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Discrete & Continuous Dynamical Systems - B

2014 , Volume 19 , Issue 9

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On optimal control of a sweeping process coupled with an ordinary differential equation
Lukáš Adam and  Jiří Outrata
2014, 19(9): 2709-2738 doi: 10.3934/dcdsb.2014.19.2709 +[Abstract](39) +[PDF](533.6KB)
We study a special case of an optimal control problem governed by a differential equation and a differential rate--independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process.
    We perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.
Transport semigroup associated to positive boundary conditions of unit norm: A Dyson-Phillips approach
Luisa Arlotti and  Bertrand Lods
2014, 19(9): 2739-2766 doi: 10.3934/dcdsb.2014.19.2739 +[Abstract](30) +[PDF](615.4KB)
We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\mathcal{T}_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\mathcal{T}_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t))_{t\geq 0}$. We show here that $(V_H(t))_{t\geq 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t))_{t \geq 0}$ as the sum of an expansion series similar to Dyson-Phillips series. We develop an honesty theory for such boundary perturbations that allows to consider the honesty of trajectories on subintervals $J \subseteq [0,\infty)$. New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.
Straightforward approximation of the translating and pulsating free surface Green function
Zhi-Min Chen
2014, 19(9): 2767-2783 doi: 10.3934/dcdsb.2014.19.2767 +[Abstract](37) +[PDF](767.6KB)
The translating and pulsating free surface Green function represents the velocity potential of a three-dimensional free surface source advancing in waves. This function involves singular wave integral, which is troublesome in numerical computation. In the present study, a regular wave integral approach is developed for the discretisation of the singular wave integral in a whole space harmonic function expansion, which permits the free surface wave produced by the fluid motion to be decomposed by plane regular propagation waves. This approximation gives rise to a simple and straightforward evaluation of the Green function. The algorithm is validated from comparisons between present numerical results and existing numerical data.
Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds
Simone Fiori
2014, 19(9): 2785-2808 doi: 10.3934/dcdsb.2014.19.2785 +[Abstract](32) +[PDF](1491.6KB)
The present research paper proposes an extension of the classical scalar Auto-Regressive Moving-Average (ARMA) model to real-valued Riemannian matrix manifolds. The resulting ARMA model on matrix manifolds is expressed as a non-linear discrete-time dynamical system in state-space form whose state evolves on the tangent bundle associated with the underlying manifold. A number of examples are discussed within the present contribution that aim at illustrating the numerical behavior of the proposed ARMA model. In order to measure the degree of temporal dependency between the state-values of the ARMA model, an extension of the classical autocorrelation function for scalar sequences is suggested on the basis of the geometrical features of the underlying real-valued matrix manifold.
Dynamic transition and pattern formation for chemotactic systems
Tian Ma and  Shouhong Wang
2014, 19(9): 2809-2835 doi: 10.3934/dcdsb.2014.19.2809 +[Abstract](27) +[PDF](453.6KB)
The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$, which is derived by incorporating the nonlinear interactions of both stable and unstable modes. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.
Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative
Imen Manoubi
2014, 19(9): 2837-2863 doi: 10.3934/dcdsb.2014.19.2837 +[Abstract](24) +[PDF](540.3KB)
In this paper, we study the water wave model with a nonlocal viscous term \begin{equation*} u_t + u_x + \beta u_{x x x} + \frac{\sqrt \nu}{\sqrt \pi}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds + u u_x = v u_{xx}, \end{equation*} where $\frac{1}{\sqrt \pi}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds $ is the Riemann-Liouville half derivative. We prove the well-posedness of the equation and we investigate theoretically and numerically the asymptotical behavior of the solutions. Also, we compare our theoretical and numerical results with those given in [4] for a similar equation.
Stability criteria for SIS epidemiological models under switching policies
Mustapha Ait Rami , Vahid S. Bokharaie , Oliver Mason and  Fabian R. Wirth
2014, 19(9): 2865-2887 doi: 10.3934/dcdsb.2014.19.2865 +[Abstract](31) +[PDF](466.7KB)
We study the spread of disease in an SIS model for a structured population. The model considered is a time-varying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a threshold parameter for this model in the spirit of the basic reproduction number for time-invariant models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model.
Stochastically perturbed sliding motion in piecewise-smooth systems
