Discrete and Continuous Dynamical Systems - B
May 2022 , Volume 27 , Issue 5
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A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.
We consider a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects which is introduced in [
We provide a new multiplicity result for a weighted
The aim of this paper is to develop a center manifold theory for a class of stochastic partial differential equations with a non-dense domain through the Lyapunov-Perron method. We construct a novel variation of constants formula by the resolvent operator to formulate the integrated solutions. Moreover, we impose an additional condition involving a non-decreasing map to deduce the required estimate since Young's convolution inequality is not applicable. As an application, we present a stochastic parabolic equation to illustrate the obtained results.
We consider the susceptible-infected-removed (SIR) epidemic model and apply optimal control to it successfully. Here two control inputs are considered, so that the infection rate is decreased and infected individuals are removed. Our approach is to reduce computation of the optimal control input to that of the stable manifold of an invariant manifold in a Hamiltonian system. The situation in which the target equilibrium has a center direction is very different from similar previous work. Some numerical examples in which the computer software AUTO is used to numerically compute the stable manifold are given to demonstrate the usefulness of our approach for the optimal control in the SIR model. Our study suggests how we can decrease the number of infected individuals quickly before a critical situation occurs while keeping social and economic burdens small.
This paper is devoted to the study of the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version. We show the appropriate manner to apply Kato's strategy and this context, with initial conditions in the divergence-free Lebesgue space
This work deals with the Cauchy problem and the asymptotic behavior of the solution of the fermion equation in the Sobolev spaces with a polynomial weight in the torus. We first investigate the linearized equation and obtain the optimal exponential decay rate for the associated semigroup. Our strategy is taking advantage of quantitative spectral gap estimates in smaller reference Hilbert space, the factorization method and the enlargement of the functional space. We then turn to the nonlinear equation and prove the global existence and uniqueness of solutions in a close-to-equilibrium regime. Moreover, we prove an exponential stability for such a solution with the optimal decay rate given by the semigroup decay of the linearized equation.
We perturb with an additive noise the Hamiltonian system associated to a cubic anharmonic oscillator. This gives rise to a system of stochastic differential equations with quadratic drift and degenerate diffusion matrix. Firstly, we show that such systems possess explosive solutions for certain initial conditions. Then, we carry a small noise expansion's analysis of the stochastic system which is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We then investigate the probabilistic properties of the sequence of coefficients which turn out to be the unique strong solutions of stochastic perturbations of the well-known Lamé's equation. We also obtain explicit expressions of these in terms of Jacobi elliptic functions. Furthermore, we prove, in the case of Brownian noise, a lower bound for the probability that the truncated expansion stays close to the solution of the deterministic problem. Lastly, when the noise is bounded, we provide conditions for the almost sure convergence of the global expansion.
In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.
In this article a model of tumor growth is considered. The model is based on the reaction-diffusion equation that describes the distribution of nutrients within the tissue. Our aim was to predict the influence of nutrients on the tumor development. In the tissue the nutrients are transformed into energy, which supports the transfer of chemical and electrical signals and also transfer and copy the information in the tumor cells. We investigate, from a mathematical point of view, under which conditions this process takes place and how it affects the evolution of the tumor.
We study a normal form of the subcritical Hopf bifurcation subjected to time-delayed feedback. An unstable periodic orbit is born at the bifurcation in the normal form without the delay and it can be stabilized by the time-delayed feedback. We show that there exist finite time blow-up solutions for small initial functions, near the bifurcation point, when the feedback gains are small. This can happen even if the origin is stable or the unstable periodic orbit of the normal form is stabilized by the delay feedback. We give numerical examples to illustrate the theoretical result.
This paper is concerned with the wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse. We first show the well-posedness of solutions for such a problem. Then, in terms of the basic reproduction number and the wave speed, we establish a threshold result which reveals the existence and non-existence of the strong traveling waves accounting for phase transitions between the disease-free equilibrium and the endemic steady state. Further, we clarify and characterize the minimal wave speed of traveling waves. Finally, numerical simulations and discussions are also given to illustrate the analytical results. Our result indicates that the relapse can encourage the spread of the disease.
We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals of Crank-Nicolson type. The feature of this functional is that it enjoys energy conservation in the absence of free boundaries, which is an essential property for numerical calculations. The existence and regularity of minimizers is shown and an energy estimate is derived. These results are then used to show the existence of a weak solution to the free boundary problem in the 1-dimensional setting.
We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in [
In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the
We consider the fourth-order Schrödinger equation
In this paper, using the Poincaré compactification technique we classify the topological phase portraits of a special kind of quadratic differential system, the Abel quadratic equations of third kind. In [
In this study, we develop a diffusive HIV-1 infection model with intracellular invasion, production and latent infection distributed delays, nonlinear incidence rate and nonlinear CTL immune response. The well-posedness, local and global stability for the model proposed are carefully investigated in spite of its strong nonlinearity and high dimension. It is revealed that its threshold dynamics are fully determined by the viral infection reproduction number
The manuscript aims to investigate the qualitative analysis of a plankton-fish interaction with food limited growth rate of plankton population and non-constant harvesting of fish population. The ecological feasibility of population densities of both plankton and fish in terms of positivity and boundedness of solutions is shown. The conditions for the existence of various equilibrium points and their stability are derived thoroughly. This study mainly focuses on how the harvesting affects equilibrium points, their stability, periodic solutions and bifurcations in the proposed system. It is shown that the system exhibits saddle-node bifurcation in the form of a collision of two interior equilibrium points. Existence conditions for the occurrence of Hopf-bifurcation around interior equilibrium points are discussed. Lyapunov coefficients are examined to check the stability properties of these periodic solutions. We have also plotted the bifurcation diagrams for saddle-node, transcritical and Hopf bifurcations. A detailed algorithm for the occurrence of Bogdanov-Takens bifurcation is derived and finally some numerical simulations are also carried out to validate the theoretical results. This work suggests that the harvesting of fish population can change the dynamics of the system, which may be useful for the ecological management.
We consider the Cournot-Theocharis oligopoly model, where firms make their choices under adaptive expectations. Following [
We consider the logistic family
In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The problem is written on mixed form using nonconforming
In this note, we consider generalizations of the Cucker-Smale dynamical system and we derive rigorously in Wasserstein's type topologies the mean-field limit (and propagation of chaos) to the Vlasov-type equation introduced in [
Based on the method of compression and pull forming mechanism (CAP), the authors in a well-known paper proposed and analyzed the Lü-like system:
The paper studies the pattern formation dynamics of a discrete in time and space model with nonlocal resource competition and dispersal. Our model is generalized from the metapopulation model proposed by Doebeli and Killingback [2003. Theor. Popul. Biol. 64, 397-416] in which competition for resources occurs only between neighboring populations. Our study uses symmetric discrete probability kernels to model nonlocal interaction and dispersal. A linear stability analysis of the model shows that solutions to this equation exhibits pattern formation when the dispersal rate is sufficiently small and the discrete interaction kernel satisfies certain conditions. Moreover, a weakly nonlinear analysis is used to approximate stationary patterns arising from the model. Numerical solutions to the model and the approximations obtained through the weakly nonlinear analysis are compared.
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