Discrete and Continuous Dynamical Systems - B
August 2022 , Volume 27 , Issue 8
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This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the
We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation
The main purpose of the present paper is to study the blow-up problem of a weakly coupled quasilinear parabolic system as follows:
This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on
This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when
We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type
In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.
We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian
In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns
This paper examines the stabilization problem of the axially moving Kirchhoff beam. Under the nonlinear damping criterion established by the slope-restricted condition, the existence and uniqueness of solutions of the closed-loop system equipped with nonlinear time-delay disturbance at the boundary is investigated via the Faedo-Galerkin approximation method. Furthermore, the solution is continuously dependent on initial conditions. Then the exponential stability of the closed-loop system is established by the direct Lyapunov method, where a novel energy function is constructed.
In this paper, we obtain the uniform estimates with respect to the Knudsen number
In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.
In this paper, we study an elliptic system arising from the U(1)
which are defined on a parallelogram
Whether increasing biodiversity will lead to a promotion (amplification effect) or inhibition (dilution effect) in the transmission of infectious diseases remains to be discovered. In vector-borne infectious diseases, Lyme Disease (LD) and West Nile Virus (WNV) have become typical examples of the dilution effect of biodiversity. Thus, as a vector-borne disease, biodiversity may also play a positive role in the control of the Zika virus. We developed a Zika virus model affected by biodiversity through a competitive mechanism. Through the qualitative analysis of the model, the stability condition of the disease-free equilibrium point and the control threshold of the disease - the basic reproduction number is given. Not only has the numerical analysis verified the inference results, but also it has shown the regulatory effect of the competition mechanism on Zika virus transmission. As competition limits the size of the vector population, the number of final viral infections also decreases. Besides, we also find that under certain parameter conditions, the dilution effect may disappear because of the different initial values. Finally, we emphasized the impact of human activities on biological diversity, to indirectly dilute the abundance of diversity and make the virus continuously spread.
In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.
This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an
HIV infects active uninfected CD4
In this paper, we propose a time-delayed West Nile virus (WNv) model with impulsive culling of mosquitoes. The mathematical difficulty lies in how to choose a suitable phase space and deal with the interaction of delay and impulse. By the recent theory developed in [
This work considers a pursuit-evasion model
with positive parameters
The bistable dynamics of a modified Nicholson's blowflies delay differential equation with Allee effect is analyzed. The stability and basins of attraction of multiple equilibria are studied by using Lyapunov-LaSalle invariance principle. The existence of multiple periodic solutions are shown using local and global Hopf bifurcations near positive equilibria, and these solutions generate long transient oscillatory patterns and asymptotic stable oscillatory patterns.
We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.
In the present study, a nonlinear model is formulated to demonstrate crop - weed interactions, when they both grow together on agricultural land and compete with each other for the same resources like sunlight, water, nutrients etc., under the aegis of herbicides. The developed model is mathematically analyzed through qualitative theory of differential equations to demonstrate rich dynamical characteristics of the system, which are important to be known for maximizing crop yield. The qualitative results reveal that the system not only exhibits stability of more than one equilibrium states, but also undergoes saddle - node, transcritical and Hopf bifurcations, however, depending on parametric combinations. The results of saddle - node and transcritical bifurcations help to plan strategies for maximum crop yield by putting check over the parameters responsible for the depletion of crops due to their interaction with weeds and herbicides. Hopf - bifurcation shows bifurcation of limit cycle through Hopf - bifurcation threshold, which supports that crop - weed interactions are not always of regular type, but they can also be periodic.
The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.
In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns.
The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smale-horseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smale-horseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasi-periodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasi-periodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.
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