The existence of quasiperiodic solutions for coupled Duffing-type equations

In this paper, we prove the existence of infinitely many quasiperiodic solutions for a class of coupled Duffing-type equations via KAM theorem. Moreover, the set of quasiperiodic solutions is of infinitely Lebesgue measure in the phase space.     

Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems

By using the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order self-adjoint discrete Hamiltonian system Δ[p(n)Δu(n−1)]−L(n)u(n)+∇W(n,u(n))=0 has infinitely many homoclinic orbits, where n∈Z, u∈RN, p,L:Z→RN×N and W:Z×RN→R are no periodic in n. Our conditions on the potential W(n,x) are rather relaxed.     

Two families of Liouville integrable lattice equations

By virtue of zero curvature representations, we are successful to generate the Lax representations of two hierarchies of discrete lattice equations respectively, which are derived from two new and interesting 3 × 3 matrix spectral problems. Moreover, by using the trace identity, the bi-Hamiltonian structures of the above systems are given, and it is shown that they are integrable in the Liouville sense. Finally, infinitely many conservation laws for the second hierarchy of lattice equations are given by a direct method.     

Defect indices and definiteness conditions for a class of discrete linear Hamiltonian systems

This paper is concerned with a class of discrete linear Hamiltonian systems in finite or infinite intervals. A definiteness condition and its equivalent statements are discussed and three sufficient conditions for the definiteness condition are given. A precise relationship between the defect index of the minimal subspace generated by the system and the number of linearly independent square summable solutions of the system is established. In particular, they are equal if and only if the definiteness condition is satisfied. Finally, two criteria for the limit point case and one criterion for the limit circle case are obtained.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.     

On the sizes of graphs and their powers: The undirected case

Let G be an undirected graph and G^r be its r-th power. We study different issues dealing with the number of edges in G and G^r. In particular, we answer the following question: given an integer r≥2 and all connected graphs G of order n such that G^r≠K_n, what is the minimum number of edges that are added when going from G to G^r, and which are the graphs achieving this bound?     

Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach

The purpose of this paper is to apply the Hamiltonian approach to nonlinear oscillators. The Hamiltonian approach is applied to derive highly accurate analytical expressions for periodic solutions or for approximate formulas of frequency. A conservative oscillator always admits a Hamiltonian invariant, H , which stays unchanged during oscillation. This property is used to obtain approximate frequency–amplitude relationship of a nonlinear oscillator with high accuracy. A trial solution is selected with unknown parameters. Next, the Ritz–He method is used to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations. In contrast with the traditional methods, the proposed method does not require any small parameter in the equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Infinitely many homoclinic solutions for some second order Hamiltonian systems

In this paper, we investigate the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems. By using fountain theorem due to Zou, we obtain two new criteria for guaranteeing that second order Hamiltonian systems have infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.     

Effective Hamiltonian in the extended Krenciglowa-Kuo method

The extended Krenciglowa-Kuo (EKK) method allows us to calculate the effective Hamiltonian in a non-degenerate model space. We show that the EKK method can be implemented numerically in two iterative schemes, which are explained in detail with emphasis on convergence conditions. Using test calculations in a simple model, we clarify how and on what conditions we can calculate the effective Hamiltonian.