American Institute of Mathematical Sciences

Toric differential inclusions play a pivotal role in providing a rigorous interpretation of the connection between weak reversibility and the persistence of mass-action systems and polynomial dynamical systems. We introduce the notion of quasi-toric differential inclusions, which are strongly related to toric differential inclusions, but have a much simpler geometric structure. We show that every toric differential inclusion can be embedded into a quasi-toric differential inclusion and that every quasi-toric differential inclusion can be embedded into a toric differential inclusion. In particular, this implies that weakly reversible dynamical systems can be embedded into quasi-toric differential inclusions.

This paper proposes a new scheme generalized hybrid projective synchronization for two different chaotic systems using adaptive control, where the master and slave systems do not necessarily have the same number of uncertain parameters. In this method the master system is synchronized by the sum of hybrid state variables for the slave system. Based on Lyapunov stability theory, an adaptive controller for the synchronization of two different chaotic systems is proposed, This method is also applicable if the master and slave systems are identical. As example the generalized hybrid projective synchronization between Vaidyanathan and Zeraoulia chaotic systems are discussed. Numerical simulation are provided to demonstrate the effectiveness of the proposed method.

The paper concerns the construction of a compressible liquid-vapor relaxation model which is able to capture the metastable states of the non isothermal van der Waals model as well as saturation states. Starting from the Gibbs formalism, we propose a dynamical system which complies with the second law of thermodynamics. Numerical simulations illustrate the expected behaviour of metastable states: an initial metastable condition submitted to a certain perturbation may stay in the metastable state or reaches a saturation state. The dynamical system is then coupled to the dynamics of the compressible fluid using an Euler set of equations supplemented by convection equations on the fractions of volume, mass and energy of one of the phases.

In this article, we propose a new over-penalized weak Galerkin (OPWG) method with a stabilizer for second-order elliptic problems. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements \begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{d}) $\end{document}, or \begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k-1}, [\mathbb{P}_{k-1}]^{d}) $\end{document}, with dimensions of space \begin{document}$ d = 2, \; 3 $\end{document}. The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shape-regularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are \begin{document}$ O(h^{-\beta_{0}(d-1)-1}) $\end{document}, \begin{document}$ \beta_{0} $\end{document} being the penalty exponent. Therefore we introduce a new mini-block diagonal preconditioner, which is proven to be theoretically and numerically effective in reducing the condition numbers of stiffness matrices to the magnitude of \begin{document}$ O(h^{-2}) $\end{document}. Optimal error estimates in a discrete \begin{document}$ H^1 $\end{document}-norm and \begin{document}$ L^2 $\end{document}-norm are established, from which the optimal penalty exponent can be easily chosen. Several numerical examples are presented to demonstrate flexibility, effectiveness and reliability of the new method.

In this paper, we investigate an accurate and efficient method for nonlinear Maxwell's equation. DG method and Crank-Nicolson scheme are employed for spatial and time discretization, respectively. A semi-explicit extrapolation technique is adopted for the linearization of the nonlinear term. Since the proposed scheme is semi-implicit, only a linear system needs to be solved at each time step. Optimal convergent order of \begin{document}$ O(\tau^2+h^{p+\frac{1}{2}}) $\end{document} is proved under time step size condition \begin{document}$ \tau\leq h^{d/4} $\end{document}. Finally, 2D and 3D numerical examples are provided to validate the theoretical convergence rate.

Concerning a class of diffusive logistic equations, Ni [1,Abstract] proposed an optimization problem to consider the supremum of the ratio of the \begin{document}$ L^1 $\end{document} norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is \begin{document}$ 3 $\end{document} in the one-dimensional case. In [1], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an \begin{document}$ L^1 $\end{document} unbounded sequence of sub-solutions.

In this paper, we consider a microscopic semilinear elliptic equation posed in periodically perforated domains and associated with the Fourier-type condition on internal micro-surfaces. The first contribution of this work is the construction of a reliable linearization scheme that allows us, by a suitable choice of scaling arguments and stabilization constants, to prove the weak solvability of the microscopic model. Asymptotic behaviors of the microscopic solution with respect to the microscale parameter are thoroughly investigated in the second theme, based upon several cases of scaling. In particular, the variable scaling illuminates the trivial and non-trivial limits at the macroscale, confirmed by certain rates of convergence. Relying on classical results for homogenization of multiscale elliptic problems, we design a modified two-scale asymptotic expansion to derive the corresponding macroscopic equation, when the scaling choices are compatible. Moreover, we prove the high-order corrector estimates for the homogenization limit in the energy space \begin{document}$ H^1 $\end{document}, using a large amount of energy-like estimates. A numerical example is provided to corroborate the asymptotic analysis.

A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.

