# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

November 2017 , Volume 22 , Issue 9

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2017, 22(9): 3235-3258 doi: 10.3934/dcdsb.2017090 +[Abstract](2906) +[HTML](61) +[PDF](1909.6KB)
Abstract:

In this paper, we focus on production network models based on ordinary and partial differential equations that are coupled to semi-Markovian failure rates for the processor capacities. This modeling approach allows for intermediate capacity states in the range of total breakdown to full capacity, where operating and down times might be arbitrarily distributed. The mathematical challenge is to combine the theory of semi-Markovian processes within the framework of conservation laws. We show the existence and uniqueness of such stochastic network solutions, present a suitable simulation method and explain the link to the common queueing theory. A variety of numerical examples emphasizes the characteristics of the proposed approach.

2017, 22(9): 3259-3272 doi: 10.3934/dcdsb.2017136 +[Abstract](2322) +[HTML](63) +[PDF](393.5KB)
Abstract:

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems \begin{document} $\dot{x}=-y+x^2, \;\dot{y}=x+xy$ \end{document}, and \begin{document} $\dot{x}=-y+x^2y, \;\dot{y}=x+xy^2$ \end{document}, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively.

Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively.

2017, 22(9): 3273-3294 doi: 10.3934/dcdsb.2017137 +[Abstract](2221) +[HTML](64) +[PDF](438.7KB)
Abstract:

In this article, we study a coupled Allen-Cahn-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. We prove the existence and uniqueness of the weak and strong solution when the external force contains some delays. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions.

2017, 22(9): 3295-3316 doi: 10.3934/dcdsb.2017138 +[Abstract](2578) +[HTML](77) +[PDF](454.8KB)
Abstract:

This paper is concerned with a prey-predator model with sign-changing intrinsic growth rate in heterogeneous time-periodic environment, where the prey species lives in the whole space but the predator species lives in a region enclosed by a free boundary. It is shown that the results for the case of the non-periodic environment remain true in time-periodic environment. In fact, we first establish a similar spreading-vanishing dichotomy, which implies that if the predator species could spread successfully, then the two species will coexist, and this is certainly for the situation that the predation is relatively weak. Furthermore, some criteria are also obtained for spreading and vanishing. At last, some rough estimates of the asymptotic spreading speed are given if spreading occurs.

2017, 22(9): 3317-3340 doi: 10.3934/dcdsb.2017139 +[Abstract](2417) +[HTML](60) +[PDF](2326.1KB)
Abstract:

This work is concerned with the study of the scalar delay differential equation

motivated by a simple car-following model on an unbounded straight line. Here, the positive real \begin{document} $h$ \end{document} denotes some parameter, and \begin{document} $V$ \end{document} is some so-called optimal velocity function of the traffic model involved. We analyze the existence and local stability properties of solutions \begin{document} $z(t)=c\,t+d$ \end{document}, \begin{document} $t∈\mathbb{R}$ \end{document}, with \begin{document} $c,d∈\mathbb{R}$ \end{document}. In the case \begin{document} $c\not=0$ \end{document}, such a solution of the differential equation forms a wavefront solution of the car-following model where all cars are uniformly spaced on the line and move with the same constant velocity. In particular, it is shown that all but one of these wavefront solutions are located on two branches parametrized by \begin{document} $h$ \end{document}. Furthermore, we prove that along the one branch all solutions are unstable due to the principle of linearized instability, whereas along the other branch some of the solutions may be stable. The last point is done by carrying out a center manifold reduction as the linearization does always have a zero eigenvalue. Finally, we provide some numerical examples demonstrating the obtained analytical results.

2017, 22(9): 3341-3367 doi: 10.3934/dcdsb.2017140 +[Abstract](2022) +[HTML](63) +[PDF](4358.7KB)
Abstract:

Stable fiber bundles are important structures for understanding nonautonomous dynamics. These sets have a hierarchical structure ranging from stable to strong stable fibers. First, we compute corresponding structures for linear systems and prove an error estimate. The spectral concept of choice is the Sacker-Sell spectrum that is based on exponential dichotomies. Secondly, we tackle the nonlinear case and propose an algorithm for the numerical approximation of stable hierarchies in nonautonomous difference equations. This method generalizes the contour algorithm for computing stable fibers from [38,39]. It is based on Hadamard's graph transform and approximates fibers of the hierarchy by zero-contours of specific operators. We calculate fiber bundles and illustrate errors involved for several examples, including a nonautonomous Lorenz model.

2017, 22(9): 3369-3378 doi: 10.3934/dcdsb.2017141 +[Abstract](2129) +[HTML](74) +[PDF](377.6KB)
Abstract:

The fully parabolic Keller-Segel system with logistic source

is considered in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N≥ 1$) under Neumann boundary conditions, where $κ∈\mathbb{R}$, $μ>0$, $χ>0$ and $τ>0$. It is shown that if the ratio $\frac{χ}{μ}$ is sufficiently small, then any global classical solution $(u, v)$ converges to the spatially homogenous steady state $(\frac{κ_+}{μ}, \frac{κ_+}{μ})$ in the large time limit. Here we use an approach based on maximal Sobolev regularity and thus remove the restrictions $τ=1$ and the convexity of $\Omega$ required in [17].

2017, 22(9): 3379-3407 doi: 10.3934/dcdsb.2017142 +[Abstract](2244) +[HTML](63) +[PDF](658.2KB)
Abstract:

In this paper, for non-autonomous RDS we study cocycle attractors with autonomous attraction universes, i.e. pullback attracting some autonomous random sets, instead of non-autonomous ones. We first compare cocycle attractors with autonomous and non-autonomous attraction universes, and then for autonomous ones we establish some existence criteria and characterization. We also study for cocycle attractors the continuity of sections indexed by non-autonomous symbols to find that the upper semi-continuity is equivalent to uniform compactness of the attractor, while the lower semi-continuity is equivalent to an equi-attracting property under some conditions. Finally, we apply these theoretical results to 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing.

2017, 22(9): 3409-3420 doi: 10.3934/dcdsb.2017172 +[Abstract](2473) +[HTML](77) +[PDF](315.2KB)
Abstract:

This paper is devoted to study the dynamical properties of a Leslie-Gower prey-predator system with strong Allee effect in prey. We first gives some estimates, and then study the dynamical properties of solutions. In particular, we mainly investigate the unstable and stable manifolds of the positive equilibrium when the system has only one positive equilibrium.

2017, 22(9): 3421-3438 doi: 10.3934/dcdsb.2017173 +[Abstract](2582) +[HTML](76) +[PDF](1769.8KB)
Abstract:

In this paper, we present the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. The Galerkin mixed finite element satisfying inf-sup condition is used for the spatial discretization and the temporal treatment is implicit/explict scheme, which is Euler implicit scheme for the linear terms and explicit scheme for the nonlinear term. We prove that this method is almost unconditionally convergent and obtain the optimal \begin{document} $H^1-L^2$ \end{document} error estimate of the numerical velocity-pressure under the hypothesis of \begin{document} $H^2$ \end{document}-regularity of the solution for the three dimensional nonstationary Navier-Stokes equations. Finally some numerical experiments are carried out to demonstrate the effectiveness of the method.

2017, 22(9): 3439-3458 doi: 10.3934/dcdsb.2017174 +[Abstract](2795) +[HTML](80) +[PDF](597.3KB)
Abstract:

This is the second part of a series papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high-dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high-dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. The sparse grid Gaussian-Hermite quadrature rule is used to approximate the conditional expectations. And for the associated high-dimensional interpolations, we adopt a spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and highly accurate approximations in high dimensions. Several numerical examples are presented to demonstrate the efficiency of the proposed methods.

2017, 22(9): 3459-3481 doi: 10.3934/dcdsb.2017175 +[Abstract](1814) +[HTML](57) +[PDF](1145.5KB)
Abstract:

A numerical procedure based on the renormalization group (RG) is presented. This procedure will compute the spatial profile and blow up time for self-similar behavior. This will be generated by a family of parabolic IVPs, which includes the semilinear heat equation. This procedure also handles different diffusion structures, finite time extinction problems and exponential absorption with trivial modifications to the power law version. Convergence of the procedure is proved for the semilinear heat equation, which is a marginal perturbation to the heat equation, with a typical class of initial data. Numerical experiments show the accuracy of the method for various related problems. The main feature is simplicity: it will be shown that an explicit numerical method with a fixed mesh size is capable of computing very sharp approximations to these behaviors. This will include a priori detection of the logarithmic correction in the case of the semilinear heat equation.

2017, 22(9): 3483-3498 doi: 10.3934/dcdsb.2017176 +[Abstract](3074) +[HTML](77) +[PDF](3732.3KB)
Abstract:

In this paper we deal with regime switching predator-prey models perturbed by white noise. We give a threshold by which we know whenever a switching predator-prey system is eventually extinct or permanent. We also give some numerical solutions to illustrate that under the regime switching, the permanence or extinction of the switching system may be very different from the dynamics in each fixed state.

