Exponentially small splittings in Hamiltonian systems

Families of area preserving analytical maps, depending on a small parameter epsilon, are considered, with the case epsilon=0 corresponding to an integrable map. The asymptotic formulas for the splittings of separatrices are derived by the method of analytical continuation of the separatrices to the complex domain. The main terms of the asymptotics are exponentially small with respect to the size of the perturbation. As epsilon tends to zero, the intersection angle of the separatrices can oscillate. The exponent and the oscillatory multiplier of the asymptotic formulas are determined by the position of poles of the homoclinic (heteroclinic) orbit of the limiting flow. Pre-exponential coefficients in the asymptotic formulas contain a multiplier obtained by the numerical study of separatrices of "model" maps in the complex domain.     

Telomerization of butadiene with diethylamine catalyzed by Ni, Pd and Pt allyl halide complexes

The telomerization of butadiene with diethylamine with the formation of the tertiary amine (C₂H₅)₂NC₈H₁₃ is catalyzed by Ni, Pd and Pt allyl halide complexes. Triphenylphosphine increases the activity of such catalysts. The rate of telomerization depends strongly on the [PPh₃]/[M] ratio (M=Pd and Pt) and increases in the series of metals: Ni3H₅PdCl)₂ exceeds that of (-C₃H₅)₂ Pd by more than two orders of magnitude.

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Ni, Pd Pt (C₂H₅)₂NC₈H₁₃. . [PPh₃]/[M], M=Pd Pt, : Ni3H₅PdCl)₂ (-C₃H₅)₂Pd.

     

Percival variational principle for invariant measures and commensurate-incommensurate phase transitions in one-dimensional chains

We prove for a one-dimensional system of classical particles with potential energy, $$U_{\alpha ,\gamma } = \sum\limits_n {\left[ {\alpha V(x_n ) + F(x_{n + 1} - x_n - \gamma )} \right]} $$ , the existence of such a smooth function γ(α), 0≦α≦α₀(ω) that the system with potential energyU _αγ(α) has the equilibrium state at the temperatureT=0. This is the incommensurate phase with the ratio of periods equal to the prescribed irrational number ω, badly approximated by rational ones. A simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing Percival's energy functional.     

A counterexample to the hypothesis of V. I. Arnol'd regarding the proper vibrations of the membrane

It is shown that the hypothesis of transversality formulated by V. I. Arnol'd is not satisfied for all domains. Examples are presented of domains for which the variations of the eigenvalues of the Laplace operator for small perturbations of the domain are linearly dependent.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 134–137, 1975.