Kazuyuki Yagasaki

We consider atomic force microscopy cantilevers tapping on samples and provide theoretical explanations for main findings of numerical computations and experimental measurements by Lee et al. [Phys. Rev. B 66, 115409 (2002)] when the van der Waals force has only a secondary influence on their dynamics. To this end we use the averaging method and an extended version of the subharmonic Melnikov method. Necessary computations for the subharmonic Melnikov method are performed numerically. An analyti

We develop a Melnikov-type global perturbation technique for detecting the existence of transverse homoclinic orbits and occurrence of homoclinic bifurcations in periodic perturbations of multi-degree-of-freedom Hamiltonian systems. The unperturbed system is assumed to have a saddle-centre whose stable and unstable manifolds do not coincide but intersect in a lower-dimensional manifold, and does not have to be completely integrable. Other Melnikov-type methods do not apply in this situation. We

The subharmonic Melnikov theory for periodic perturbations of planar Hamiltonian systems is improved. An approximation to the associated Poincaré map in action-angle coordinates is explicitly constructed, and existence, stability, and bifurcation theorems for subharmonics are obtained. In particular, simple formulas for determining the stability of subharmonics and invariant circles bifurcating from them at Hopf bifurcations are obtained, and a degenerate resonance case, which was not appropriat

In this paper we study the dynamics of a weakly nonlinear single-degree-of-freedom system subjected to combined parametric and external excitation. The averaging method is used to establish the existence of invariant tori and analyze their stability. Furthermore, by applying the Melnikov technique to the average system it is shown that there exist transverse homoclinic orbits resulting in chaotic dynamics. Numerical simulation results are also given to demonstrate the theoretical results.

This paper describes a driver HomMap to the standard local bifurcation software AUTO for numerical analysis of homoclinic and heteroclinic bifurcations in maps and periodically forced systems. The driver can detect loci of homoclinic points if the unstable or stable manifolds are of dimension one, and treat problems of two or more dimensions. The algorithms used and their implementations in AUTO with HomMap are explained. Three examples are given, namely the Hénon map and the single and coupled