Chaotic dynamics of charged particles in the field of two monochromatic waves in a magnetized plasma

We study the dynamics of charged particles in the presence of two electrostatic waves propagating obliquely to an ambient magnetic field. The presence of a second wave makes the problem a two-dimensional and time-dependent one with a complicated phase space behavior. We derive a set of difference equations (maps) for the nonrelativistic particle motion limit and numerically study them to elucidate the various aspects of the phase space dynamics. For the general case of oblique propagation, we observe synergistic effects leading to the lowering of the stochasticity threshold and the concomitant reduction in electric field amplitudes for particle heating applications. These results can be understood in terms of the resonance structures associated with the two waves and we obtain approximate analytic expressions for the thresholds. For the degenerate case of omega(1)=nOmega,omega(2)=mOmega (where omega(1),omega(2) are the frequencies of the two waves, Omega is the cyclotron frequency and n,m are integers) and strictly perpendicular propagation, the problem simplifies to a one-and-one-half-dimensional one. We observe the presence of stochastic webs in this situation. (c) 1996 American Institute of Physics.     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

Scaling of decay lengths for surface states in the gaps of one-dimensional semi-infinite superlattices

We look at some one-dimensional semi-infinite superlattices with an underlying Hamiltonian that is of the nearest neighbour, tight binding type. A real space rescaling procedure which is exact in one dimension is applied to obtain the location of the subbands. It has been found that these subbands never overlap in 1D, and we interpret this as a band repulsion effect. Relevance in the case of a disordered system where this band repulsion crosses over to the well-known level repulsion is discussed. Then with a proper matching at the boundary we solve for the sets of denumerably infinite number of decaying solutions (the surface states) in the gaps. These types of states have been proposed quite some time ago. We look at detail theirexact analytical solutions in 1D and find that their decay lengths near the band edges diverge as |E–E _b|^–v, wherev=1/2 andE _b is the nearest band edge. The decay lengths and their divergence exponent match extremely well with those obtained from transfer matrix method. Some recent experiments on quantum well structures seem to have observed such states.     

Indirect complexometric titration of sodium and potassium with EDTA

Summary Sodium is precipitated as sodium zinc uranyl acetate, filtered and dissolved in water, and zinc is titrated with EDTA using eriochrome black T as indicator. Potassium is precipitated as potassium sodium cobaltinitrite and dissolved in hot water containing little hydrochloric acid. The blue colored solution produced by cobalt in solution, with ammonium thiocyanate and acetone was titrated with EDTA until colorless. The results are good.The author is grateful to Prof. Philip W. West, Boyd Professor of Chemistry, Louisiana State University, for kindly providing the facilities to carry out the investigation.