Xylogalacturonan-enriched pectin from the fruit pulp of Adansonia digitata: Structural characterization and antidepressant-like effect.

The low methyl-esterified and acetylated xylogalacturonan (DM 20 %, DA 2 %, Mw ∼ 58 kDa) was isolated by water extraction for 4 h × 2 at 50 °C (yield 23 %) from the pulp of baobab fruit (Adansonia digitata L.). Subsequent tightening of the conditions for water extraction by mean increasing the temperature to 70 °C and time to 12 h led to the co-extraction of small amounts of starch components and RG I with xylogalacturonan. Structural analysis (DEAE-cellulose ion-exchange chromatography, HPSEC, monosaccharide analysis, NMR spectroscopy) revealed that about 12 mol. % of 1,4-linked α-GalpA residues were substituted by single ß-Xylp residues at the O-3 position. The xylogalacturonan was found to possess an antidepressant-like effect in mice. The study offers using the baobab fruit as a rich source of soluble dietary fiber - water-soluble pectin with beneficial physiological effect.

Smale-Williams solenoids in autonomous system with saddle equilibrium.

We construct an autonomous low-dimensional system of differential equations by replacement of real-valued variables with complex-valued variables in a self-oscillating system with homoclinic loops of a saddle. We provide analytical and numerical indications and argue that the emerging chaotic attractor is a uniformly hyperbolic chaotic attractor of Smale-Williams type. The four-dimensional phase space of the flow consists of two parts: a vicinity of a saddle equilibrium with two pairs of equal eigenvalues where the angular variable undergoes a Bernoulli map, and a region which ensures that the trajectories return to the origin without angular variable changing. The trajectories of the flow approach and leave the vicinity of the saddle equilibrium with the arguments of complex variables undergoing a Bernoulli map on each return. This makes possible the formation of the attractor of a Smale-Williams type in Poincaré cross section. In essence, our model resembles complex amplitude equations governing the dynamics of wave envelops or spatial Fourier modes. We discuss the roughness and generality of our scheme.

Quantum Lyapunov exponents beyond continuous measurements.

Quantum systems, when interacting with their environments, may exhibit nonequilibrium states that are tempting to be interpreted as quantum analogs of chaotic attractors. However, different from the Hamiltonian case, the toolbox for quantifying dissipative quantum chaos remains limited. In particular, quantum generalizations of Lyapunov exponents, the main quantifiers of classical chaos, are established only within the framework of continuous measurements. We propose an alternative generalization based on the unraveling of quantum master equation into an ensemble of "quantum trajectories," by using the so-called Monte Carlo wave-function method. We illustrate the idea with a periodically modulated open quantum dimer and demonstrate that the transition to quantum chaos matches the period-doubling route to chaos in the corresponding mean-field system.

Hyperbolic chaos in the klystron-type microwave vacuum tube oscillator.

The ring-loop oscillator consisting of two coupled klystrons which is capable of generating hyperbolic chaotic signal in the microwave band is considered. The system of delayed-differential equations describing the dynamics of the oscillator is derived. This system is further reduced to the two-dimensional return map under the assumption of the instantaneous build-up of oscillations in the cavities. The results of detailed numerical simulation for both models are presented showing that there exists large enough range of control parameters where the sustained regime corresponds to the structurally stable hyperbolic chaos.

Mandelbrot set in coupled logistic maps and in an electronic experiment.

We suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps. The idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one-dimensional quadratic maps with a particular coupling. Hence, dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems, which relate to the Feigenbaum universality class. To substantiate the feasibility of this concept, we consider an electronic system, which exhibits dynamical phenomena intrinsic to complex analytic maps. Experimental results are presented, providing the Mandelbrot set in the parameter plane of this physical system.

Scaling properties of bicritical dynamics in unidirectionally coupled period-doubling systems in the presence of noise.

We study scaling regularities associated with the effects of additive noise on the bicritical behavior of a system of two unidirectionally coupled quadratic maps. A renormalization group analysis of the effects of noise is developed. We outline the qualitative and quantitative differences between the response of the system to random perturbations added to the master subsystem or the slave subsystem. The universal constants determining the rescaling rules for the intensity of the noise sources in the master and slave subsystems are found to be gamma=6.619036... and nu=2.713708..., respectively. A number of computer graphical illustrations for the scaling regularities is presented. We discuss the smearing of the fine structure of the bicritical attractor and the Fourier spectra in the presence of noise, the self-similar structure of the Lyapunov charts on the parameter plane near the bicritical point, and the shift of the threshold of hyperchaos in dependence of the noise intensity.

Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rössler oscillator.

Universal behavior discovered earlier in two-dimensional noninvertible maps is found numerically in a periodically driven Rössler system. The critical behavior is associated with the limit of a period-doubling cascade at the edge of the Arnold tongue, and may be reached by variation of two control parameters. The corresponding scaling regularities, distinct from those of the Feigenbaum cascade, are demonstrated. Presence of a critical quasiattractor, an infinite set of stable periodic orbits of quadrupled periods, is outlined. As argued, this type of critical behavior may occur in a wide class of periodically driven period-doubling systems.

Complex periodic orbits, renormalization, and scaling for quasiperiodic golden-mean transition to chaos.

At the critical point of the golden-mean quasiperiodic transition to chaos we show the presence of an infinite sequence of unstable orbits in complex domain with periods given by the Fibonacci numbers. The Floquet eigenvalues (multipliers) are found to converge fast to a universal complex constant. We explain this result on the basis of the renormalization group approach and suggest using it for accurate estimates of the location of the golden-mean critical points in parameter space for a class of nonlinear dissipative systems defined analytically. As an example, we obtain data for the golden-mean critical point in the two-dimensional dissipative invertible map of Zaslavsky. We give a set of graphical illustrations for the scaling properties and emphasize that demonstration of self-similarity on two-dimensional diagrams of Arnold tongues requires the use of a properly chosen curvilinear coordinate system. We discuss a procedure of construction of the appropriate local coordinate system in the parameter plane and present the corresponding data for the circle map and Zaslavsky map.