Torsion and attractors in the Kolmogorov hydrodynamical system

Geometrical unified theory of Rikitake system and KCC-theory

“…Then, by using the decomposition factors (27), (28) and the relation (30), the Euler-Schouten tensor is expressed by the mutual-inductance, i.e. H σ ρ κ = G κ σ ρ = M κ σ ρ /2.…”

Energy cycle for the Lorenz attractor

“…In the spirit of 1955 Lorenz work on general circulation of the atmosphere [1], we introduce for system (6) quantities respectively known as available potential energy max min APE U U = − and unavailable potential energy min UPE U = ; they represent, respectively, the portion of potential energy that can be converted into kinetic energy, and the portion that cannot. In atmospheric science APE is a very important subject since its variability determines transitions in the atmospheric circulation.…”

“…Such a formalism, as mentioned before, is particularly useful in fluid dynamics [5], where Navier-Stokes equations show interesting properties in their Hamiltonian part (Euler equations). Moreover, finite dimensional systems as (1) represent the proper reduction of fluid dynamical equations [6], in terms of conservation of the symplectic structures in the infinite domain [7]. Method of reduction, contrary to the classical truncation one, leads to the study of dynamics on Lie algebras, i.e to the study of Lie-Poisson equations on them, which are extremely interesting from the physical viewpoint and with a mathematical aesthetical appeal [8,9].…”

Oscillating forcings and new regimes in the Lorenz system: a four-lobe attractor

“…Furthermore, finite-dimensional systems like Eq. (1) represent the proper reduction of fluid dynamical equations (Pasini et al, 1998) in terms of conservation of the symplectic structures in the infinite domain (Zeitlin, 2004). This method, contrary to the classical truncation one, leads to the study of dynamics on Lie algebra, also known as LiePoisson equations, which is extremely interesting from the physical viewpoint and has a mathematical aesthetical appeal (Pelino and Pasini, 2001;Pelino and Maimone, 2007).…”

“…In our case, we use the Lie-Poisson structure of Lorenz system, which corresponds to the simplest Nambu dynamics. In what follows, we adopt a unified formalism previously developed (Pasini et al, 1998;Pasini and Pelino, 2000;Pelino and Pasini, 2001;Pasini et al, 2010;Gianfelice et al, 2012) in order to clearly discuss the effect of different forcings and test some general cases, finally reaching the evidence of creation of new regimes and a four lobe Lorenz attractor.…”