Exact finite-difference schemes for two-dimensional linear systems with constant coefficients

An exact finite-difference scheme for a system of two linear differential equations with constant coefficients, (d/dt)x(t)=Ax(t)$(\mathrm{d}/\mathrm{d}t)\mathbf{x}(t)=A\mathbf{x}(t)$, is proposed. The scheme is different from what was proposed by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, New Jersey, 1994, p. 147], in which the derivatives of the two equations are formed differently. Our exact scheme is in the form of (1/φ(h))(_xk+1-_xk)=A[θ_xk+1+(1-θ)_xk]$(1/\phi (h))({\mathbf{x}}_{k+1}-{\mathbf{x}}_k)=A[\theta {\mathbf{x}}_{k+1}+(1-\theta ){\mathbf{x}}_k]$; both derivatives are in the same form of (_xk+1-_xk)/φ(h)$(x_{k+1}-x_k)/\phi (h)$.     

Simultaneous bifurcation of limit cycles from a linear center with extra singular points

The period annuli of the planar vector field x^′=−yF(x,y)$x^{\prime }=-yF(x,y)$, y^′=xF(x,y)$y^{\prime }=xF(x,y)$, where the set {F(x,y)=0}$\{F(x,y)=0\}$ consists of k   different isolated points, is defined by k+1$k+1$ concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n  . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1$k=1$, the provided upper bound is reached. Finally, the case k=2$k=2$ is also treated.     

An application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations

In this paper, a class of systems of nonlinear differential equations at resonance is considered. With the use of a global inversion theorem which is an extended form of a non-variational version of a max–min principle, we prove that this class of equations possesses a unique 2π$2\pi$-periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li, Periodic solution for 2k$2k$th boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661–671; F. Cong, Periodic solutions for 2k$2k$th order ordinary differential equations with nonresonance, Nonlinear Anal. 32 (1998) 787–793; F. Cong, Periodic solutions for second order differential equations, Appl. Math. Lett. 18 (2005) 957–961; W. Li, Periodic solutions for 2k$2k$th order ordinary differential equations with resonance, J. Math. Anal. Appl. 259 (2001) 157–167; W. Li, H. Li, A min–max theorem and its applications to nonconservative systems, Int. J. Math. Math. Sci. 17 (2003) 1101–1110; W. Li, Z. Shen, A constructive proof of existence and uniqueness of 2π$2\pi$-periodic solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209–1220]. This result extends the results known so far.     

Finite time blow-up in nonlinear suspension bridge models

This paper settles a conjecture by Gazzola and Pavani [10] regarding solutions to the fourth order ODE w⁽⁴⁾+kw^″+f(w)=0$w^{(4)}+k{w}^{″}+f(w)=0$ which arises in models of traveling waves in suspension bridges when k>0$k>0$. Under suitable assumptions on the nonlinearity f   and initial data, we demonstrate blow-up in finite time. The case k≤0$k\le 0$ was first investigated by Gazzola et al., and it is also handled here with a proof that requires less differentiability on f. Our approach is inspired by Gazzola et al. and exhibits the oscillatory mechanism underlying the finite-time blow-up. This blow-up is nonmonotone, with solutions oscillating to higher amplitudes over shrinking time intervals. In the context of bridge dynamics this phenomenon appears to be a consequence of mutually-amplifying interactions between vertical displacements and torsional oscillations.     

A combinatorial proof of the skew K-saturation theorem

We give a combinatorial proof of the skew version of the K-saturation theorem. More precisely, for any positive integer k$k$, we give an explicit injection from the set of skew semistandard Young tableaux with skew shape kλ/kμ$k\lambda /k\mu$ and type kν$k\nu$, to the set of skew semistandard Young tableaux of shape λ/μ$\lambda /\mu$ and type ν$\nu$.     

Periodic solutions for a Liénard type p-Laplacian differential equation

By using topological degree theory and some analysis skill, we obtain sufficient conditions for the existence and uniqueness of periodic solutions for Liénard type p$p$-Laplacian differential equation.     

The effect of predator competition on positive solutions for a predator–prey model with diffusion

A diffusive predator–prey model with predator competition is considered under Dirichlet boundary conditions. Some existence and non-existence results are firstly obtained. Then by investigating the bifurcation of positive solutions, the multiplicity of positive solutions is established for suitably large m$m$. Furthermore, by meticulously analyzing the asymptotic behaviors of positive solutions when k$k$ goes to ∞$\infty$, we find that there is at most a positive solution for any c∈R$c\in R$ when k$k$ is sufficiently large. At last, some numerical simulations are presented to supplement the analytic results in one dimension.