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Discrete & Continuous Dynamical Systems - B

2012 , Volume 17 , Issue 5

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Digraphs vs. dynamics in discrete models of neuronal networks
Sungwoo Ahn and Winfried Just
2012, 17(5): 1365-1381 doi: 10.3934/dcdsb.2012.17.1365 +[Abstract](429) +[PDF](394.5KB)
Abstract:
It has recently been shown that discrete-time finite-state models can reliably reproduce the ordinary differential equation (ODE) dynamics of certain neuronal networks. We study which dynamics are possible in these discrete models for certain types of network connectivities. In particular we are interested in the number of different attractors and bounds on the lengths of attractors and transients. We completely characterize these properties for cyclic connectivities and derive additional results on the lengths of attractors in more general classes of networks.
Convergence results for the vector penalty-projection and two-step artificial compressibility methods
Philippe Angot and Pierre Fabrie
2012, 17(5): 1383-1405 doi: 10.3934/dcdsb.2012.17.1383 +[Abstract](439) +[PDF](404.4KB)
Abstract:
In this paper, we propose and analyze a new artificial compressibility splitting method which is issued from the recent vector penalty-projection method for the numerical solution of unsteady incompressible viscous flows introduced in [1], [2] and [3]. This method may be viewed as an hybrid two-step prediction-correction method combining an artificial compressibility method and an augmented Lagrangian method without inner iteration. The perturbed system can be viewed as a new approximation to the incompressible Navier-Stokes equations. In the main result, we establish the convergence of solutions to the weak solutions of the Navier-Stokes equations when the penalty parameter tends to zero.
Time-averaging and permanence in nonautonomous competitive systems of PDEs via Vance-Coddington estimates
Joanna Balbus and Janusz Mierczyński
2012, 17(5): 1407-1425 doi: 10.3934/dcdsb.2012.17.1407 +[Abstract](433) +[PDF](573.1KB)
Abstract:
One of the mathematically challenging problems in the population dynamics is finding conditions under which all of the populations coexist. A mathematical formulation of this notion is the concept of permanence, sometimes called also uniform persistence. In this article we give conditions for permanence in nonautonomous competitive Kolmogorov systems of reaction-diffusion equations. Those conditions are in a form of inequalities involving time-averages of intrinsic growth rates, as well as interaction coefficients, migration rates and principal eigenvalues. The proofs use estimates due to R. R. Vance and E. A. Coddington. Connections with invasibility via the principal spectrum theory are also investigated.
A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations
Laurent Boudin, Bérénice Grec and Francesco Salvarani
2012, 17(5): 1427-1440 doi: 10.3934/dcdsb.2012.17.1427 +[Abstract](529) +[PDF](570.8KB)
Abstract:
We consider the Maxwell-Stefan model of diffusion in a multicomponent gaseous mixture. After focusing on the main differences with the Fickian diffusion model, we study the equations governing a three-component gas mixture. Mostly in the case of a tridiagonal diffusion matrix, we provide a qualitative and quantitative mathematical analysis of the model. We develop moreover a standard explicit numerical scheme and investigate its main properties. We eventually include some numerical simulations underlining the uphill diffusion phenomenon.
Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions
Guanggan Chen and Jian Zhang
2012, 17(5): 1441-1453 doi: 10.3934/dcdsb.2012.17.1441 +[Abstract](482) +[PDF](392.5KB)
Abstract:
This work is concerned with the asymptotic dynamical behavior for a weakly damped stochastic nonlinear wave equation with dynamical boundary conditions. The white noises appear both in the model and in the dynamical boundary condition. Since the energy relation of this stochastic system does not directly imply the a priori estimate of the solution, we propose a pseudo energy equation to infer almost sure boundedness of the solution. Then a unique invariant measure is shown to exist for the system.
The Euler-Maruyama approximation for the absorption time of the CEV diffusion
Pavel Chigansky and Fima C. Klebaner
2012, 17(5): 1455-1471 doi: 10.3934/dcdsb.2012.17.1455 +[Abstract](533) +[PDF](435.1KB)
Abstract:
The standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires non-degeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the Euler-Maruyama scheme.
Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's
Konstantinos Chrysafinos and Efthimios N. Karatzas
2012, 17(5): 1473-1506 doi: 10.3934/dcdsb.2012.17.1473 +[Abstract](535) +[PDF](544.5KB)
Abstract:
A discontinuous Galerkin finite element method for an optimal control problem having states constrained to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. It is shown that under suitable assumptions, the error estimates of the corresponding optimality system are of the same order to the standard linear (uncontrolled) parabolic problem. These estimates have symmetric structure and are also applicable for higher order elements.
