Methoxide of an Anderson-type polyoxometalate and its conversion to a new type of species, [IMo9O32(OH)(OH2)3]4-

A totally new type of polyoxometalate, [IMo(9)O(32)(OH)(OH(2))(3)](4)(-), has been synthesized by reacting [IMo(6)O(22)(OMe)(2)](3)(-) with water. The [IMo(9)O(32)(OH)(OH(2))(3)](4)(-) anion further transforms into [(IMo(7)O(26))(2)](6)(-), a molecular oxide that has a rutile core, in dry acetonitrile, while it stays intact for several hours in wet acetonitrile.     

Intramolecular rearrangements of cyclooctadienerhodium(I) fragments supported on a vanadium oxide cluster as evidenced by dynamic17O NMR spectroscopy

Variable-temperature¹⁷O NMR together with⁵¹V and¹⁰³Rh NMR studies on newly prepared vanadium oxide-supported organorhodium(I) fragment(s), [(RhCOD) _n (V₄O₁₂)]^(4–n)– (n = 1, 2; COD = ⁴-1,5-cyclooctadiene) indicate that intramolecular rearrangements of RhCOD fragment(s) on a vanadium oxide surface occur in solution.     

Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5n or 6n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic.      

(IMo(7)O(26))(2)](6)(-): A missing link between molecular and solid oxides

A new molecular oxide, [(IMo(7)O(26))(2)](6)(-), that has a self-contained structure has been synthesized. Its structural relevance both to the rutile structure and several molecular oxides that had been classified as "strange ones with odd structures" has given some insights as to why those species assume such structures. The novel yet self-contained nature of the structure suggests the existence of a new class of molecular oxides of related structures.     

Reversible dimerization of decaniobate

Three different solvates of TBA₆[Nb₁₀O₂₈] (TBA = tetra-n-butylammonium) were structurally characterized. The results revealed that two water molecules are hydrogen-bonded to the terminal oxygens of the [Nb₁₀O₂₈]⁶⁻ anion in the same manner in all of the solvates. Decaniobate [Nb₁₀O₂₈]⁶⁻ dimerizes by the action of HCl to form icosaniobate [Nb₂₀O₅₄]⁸⁻, while icosaniobate breaks up into decaniobate [Nb₁₀O₂₈]⁶⁻ by the action of TBAOH. Decaniobate also dimerizes spontaneously to form icosaniobate [Nb₂₀O₅₄]⁸⁻ in CH₂Cl₂ even if no acid is added to the solution. The reaction was followed by IR spectroscopy, and the results suggested the reaction is second order with respect to the concentration of [Nb₁₀O₂₈]⁶⁻.     

A simple feedback control system: Bifurcations of periodic orbits and chaos

We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikov's method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikov's method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.     

Periodic and Homoclinic Motions in Forced,Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.