Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model

We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich-Morioka-Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof of the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an open set of parameter values in a 4-parameter family of three-dimensional Henon-like diffeomorphisms

Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation

We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers (\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma), where $0<\lambda<1<|\gamma|$ and $|\lambda^2\gamma|=1$. We show that in a three-parameter family, g_{\eps}, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near \eps =0 where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional H\'enon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional H\'enon maps occupy in the class of quadratic volume-preserving diffeomorphisms.Comment: laTeX, 25 pages, 6 eps figure

On the hierarchy of partially invariant submodels of differential equations

It is noticed, that partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduce a hierarchic structure in the set of all PISs of a given system of differential equations. By using this structure one can significantly decrease an amount of calculations required in enumeration of all PISs for a given system of partially differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. In this framework the complete classification of regular partially invariant solutions of ideal MHD equations is given

Bifurcation to Chaos in the complex Ginzburg-Landau equation with large third-order dispersion

We give an analytic proof of the existence of Shilnikov chaos in complex Ginzburg-Landau equation subject to a large third-order dispersion perturbation

Synchrotron X-ray Diffraction Study of BaFe2As2 and CaFe2As2 at High Pressures up to 56 GPa: Ambient and Low-Temperatures Down to 33 K

We report high pressure powder synchrotron x-ray diffraction studies on MFe2As2 (M=Ba, Ca) over a range of temperatures and pressures up to about 56 GPa using a membrane diamond anvil cell. A phase transition to a collapsed tetragonal phase is observed in both compounds upon compression. However, at 300 (33) K in the Ba-compound the transition occurs at 26 (29) GPa, which is a much higher pressure than 1.7 (0.3) GPa at 300 (40) K in the Ca-compound, due to its larger volume. It is important to note that the transition in both compounds occurs when they are compressed to almost the same value of the unit cell volume and attain similar ct/at ratios. We also show that the FeAs4 tetrahedra are much less compressible and more distorted in the collapsed tetragonal phase than their nearly regular shape in the ambient pressure phase. We present a detailed analysis of the pressure dependence of the structures as well as equation of states in these important BaFe2As2 and CaFe2As2 compounds.Comment: 26 pages, 12 figure

Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions

For the Euler equations governing compressible isentropic fluid flow with a barotropic equation of state (where pressure is a function only of the density), local conservation laws in $n>1$ spatial dimensions are fully classified in two primary cases of physical and analytical interest: (1) kinematic conserved densities that depend only on the fluid density and velocity, in addition to the time and space coordinates; (2) vorticity conserved densities that have an essential dependence on the curl of the fluid velocity. A main result of the classification in the kinematic case is that the only equation of state found to be distinguished by admitting extra $n$-dimensional conserved integrals, apart from mass, momentum, energy, angular momentum and Galilean momentum (which are admitted for all equations of state), is the well-known polytropic equation of state with dimension-dependent exponent $\gamma=1+2/n$. In the vorticity case, no distinguished equations of state are found to arise, and here the main result of the classification is that, in all even dimensions $n\geq 2$, a generalized version of Kelvin's two-dimensional circulation theorem is obtained for a general equation of state.Comment: 24 pages; published version with misprints correcte