D. J. W. Simpson and  R. Kuske
2014, 19(9): 2889-2913 doi: 10.3934/dcdsb.2014.19.2889 +[Abstract](50) +[PDF](825.1KB)
Sliding motion is evolution on a switching manifold of a discontinuous, piecewise-smooth system of ordinary differential equations. In this paper we quantitatively study the effects of small-amplitude, additive, white Gaussian noise on stable sliding motion. For equations that are static in directions parallel to the switching manifold, the distance of orbits from the switching manifold approaches a quasi-steady-state density. From this density we calculate the mean and variance for the near sliding solution. Numerical results of a relay control system reveal that the noise may significantly affect the period and amplitude of periodic solutions with sliding segments.
Global dynamics of a piece-wise epidemic model with switching vaccination strategy
Aili Wang , Yanni Xiao and  Robert A. Cheke
2014, 19(9): 2915-2940 doi: 10.3934/dcdsb.2014.19.2915 +[Abstract](38) +[PDF](2710.5KB)
A piece-wise epidemic model of a switching vaccination program, implemented once the number of people exposed to a disease-causing virus reaches a critical level, is proposed. In addition, variation or uncertainties in interventions are examined with a perturbed system version of the model. We also analyzed the global dynamic behaviors of both the original piece-wise system and the perturbed version theoretically, using generalized Jacobian theory, Lyapunov constants for a non-smooth vector field and a generalization of Dulac's criterion. The main results show that, as the critical value varies, there are three possibilities for stabilization of the piece-wise system: (i) at the disease-free equilibrium; (ii) at the endemic states for the two subsystems or (iii) at a generalized equilibrium which is a novel global attractor for non-smooth systems. The perturbed system exhibits new global attractors including a pseudo-focus of parabolic-parabolic (PP) type, a pseudo-equilibrium and a crossing cycle surrounding a sliding mode region. Our findings demonstrate that an infectious disease can be eradicated either by increasing the vaccination rate or by stabilizing the number of infected individuals at a previously given level, conditional upon a suitable critical level and the parameter values.
On the steady state of a shadow system to the SKT competition model
Qi Wang
2014, 19(9): 2941-2961 doi: 10.3934/dcdsb.2014.19.2941 +[Abstract](65) +[PDF](702.0KB)
We study a boundary value problem with an integral constraint that arises from the modelings of species competition proposed by Lou and Ni in [10]. Through local and global bifurcation theories, we obtain the existence of non-constant positive solutions to this problem, which are small perturbations from its positive constant solution, over a one-dimensional domain. Moreover, we investigate the stability of these bifurcating solutions. Finally, for the diffusion rate being sufficiently small, we construct infinitely many positive solutions with single transition layer, which is represented as an approximation of a step function. The transition-layer solution can be used to model the segregation phenomenon through interspecific competition.
Second moment boundedness of linear stochastic delay differential equations
Zhen Wang , Xiong Li and  Jinzhi Lei
2014, 19(9): 2963-2991 doi: 10.3934/dcdsb.2014.19.2963 +[Abstract](32) +[PDF](561.3KB)
This paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework--for general $\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay--of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a specific case of a type of $2$-dimensional equation that the stochastic terms are decoupled. For the $2$-dimensional equation, we obtain the characteristic function that is explicitly given by equation coefficients, and the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded.
A reaction-diffusion model of dengue transmission
Zhiting Xu and  Yingying Zhao
2014, 19(9): 2993-3018 doi: 10.3934/dcdsb.2014.19.2993 +[Abstract](36) +[PDF](930.4KB)
This paper is devoted to the mathematical analysis of a reaction-diffusion model of dengue transmission. In the case of a bounded spatial habitat, we obtain the local stability as well as the global stability of either disease-free or endemic steady state in terms of the basic reproduction number $\mathcal{R}_0$. In the case of an unbounded spatial habitat, we establish the existence of the traveling wave solutions connecting the two constant steady states when $\mathcal{R}_0>1$, and the nonexistence of the traveling wave solutions that connect the disease-free steady state itself when $\mathcal{R}_0<1$. Numerical simulations are performed to illustrate the main analytic results.
Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source
Zhengce Zhang and  Yan Li
2014, 19(9): 3019-3029 doi: 10.3934/dcdsb.2014.19.3019 +[Abstract](35) +[PDF](408.7KB)
Throughout this paper, we consider the equation \[u_t - \Delta u = e^{|\nabla u|}\] with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.

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