We examine a Wong-Zakai type approximation of a family of stochastic differential equations driven by a general càdlàg semimartingale. For such an approximation, compared with the pointwise convergence result by Kurtz, Pardoux and Protter [10,Theorem 6.5], we establish stronger convergence results under the Skorokhod \begin{document}$ M_1 $\end{document}-topology, which, among other possible applications, implies the convergence of the first passage time of the solution to the approximating stochastic differential equation.

We consider a problem that describes the motion of a viscous incompressible and heat-conducting micropolar fluids in a bounded domain \begin{document}$ \Omega \subset \mathbb{R}^3 $\end{document}. We use an iterative method to analyze the existence, uniqueness, and regularity of the solutions. We also determine the convergence rates in several norms.

In this paper we consider the following chemotaxis-growth system with nonlinear signal production and logistic source

\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi\nabla\cdot( u\nabla v)+\lambda u-\mu u^{\alpha}, \quad &x\in \Omega, t>0, \\ 0 = \Delta v-\mu (t)+f(u), \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}f(u(\cdot, t)), \quad &x\in \Omega, t>0, \ \end{array} \right. \end{eqnarray*} $\end{document}

with homogeneous Neumann boundary conditions in the ball \begin{document}$ \Omega = B_R(0)\subset \mathbb{R}^n $\end{document} for \begin{document}$ n\geq 1 $\end{document} and \begin{document}$ R>0 $\end{document}, where \begin{document}$ \chi, \lambda, \mu>0 $\end{document}, \begin{document}$ \alpha>1 $\end{document}, and \begin{document}$ f $\end{document} is an appropriate regular function satisfying \begin{document}$ f(u)\geq ku^{\kappa} $\end{document} for all \begin{document}$ u\geq1, \kappa>0 $\end{document} with some constants \begin{document}$ k>0 $\end{document}. If the number \begin{document}$ \kappa $\end{document} and \begin{document}$ \alpha $\end{document} satisfy

\begin{document}$ \begin{equation*} \kappa+1>\alpha\left(\frac{2}{n}+1\right), \end{equation*} $\end{document}

then there exists appropriate initial data such that the corresponding solution \begin{document}$ (u, v) $\end{document} of the system blow up in finite time. This result extends the blow-up result of the chemotaxis model without logistic cell kinetics in [45]. Apparently, for the case \begin{document}$ \kappa = 1 $\end{document}, this provides a rigorous detection for blow-up of solution in spaces-dimensions three and four.

In this paper, we model a mosquito plasticity problem and investigate the large time behavior of matured population under different control strategies. We prove that when the control is small, then the matured population will become large for large time and when the control is large, then the matured population will become small for large time. In the intermediate case, we derive a time-delayed model for the matured population which can be governed by a sub-equation and a super-equation. We prove the existence of traveling fronts for the sub-equation and use it to prove that the matured population will finally be between the positive states of the sub-equation and super-equation. At last, we present numerical simulations.

We introduce an over-penalized weak Galerkin method for elliptic interface problems with non-homogeneous boundary conditions and discontinuous coefficients, where the method combines a weak Galerkin stabilizer with interior penalty terms. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements \begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{2}) $\end{document}. As an advantage of the method, elliptic interface problems with low regularity are approximated well. The over-penalized weak Galerkin method is based on weak functions whose edge part is double-valued on each interior edge sharing by two neighboring elements. Jumps between the edge parts are naturally used to define penalty terms. The over-penalized weak Galerkin method allows to use arbitrary meshes, even for low regularity solutions. These features make the new method more flexible and efficient for solving interface equations. Furthermore, a priori error estimates in energy and \begin{document}$ L^{2} $\end{document} norms are derived rigorously, and numerical results confirm the effectiveness of the method.

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive Lotka-Volterra type competition system with advection terms under the homogeneous Dirichlet boundary condition. First, we obtain the existence, multiplicity and explicit structure of the spatially nonhomogeneous steady-state solutions by using implicit function theorem and Lyapunov-Schmidt reduction method. Secondly, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of spatially nonhomogeneous positive steady-state solutions and the non-existence of Hopf bifurcations at spatially nonhomogeneous positive steady-state solutions are given. Finally, two concrete examples are provided to support our previous theoretical results. It should be noticed that an elliptic operator with advection term is not self-adjoint, which causes some trouble in the spatial decomposition, explicit expressions of steady-state solutions and some deductive processes related to infinitesimal generators. Moreover, unlike other work, the advection rate here depends on the spatial position, which increases some difficulties in the investigation of the principal eigenvalue.

We consider the Cauchy problem of two-dimensional chemotaxis-shallow water system in the present paper. For regular initial data with small energy but possibly large oscillations, we prove the global well-posedness of classical solution. Then, we show the large-time behavior of the solution using the time-independent lower-order estimates as well.

In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions. Next, we obtain some sufficient conditions ensuring the uniform persistence. Finally, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions.