2017, 22(9): 3499-3528 doi: 10.3934/dcdsb.2017177 +[Abstract](2225) +[HTML](69) +[PDF](1424.5KB)
Abstract:

Focusing on stochastic dynamics involving continuous states as well as discrete events, this paper investigates dynamical behaviors of stochastic multi-group Lotka-Volterra model with regime switching. The contributions of the paper lie on: (a) giving the sufficient conditions of stochastic permanence for generic stochastic multi-group Lotka-Volterra model, which are much weaker than the existing results in the literature; (b) obtaining the stochastic strong permanence and ergodic property for the mutualistic systems; (c) establishing the almost surely asymptotic estimate of solutions. These can specify some realistic recurring phenomena and reveal the fact that regime switching can suppress the impermanence. A couple of examples and numerical simulations are given to illustrate our results.

2017, 22(9): 3529-3545 doi: 10.3934/dcdsb.2017178 +[Abstract](2489) +[HTML](83) +[PDF](417.6KB)
Abstract:

We develop a fast Hermite finite element method for a one-dimensional space-fractional diffusion equation, by proving that the stiffness matrix of the method can be expressed as a Toeplitz block matrix. Then a block circulant preconditioner is presented. Numerical results are presented to show the utility of the fast method.

2017, 22(9): 3547-3574 doi: 10.3934/dcdsb.2017179 +[Abstract](2867) +[HTML](71) +[PDF](865.2KB)
Abstract:

This paper investigates the formation of time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model with a focus on the effect of cellular growth. We carry out rigorous Hopf bifurcation analysis to obtain the bifurcation values, spatial profiles and time period associated with these oscillating patterns. Moreover, the stability of the periodic solutions is investigated and it provides a selection mechanism of stable time-periodic mode which suggests that only large domains support the formation of these periodic patterns. Another main result of this paper reveals that cellular growth is responsible for the emergence and stabilization of the oscillating patterns observed in the 3×3 system, while the system admits a Lyapunov functional in the absence of cellular growth. Global existence and boundedness of the system in 2D are proved thanks to this Lyapunov functional. Finally, we provide some numerical simulations to illustrate and support our theoretical findings.

2017, 22(9): 3575-3589 doi: 10.3934/dcdsb.2017180 +[Abstract](1922) +[HTML](58) +[PDF](660.9KB)
Abstract:

The Lyapunov function is a very useful tool in the theory of dynamical systems, in particular in the study of the stability of an equilibrium point. In this paper we construct a locally Hölder continuous Lyapunov function for a uniformly expansive set for a map f in a metric space X. In the construction a basic role is played by the functions defining the stable and unstable cone-fields. As a tool we also use the approximately quasiconvex functions.

2017, 22(9): 3591-3614 doi: 10.3934/dcdsb.2017181 +[Abstract](2260) +[HTML](64) +[PDF](546.7KB)
Abstract:

In this paper, a general class of nonlinear dynamical systems described by retarded differential equations with discontinuous right-hand sides is considered. Under the extended Filippov-framework, we investigate some basic stability problems for retarded differential inclusions (RDI) with given initial conditions by using the generalized Lyapunov-Razumikhin method. Comparing with the previous work, the main results given in this paper show that the Lyapunov-Razumikhin function is allowed to have a indefinite or positive definite derivative for almost everywhere along the solution trajectories of RDI. However, in most of the existing literature, the derivative (if it exists) of Lyapunov-Razumikhin function is required to be negative or semi-negative definite for almost everywhere. In addition, the Lyapunov-Razumikhin function in this paper is allowed to be non-smooth. To deal with the stability, we also drop the specific condition that the Lyapunov-Razumikhin function should have an infinitesimal upper limit. Finally, we apply the proposed Razumikhin techniques to handle the stability or stabilization control of discontinuous time-delayed neural networks. Meanwhile, we present two examples to demonstrate the effectiveness of the developed method.

2017, 22(9): 3615-3628 doi: 10.3934/dcdsb.2017182 +[Abstract](2630) +[HTML](72) +[PDF](374.8KB)
Abstract:

The time evolution of probability densities for solutions to stochastic differential equations (SDEs) without delay is usually described by Fokker-Planck equations, which require the adjoint of the infinitesimal generator for the solutions. However, Fokker-Planck equations do not exist for stochastic delay differential equations (SDDEs) since the solutions to SDDEs are not Markov processes and have no corresponding infinitesimal generators. In this paper, we address the open question of finding governing equations for probability densities of SDDEs with discrete time delays. In the governing equation, densities for SDDEs with discrete time delays are expressed in terms of those for SDEs without delay. The latter have been well studied and can be obtained by solving the corresponding Fokker-Planck equations. The governing equation is given in a simple form that facilitates theoretical analysis and numerical computation. Some example are presented to illustrate the proposed governing equations.

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