Period doubling and reducibility in the quasi-periodically forced logistic map
Àngel Jorba, Pau Rabassa and Joan Carles Tatjer
2012, 17(5): 1507-1535 doi: 10.3934/dcdsb.2012.17.1507 +[Abstract](497) +[PDF](3216.1KB)
Abstract:
We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, such as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and reducibility. This reveals that the parameter values for which the invariant curve doubles its period are contained in regions of the parameter space where the invariant curve is reducible. Then we present two additional studies to explain this fact. In first place we consider the images and the preimages of the critical set (the set where the derivative of the map w.r.t the non-periodic coordinate is equal to zero). Studying these sets we construct constrains in the parameter space for the reducibility of the invariant curve. In second place we consider the reducibility loss of the invariant curve as a codimension one bifurcation and we study its interaction with the period doubling bifurcation. This reveals that, if the reducibility loss and the period doubling bifurcation curves meet, they do it in a tangent way.
Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations
Qihuai Liu, Dingbian Qian and Zhiguo Wang
2012, 17(5): 1537-1550 doi: 10.3934/dcdsb.2012.17.1537 +[Abstract](511) +[PDF](389.4KB)
Abstract:
In this paper, we prove the existence of positive quasi-periodic solutions for the Lotka-Volterra competition systems with quasi-periodic coefficients by KAM technique. The result shows that, in most case, quasi-periodic solutions exist for sufficiently small quasi-periodic perturbations of the autonomous Lotka-Volterra systems. Moreover, these quasi-periodic solutions will tend to an equilibrium of the autonomous Lotka-Volterra systems.
Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows
Kun Wang, Yinnian He and Yanping Lin
2012, 17(5): 1551-1573 doi: 10.3934/dcdsb.2012.17.1551 +[Abstract](470) +[PDF](408.9KB)
Abstract:
In this article, we study the long time numerical stability and asymptotic behavior for the viscoelastic Oldroyd fluid motion equations. Firstly, with the Euler semi-implicit scheme for the temporal discretization, we deduce the global $H^2-$stability result for the fully discrete finite element solution. Secondly, based on the uniform stability of the numerical solution, we investigate the discrete asymptotic behavior and claim that the viscoelastic Oldroyd problem converges to the stationary Navier-Stokes flows if the body force $f(x,t)$ approaches to a steady-state $f_\infty(x)$ as $t\rightarrow\infty$. Finally, some numerical experiments are given to verify the theoretical predictions.
Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation
Xiangjun Wang, Jianghui Wen, Jianping Li and Jinqiao Duan
2012, 17(5): 1575-1584 doi: 10.3934/dcdsb.2012.17.1575 +[Abstract](450) +[PDF](188.6KB)
Abstract:
The Thermohaline Circulation, which plays a crucial role in the global climate, is a cycle of deep ocean due to the change in salinity and temperature (i.e., density). The effects of non-Gaussian noise on the Stommel box model for the Thermohaline Circulation are considered. The noise is represented by a non-Gaussian $\alpha$-stable Lévy motion with $0<\alpha < 2$. The $\alpha$ value may be regarded as the index of non-Gaussianity. When $\alpha=2$, the $\alpha$-stable Lévy motion becomes the usual (Gaussian) Brownian motion.
    Dynamical features of this stochastic model is examined by computing the mean exit time for various $\alpha$ values. The mean exit time is simulated by numerically solving a deterministic differential equation with nonlocal interactions. It has been observed that some salinity difference levels remain in certain ranges for longer times than other salinity difference levels, for different $\alpha$ values. This indicates a lower variability for these salinity difference levels. Realizing that it is the salinity differences that drive the thermohaline circulation, this lower variability could mean a stable circulation, which may have further implications for the global climate dynamics.
A generalized $\theta$-scheme for solving backward stochastic differential equations
Weidong Zhao, Yang Li and Guannan Zhang
2012, 17(5): 1585-1603 doi: 10.3934/dcdsb.2012.17.1585 +[Abstract](777) +[PDF](691.2KB)
Abstract:
In this paper we propose a new type of $\theta$-scheme with four parameters ($\{\theta_i\}_{i=1}^4$) for solving the backward stochastic differential equation $-dy_t=f(t,y_t,z_t) dt - z_t dW_t$. We rigorously prove some error estimates for the proposed scheme, and in particular, we show that accuracy of the scheme can be high by choosing proper parameters. Various numerical examples are also presented to verify the theoretical results.

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