Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves.     

Chemoselective Catalysis with Organosoluble Lewis Acidic Polyoxotungstates

The preparation of new organosoluble Lewis acidic polyoxometalates (POMs) is reported. These complexes were prepared by the incorporation of Zr, Sc, and Y atoms into the corresponding monolacunary Dawson [P₂W₁₇O₆₁]^10? and Keggin [PW₁₁O₃₉]^7? polyoxotungstates. The catalytic activity of these compounds was evaluated for C? C bond formation in the Diels–Alder, Mannich, and Mukaiyama‐type reactions. Comparisons with previously described Lewis acidic POMs are reported. Competitive reactions between imines and aldehydes or between various imines demonstrated that fine tuning of the reactivity could be reached by varying the metal atom incorporated into the polyanionic framework. A series of experiments that employed pyridine derivatives allowed us to distinguish between the Lewis and induced Brønsted acidity of the POMs. These catalysts activate imines in a Lewis acidic way, whereas aldehydes are activated by indirect Brønsted catalysis.     

Normal Forms, Resonances, and Meandering Tip Motions near Relative Equilibria of Euclidean Group Actions

The equivariant dynamics near relative equilibria to actions of noncompact, finite‐dimensional Lie groups G can be described by a skew‐product flow on a center manifold: with , with v in a slice transverse to the group action, and a(v) in the Lie algebra of G. We present a normal form theory near relative equilibria in this general case. For the specific case of the Euclidean groups the skew product takes the form with . We give a precise meaning to the intuitive idea of tip motion of a meandering spiral: it corresponds to the dynamics of . This clarifies the notion of meander radii and drift resonance in the plane . For illustration, we discuss the unbounded tip motions associated with a weak focus in v, on the verge of Hopf bifurcation, in the case of resonant Hopf and rotation frequencies of the spiral, and study resonant relative Hopf bifurcation. We also encounter random Brownian tip motions for trajectories which become homoclinic for . We conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the Bogdanov‐Takens bifurcation, which turn out to be small beyond any finite order. (Accepted March 30, 1998)     

Biofunctionalization of Polyoxometalates with DNA Primers,Their Use in the Polymerase Chain Reaction (PCR) and Electrochemical Detection of PCR Products

The bioconjugation of polyoxometalates (POMs), which are inorganic metal oxido clusters, to DNA strands to obtain functional labeled DNA primers and their potential use in electrochemical detection have been investigated. Activated monooxoacylated polyoxotungstates [SiW₁₁O₃₉{Sn(CH₂)₂CO}]^8? and [P₂W₁₇O₆₁{Sn(CH₂)₂CO}]^6? have been used to link to a 5′‐NH₂ terminated 21‐mer DNA forward primer through amide coupling. The functionalized primer was characterized by using a battery of techniques, including electrophoresis, mass spectrometry, as well as IR and Raman spectroscopy. The functionality of the POM‐labeled primers was demonstrated through hybridization with a surface‐immobilized probe. Finally, the labeled primers were successfully used in the polymerase chain reaction (PCR) and the PCR products were characterized by using electrophoresis.     

Discrete covalent organic-inorganic hybrids: terpyridine functionalized polyoxometalates obtained by a modular strategy and their metal complexation

The rational design and synthesis of organic-inorganic hybrids as functional molecular materials relies on both the careful conception of building-blocks and the strategy for their assembly. Three families of trialkoxo polyoxometalates (Lindqvist 2, Anderson 3, Dawson 4) grafted with remote terpyridine coordination sites have been synthesized to extend the available building-blocks. These new units can be combined with metal complexes that play a role as (i) chromophores toward charge-separated systems in light-harvesting devices and (ii) coordination motifs for metal-directed self-assembly toward multifunctional molecular hybrid materials. The X-ray crystal structures of polyoxometalate-terpyridine hybrids indicate distances of 21 ? and 19 ? between the two terpyridyl coordination sites in 2 and 3, respectively, with angles between the coordination vectors of 180° and 177.4°, respectively. Lindqvist 2 displays a reduction at -0.52 V vs SCE while Anderson 3 exhibits one reversible oxidation attributed to Mn(III)/Mn(IV) (+0.75 V vs SCE) and a broad wave at -1.28 V vs SCE assigned to the Mn(III)/Mn(II) reduction. Dawson 4 displays several processes on a wide range of potentials (+0.5 to -2.0 V vs SCE) centered on V(V), W(VI) and the organic ligand in order of decreasing potentials. The grafted terpyridine ligands in Anderson 3 and Dawson 4 were successfully coordinated to {PdCl}(+) and {RuCl(3)} moieties, respectively. The polyoxometalates and transition metal complexes retain their intrinsic properties in the final assemblies.     

Dinuclear Ru(II) complexes of bis-(dipyrid-2'-yl)triazine (bis-dpt) ligands as efficient electron reservoirs

A planar bis-dipyrid-2'-yltriazine (bis-dpt) bridging ligand forms dinuclear Ru(II) complexes able to store up to eight electrons upon electrochemical reduction.     

Unbounded sequences of stable limit cycles in the delayed Duffing equation: an exact analysis

Nonlinear Dynamics - The delayed Duffing equation $${\ddot{x}}(t)+x(t-T)+x^3(t)=0$$ is shown to possess an infinite and unbounded sequence of rapidly oscillating, asymptotically stable periodic...     

Connectivity and design of planar global attractors of Sturm type. II: Connection graphs

Based on a Morse-Smale structure we study planar global attractors Af of the scalar reaction-advection-diffusion equation ut=uxx+f(x,u,ux) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f, and hyperbolicity of equilibria. We call Af Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type.The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph Cf which we call the connection graph. Its 1-skeleton consists of the unstable manifolds (separatrices) of the index-1 Morse saddles.We present two results which completely characterize the connection graphs Cf and their 1-skeletons in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1-skeletons are characterized by the existence of cycle-free orientations with certain restrictions on their criticality. Such orientations are called bipolar in [H. de Fraysseix, P.O. de Mendez, P. Rosenstiehl, Bipolar orientations revisited, Discrete Appl. Math. 56 (1995) 157-179].In [B. Fiedler, C. Rocha, Connectivity and design of planar global attractors of Sturm type. I: Orientations and Hamiltonian paths, Crelle J. Reine Angew. Math. (2007), in press] we have shown the equivalence of the two characterizations. Moreover we have established that connection graphs of Sturm attractors indeed satisfy the required properties. In the present paper we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1-skeleton of the required properties. In [B. Fiedler, C. Rocha, Connectivity and design of planar global attractors of Sturm type. III: Small and Platonic examples, 2007, submitted for publication] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons.