Lagrange stability for impulsive Duffing equations

This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations.     

Boundedness of solutions for asymmetric Duffing equations

In this paper, we are concerned with the boundedness of all the solutions and the existence of quasiperiodic solutions for some Duffing equations , where e(t) is of period 1, and g : R → R possesses the characters: g(x) is superlinear when x ? d₀, d₀ is a positive constant and g(x) is semilinear when x ? 0.     

On the analyticity of solutions to semilinear differential equations degenerated on a submanifold

We investigate the analyticity of solutions to semilinear elliptic equations degenerated on a submanifold. We introduce a new weighted Sobolev space which is appropriate for studying such equations. The technique for linear equations using cut-off functions cannot be applied and we need to use a representation formula which requires a fundamental solution.     

On the dynamics of a higher order rational difference equations

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation

$y_{n+1}=a{y}_n+b{y}_{n?t}+c{y}_{n?l}+\frac{d{y}_{n?k}+e{y}_{n?s}}{\alpha y_{n?k}+\beta y_{n?s}},n=0,1,\dots ,$

with positive parameters and non-negative initial conditions. Numerical examples to the difference equation are given to explain our results.     

Stability of equilibrium solutions of Hamiltonian systems with n-degrees of freedom and single resonance in the critical case

In this paper we give new results for the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom. Our Main Theorem generalizes several results existing in the literature and mainly we give information in the critical cases (i.e., the condition of stability and instability is not fulfilled). In particular, our Main Theorem provides necessary and sufficient conditions for stability of the equilibrium solutions under the existence of a single resonance. Using analogous tools used in the Main Theorem for the critical case, we study the stability or instability of degenerate equilibrium points in Hamiltonian systems with one degree of freedom. We apply our results to the stability of Hamiltonians of the type of cosmological models as in planar as in the spatial case.