Symmetry justification of Lorenz’ maximum simplification

In 1960 Edward Lorenz (1917–2008) published a pioneering work on the ‘maximum simplification’ of the barotropic vorticity equation. He derived a coupled three-mode system and interpreted it as the minimum core of large-scale fluid mechanics on a ‘finite but unbounded’ domain. The model was obtained in a heuristic way, without giving a rigorous justification for the chosen selection of modes. In this paper, it is shown that one can legitimate Lorenz’ choice by using symmetry transformations of the spectral form of the vorticity equation. The Lorenz three-mode model arises as the final step in a hierarchy of models constructed via the component reduction by means of symmetries. In this sense, the Lorenz model is indeed the ‘maximum simplification’ of the vorticity equation.     

Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.     

Equivalence of Conservation Laws and Equivalence of Potential Systems

We study conservation laws and potential symmetries of (systems of) differential equations applying equivalence relations generated by point transformations between the equations. A Fokker–Planck equation and the Burgers equation are considered as examples. Using reducibility of them to the one-dimensional linear heat equation, we construct complete hierarchies of local and potential conservation laws for them and describe, in some sense, all their potential symmetries. Known results on the subject are interpreted in the proposed framework. This paper is an extended comment on the paper of Mei and Zhang [Int. J. Theor. Phys. 45: 2095–2102, 2006].     

Differential Invariants of a One-Parameter Group of Local Transformations: One Independent Variable

We propose a new procedure for the construction of an invariant differentiation operator of a one-parameter group of local transformations in the space of one independent variable and m dependent variables. We prove that, for a known universal invariant, a complete set of functionally independent differential invariants of any order of this group can be constructed by using one quadrature and differentiation. The relationship between first-order differential invariants and systems of Riccati-type equations is analyzed.     

Extreme sensitivity to detuning for globally coupled phase oscillators

We discuss the sensitivity of a population of coupled oscillators to differences in their natural frequencies, i.e., to detuning. We argue that for three or more oscillators, one can get great sensitivity even if the coupling is strong. For N globally coupled phase oscillators we find there can be bifurcation to extreme sensitivity, where frequency locking can be destroyed by arbitrarily small detuning. This extreme sensitivity is absent for N = 2, appears at isolated parameter values for N = 3 and N = 4, and can appear robustly for open sets of parameter values for N > or = 5 oscillators.     

Positive semidefinite quadratic forms on unitary matrices

Let

     

Effect of Production Conditions of Hierarchical SnAl-BEA Zeolites on Their Acidity and Catalytic Activity in Tandem Process for the Production of 4-Methoxybenzyl-sec-Butyl Ether

Theoretical and Experimental Chemistry - Hierarchical SnAl-silicate zeolites of the BEA structural type with Lewis (up to 130 μmol/g) and Brønsted (up to 330 μmol/g) acid sites, the...     

Temperature dependence features of the photoluminescence spectral distribution of Li2B4O7 and Li2B4O7:Cu

The temperature dependence of the photoluminescence (PhL) spectral distributions has been reported for the lithium tetraborate Li₂B₄O₇ (LTB) and Li₂B₄O₇:Cu (LTB:Cu) single crystals and for glassy LTB:Cu. It was found that the emission peaks of the LTB and LTB:Cu single crystals are non-elementary and splittable by temperature increase into several elementary peaks. By the sample heating the temperature quenching of PhL as well as the redistribution of the PhL intensity among elementary peaks was observed. Heating of the LTB:Cu single crystal samples caused no shift of the spectral maxima of the individual PhL peaks. The curve describing the temperature dependency of individual PhL peaks for the LTB:Cu single crystal is characterized by maxima resulting from combination of PhL and thermostimulated luminescence (TSL). The PhL intensity for glassy LTB:Cu is significantly lower than that for LTB:Cu single crystal with the equivalent copper dopant content. As compared to the LTB:Cu single crystal, the PhL spectral maximum for glassy LTB:Cu is wider and shifted to the lower energy range. Heating of the glassy LTB:Cu sample results in the PhL temperature quenching without any shift of the spectral maximum.