Matrix Quantization of Turbulence

Based on our recent work on Quantum Nambu Mechanics \cite{af2}, we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving non-dissipative sector survive for long times.Comment: 14 pages, Based on invited talks delivered at: Fifth Aegean Summer School, "From Gravity to Thermal Gauge theories and the AdS/CFT Correspondance", September 2009, Milos, Greece; the Intern. Conference on Dynamics and Complexity, Thessaloniki, Greece, 12 July 2010; Workshop on "AdS4/CFT3 and the Holographic States of Matter", Galileo Galilei Institute, Firenze, Italy, 30 October 201

Scaling and synchronization in a ring of diffusively coupled nonlinear oscillators

Chaos synchronization in a ring of diffusively coupled nonlinear oscillators driven by an external identical oscillator is studied. Based on numerical simulations we show that by introducing additional couplings at $(mN_c+1)$-th oscillators in the ring, where $m$ is an integer and $N_c$ is the maximum number of synchronized oscillators in the ring with a single coupling, the maximum number of oscillators that can be synchronized can be increased considerably beyond the limit restricted by size instability. We also demonstrate that there exists an exponential relation between the number of oscillators that can support stable synchronization in the ring with the external drive and the critical coupling strength $\epsilon_c$ with a scaling exponent $\gamma$. The critical coupling strength is calculated by numerically estimating the synchronization error and is also confirmed from the conditional Lyapunov exponents (CLEs) of the coupled systems. We find that the same scaling relation exists for $m$ couplings between the drive and the ring. Further, we have examined the robustness of the synchronous states against Gaussian white noise and found that the synchronization error exhibits a power-law decay as a function of the noise intensity indicating the existence of both noise-enhanced and noise-induced synchronizations depending on the value of the coupling strength $\epsilon$. In addition, we have found that $\epsilon_c$ shows an exponential decay as a function of the number of additional couplings. These results are demonstrated using the paradigmatic models of R\"ossler and Lorenz oscillators.Comment: Accepted for Publication in Physical Review

Direct transition to high-dimensional chaos through a global bifurcation

In the present work we report on a genuine route by which a high-dimensional (with d>4) chaotic attractor is created directly, i.e., without a low-dimensional chaotic attractor as an intermediate step. The high-dimensional chaotic set is created in a heteroclinic global bifurcation that yields an infinite number of unstable tori.The mechanism is illustrated using a system constructed by coupling three Lorenz oscillators. So, the route presented here can be considered a prototype for high-dimensional chaotic behavior just as the Lorenz model is for low-dimensional chaos.Comment: 7 page

Spontaneous peristaltic airway contractions propel lung liquid through the bronchial tree of intact and fetal lung explants

Spontaneous contractions of the fetal airways are a well recognized but poorly characterized phenomenon. In the present study spontaneous narrowing of the airways was analyzed in freshly isolated lungs from early to late gestation in fetal pigs and rabbits and in cultured fetal mouse lungs. Propagating waves of contraction traveling proximal to distal were observed in fresh lungs throughout gestation which displaced the lung liquid along the lumen. In the pseudoglandular and canalicular stages (fetal pigs) the frequency ranged from 2.3 to 3.3 contractions/min with a 39 to 46% maximum reduction of lumen diameter. In the saccular stage (rabbit) the frequency was 10 to 12/min with a narrowing of approximately 30%. In the organ cultures the waves of narrowing started at the trachea in whole lungs, or at the main bronchus in lobes (5.2 +/- 1.5 contractions/min, 22 +/- 8% reduction of lumen diameter), and as they proceeded distally along the epithelial tubes the luminal liquid was shifted toward the terminal tubules, which expanded the endbuds. As the tubules relaxed the flow of liquid was reversed. Thus the behavior of airway smooth muscle in the fetal lung is phasic in type (like gastrointestinal muscle) in contrast to that in postnatal lung, where it is tonic. An intraluminal positive pressure of 2.33 +/- 0.77 cm H(2)O was recorded in rabbit fetal trachea. It is proposed that the active tone of the smooth muscle maintains the positive intraluminal pressure and acts as a stimulus to lung growth via the force exerted across the airway wall and adjacent parenchyma. The expansion of the compliant endbuds by the fluid shifts at the airway tip may promote their growth into the surrounding mesenchyme

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

On the recurrence and robust properties of Lorenz'63 model

Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.Comment: 44 pages, 7 figures, revised version accepted for pubblicatio