space, known as the square-root singularity. In this paper we look closer into the Jacobian matrix and find out the behavior of its two parameters---the trace and the determinant, across the grazing event. We show that the determinant of the matrix remains invariant in the neighborhood of a grazing orbit, and that the singularity appears only in the trace of the matrix. Investigating the character of the trace, we show that the singularity disappears if the damped frequency of the oscillator is an integral multiple of half of the forcing frequency.
On a quasilinear hyperbolic system in blood flow modeling
Tong Li and  Kun Zhao
2011, 16(1): 333-344 doi: 10.3934/dcdsb.2011.16.333 +[Abstract](44) +[PDF](363.4KB)
This paper aims at an initial-boundary value problem on bounded domains for a one-dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that, for given smooth initial data close to a constant equilibrium state, there exists a unique global smooth solution to the model. Time asymptotically, it is shown that the solution converges to the constant equilibrium state exponentially fast as time goes to infinity due to viscous damping and boundary effects.
Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system
Yeping Li
2011, 16(1): 345-360 doi: 10.3934/dcdsb.2011.16.345 +[Abstract](41) +[PDF](394.6KB)
In this paper, we study a two- and three-dimensional bipolar Euler-Poisson system (hydrodynamic model). The system arises in mathematical modeling for semiconductors and plasmas. We are interested in the steady state isentropic case supplemented by the proper boundary conditions. We first show the existence and uniqueness of irrotational subsonic stationary solutions for the two- and three-dimensional hydrodynamic model. Next, we investigate the zero-electron-mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit for above stationary solutions, respectively. For each limit, we show the strong convergence of the sequence of solutions and give the associated convergence rates.
Derivation and stability study of a rigid lid bilayer model
Timack Ngom and  Mamadou Sy
2011, 16(1): 361-383 doi: 10.3934/dcdsb.2011.16.361 +[Abstract](28) +[PDF](416.2KB)
In this paper we present the derivation of a bilayer shallow water model with rigid lid hypothesis. We start from the incompressible Navier-Stokes equations, we introduce a small parameter $\varepsilon$ which is the ratio between the characteristic height and the characteristic length of the fluids domain. We use a formal asymptotic expansion then we resort to averaging to obtain the model. We also prove the stability of the model, in the following sense, up to a subsequence, every sequence of weak solutions converges to a solution of the model.
Positive periodic solution for Brillouin electron beam focusing system
Jingli Ren , Zhibo Cheng and  Stefan Siegmund
2011, 16(1): 385-392 doi: 10.3934/dcdsb.2011.16.385 +[Abstract](43) +[PDF](328.3KB)
An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system $x''+a(1+\cos2t)x=\frac{1}{x}$ for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.
Local and global exponential synchronization of complex delayed dynamical networks with general topology
Jin-Liang Wang , Zhi-Chun Yang , Tingwen Huang and  Mingqing Xiao
2011, 16(1): 393-408 doi: 10.3934/dcdsb.2011.16.393 +[Abstract](48) +[PDF](209.9KB)
In this paper, we consider a generalized complex network possessing general topology, in which the coupling may be nonlinear, time-varying, nonsymmetric and the elements of each node have different time-varying delays. Some criteria on local and global exponential synchronization are derived in form of linear matrix inequalities (LMIs) for the complex network by constructing suitable Lyapunov functionals. Our results show that the obtained sufficient conditions are less conservative than ones in previous publications. Finally, two numerical examples and their simulation results are given to illustrate the effectiveness of the derived results.
Unboundedness of solutions for perturbed asymmetric oscillators
Lixia Wang and  Shiwang Ma
2011, 16(1): 409-421 doi: 10.3934/dcdsb.2011.16.409 +[Abstract](33) +[PDF](346.9KB)
In this paper, we consider the existence of unbounded solutions and periodic solutions for the perturbed asymmetric oscillator with damping

$x'' + f(x )x' + ax^+ - bx^-$ $+ g(x)=p(t), $

where $x^+ =\max\{x,0\}, x^-$ $=\max\{-x,0\}$, $a$ and $b$ are two positive constants, $f(x)$ is a continuous function and $ p(t)$ is a $2\pi $-periodic continuous function, $g(x)$ is locally Lipschitz continuous and bounded. We discuss the existence of periodic solutions and unbounded solutions under two classes of conditions: the resonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\in Q$ and the nonresonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} \notin Q$. Unlike many existing results in the literature where the function $g(x)$ is required to have asymptotic limits at infinity, our main results here allow $g(x)$ be oscillatory without asymptotic limits.

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