Kinetic differential equations, being nonlinear, are capable of producing many kinds of exotic phenomena. However, the existence of multistationarity, oscillation or chaos is usually proved by numerical methods. Here we investigate a relatively simple reaction among two species consisting of five reaction steps, one of the third order. Using symbolic methods we find the necessary and sufficient conditions on the parameters of the kinetic differential equation of the reaction under which a limit cycle bifurcates from the stationary point in the positive quadrant in a supercritical Hopf bifurcation. We also performed the search for partial integrals of the system and have found one such integral. Application of the methods needs computer help (Wolfram language and the Singular computer algebra system) because the symbolic calculations to carry out are too complicated to do by hand.
Category: 2019, Vol. 1, No. 3
In  S.-N. Chow and J. A. Sanders proved that the period function is monotone for elliptic Hamiltonian of degree 3. In this paper we significantly simplify their proof, and give a new way to prove this fact, which may be used in other problems.
In this paper, we consider a generalized nonlinear forth-order dispersive-dissipative equation with a nonlocal strong generic delay kernel, which describes wave propagation in generalized nonlinear dispersive, dissipation and quadratic diffusion media. By using geometric singular perturbation theory and Fredholm alternative theory, we get a locally invariant manifold and use fast-slow system to construct the desire heteroclinic orbit. Furthermore we construct a traveling wave solution for the nonlinear equation. Some known results in the literature are generalized.
This paper is devoted to a reaction-diffusion system for a SIR epidemic model with time delay and incidence rate. Firstly, the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained. Secondly, the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations. Finally, the global asymptotical stability are obtained via Lyapunov functionals.
This paper is devoted to the asymptotic dynamics of stochastic chemostat model with Monod-Haldane response function. We first prove the existence of random attractors by means of the conjugacy method and further construct a general condition for internal structure of the random attractor, implying extinction of the species even with small noise. Moreover, we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic chemostat model in an appropriate sense.
This paper studies the factors that affect people coming to China. We select the spending of United States and South Korea, GDP, railway transportation capacity of China as observations. Based on the data from 1988 to 2015, we use the ordinary least squares analysis regression model and present a most reasonable regression equation. It is found that South Korea’s spending is negative to the number of visitors, whereas the spending of US is positive. On comparing the effect of other variables, the growth of railway transportation capacity and the overseas travel expenses of South Korea or United States, have almost no significant impact on the number of people coming to China. However, as China economic growth increasing rapidly, more and more people are willing to come.
This paper is mainly concerned with the exponential stability of a class of hybrid stochastic differential equations–stochastic differential equations with Markovian switching (SDEwMSs). It first devotes to reveal that under the global Lipschitz condition, a SDEwMS is pth (p ∈ (0, 1)) moment exponentially stable if and only if its corresponding improved EulerMaruyama(IEM) method is pth moment exponentially stable for a suitable step size. It then shows that the SDEwMS is pth(p ∈ (0, 1)) moment exponentially stable or its corresponding IEM method with small enough step sizes implies the equation is almost surely exponentially stable or the corresponding IEM method, respectively. In that sense, one can infer that the SDEwMS is almost surely exponentially stable and the IEM method, no matter whether the SDEwMS is pth moment exponentially stable or the IEM method. An example is demonstrated to illustrate the obtained results.
This paper deals with the near-invariant tori for Poisson systems. It is shown that the orbits with the initial points near the Diophantine torus approach some quasi-periodic orbits over an extremely long time. In particular, the results hold for the classical Hamiltonian system, and in this case the drift of the motions is smaller than one in the past works.
In this paper, we establish some new Lyapunov type inequalities for fractional (p, q)-Laplacian operators in an open bounded set Ω ⊂ RN , under homogeneous Dirichlet boundary conditions. Next, we use the obtained inequalities to derive some geometric properties of the generalized spectrum associated to the considered problem.
In this paper, we establish some sufficient conditions on the heat source function and the heat conduction function of the parabolic equation to guarantee that u(xx, t) blows up at finite time, and give upper and lower bounds of the blow-up time in multi-dimensional space.