In this article, we present several results on Finite-Time Stability (FTS) of impulsive differential inclusion. In order to investigate the FTS problem, a new concept of Finite-Time Stable Function Pair (FTSFP) is proposed. By virtue of average impulsive interval and FTSFP, two unified criteria on FTS of impulsive differential inclusion are obtained, which are effective for both the destabilizing impulses and the stabilizing impulses. In addition, the settling-time depends not only on the initial value, but also on the information of impulsive sequence. As an extension, a delay-independent FTS result of impulsive delayed differential inclusion is presented. Finally, the obtained results are applied to study the FTS of discontinuous impulsive neural networks.

This paper is devoted to investigate the dynamics of a stochastic susceptible-infected-susceptible epidemic model with nonlinear incidence rate and three independent Brownian motions. By defining a threshold \begin{document}$ \lambda $\end{document}, it is proved that if \begin{document}$ \lambda>0 $\end{document}, the disease is permanent and there is a stationary distribution. And when \begin{document}$ \lambda<0 $\end{document}, we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution. Moreover, the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results. As a result, appropriate intensities of white noises make the susceptible and infected individuals fluctuate around their deterministic steady–state values; the larger the intensities of the white noises are, the larger amplitude of their fluctuations; but too large intensities of white noises may make both of the susceptible and infected individuals go to extinction.

It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction \begin{document}$ \mathcal{B} $\end{document} with relatively simple geometry: the boundary of each component of \begin{document}$ \mathcal{B} $\end{document} consists of the union of two unordered curves, and the components of \begin{document}$ \mathcal{B} $\end{document} are not comparable as sets. The boundary curves are Lipschitz if the map is of class \begin{document}$ C^1 $\end{document}. Further, if a periodic point is in \begin{document}$ \partial \mathcal{B} $\end{document}, then \begin{document}$ \partial\mathcal{B} $\end{document} is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.

In this study, we consider a system of homogeneous linear differential equations, the coefficients and initial values of which are constant intervals. We apply the approach that treats an interval problem as a set of real (classical) problems. In previous studies, a system of linear differential equations with real coefficients, but with interval forcing terms and interval initial values was investigated. It was shown that the value of the solution at each time instant forms a convex polygon in the coordinate plane. The motivating question of the present study is to investigate whether the same statement remains true, when the coefficients are intervals. Numerical experiments show that the answer is negative. Namely, at a fixed time, the region formed by the solution's value is not necessarily a polygon. Moreover, this region can be non-convex.

The solution, defined in this study, is compared with the Hukuhara- differentiable solution, and its advantages are exhibited. First, under the proposed concept, the solution always exists and is unique. Second, this solution concept does not require a set-valued, or interval-valued derivative. Third, the concept is successful because it seeks a solution from a wider class of set-valued functions.

The Winfree model is the first phase model for synchronization and it exhibits diverse asymptotic patterns that cannot be observed in the Kuramoto model. In this paper, we propose a Winfree type model describing the aggregation of particles on the surface of an infinite cylinder. For a special case, our proposed model is in fact equivalent to the complex Winfree model. For the proposed model, we present a sufficient framework leading to the complete oscillator death and uniform \begin{document}$ \ell_p $\end{document}-stability in a large coupling regime. We also derive the corresponding kinetic model via uniform-in-time mean-field limit. In addition, we also provide several numerical simulations for the particle and compare them with analytical results.

This paper studies an adhesive contact model which also takes into account the damage and long memory term. The deformable body is composed of a viscoelastic material and the process is taken as quasistatic. The damage of the material caused by the compression or the tension is involved in the constitutive law and the damage function is modelled through a nonlinear parabolic inclusion. Meanwhile, the adhesion process is modelled by a bonding field on the contact surface while the contact is described by a nonmonotone normal compliance condition. The variational formulation of the model is governed by a coupled system which consists of a history-dependent hemivariational inequality for the displacement field, a nonlinear parabolic variational inequality for the damage field and an ordinary differential equation for the adhesion field. We first consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived. At the end of this paper, {we report some two-dimensional numerical simulation results} for the contact problem in order to provide numerical evidence of the theoretical results.

This paper is concerned with the study on the existence of attractors for a nonlinear porous elastic system subjected to a delay-type damping in the volume fraction equation. The study will be performed, from the point of view of quasi-stability for infinite dimensional dynamical systems and from then on we will have the result of the existence of global and exponential attractors.

In this paper, we study the Wong-Zakai approximations given by a stationary process via Euler approximation of Brownian motion and the associated long term behavior of the stochastic wave equation driven by an additive white noise on unbounded domains. We first prove the existence and uniqueness of tempered pullback attractors for stochastic wave equation and its Wong-Zakai approximation. Then, we show that the attractor of the Wong-Zakai approximate equation converges to the one of the stochastic wave equation driven by additive noise as the correlation time of noise approaches zero.

This paper is devoted to the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by Poincaré map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods.