Stagger-and-step method: detecting and computing chaotic saddles in higher dimensions

Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices. These transients are caused by the presence of chaotic saddles, and they are a common phenomenon in higher dimensional dynamical systems. For many physical systems, chaotic saddles have a big impact on laboratory measurements, but there has been no way to observe these chaotic saddles directly. We present the first general method to locate and visualize chaotic saddles in higher dimensions.     

Fixed points indices and period-doubling cascades

Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, \({F : \mathbb{R}\times\mathfrak{M} \rightarrow \mathfrak{M},}\) where \({\mathfrak{M}}\) is a locally compact manifold without boundary, typically \({\mathbb{R}^N}\). In particular, we investigate F(μ, ·) for \({\mu \in J = [\mu_{1}, \mu_{2}]}\), when F(μ ₁, ·) has only finitely many periodic orbits while F(μ ₂, ·) has exponential growth of the number of periodic orbits as a function of the period. For generic F, under additional hypotheses, we use a fixed point index argument to show that there are infinitely many “regular” periodic orbits at μ ₂. Furthermore, all but finitely many of these regular orbits at μ ₂ are tethered to their own period-doubling cascade. Specifically, each orbit ρ at μ ₂ lies in a connected component C(ρ) of regular orbits in \({J \times \mathfrak{M}}\); different regular orbits typically are contained in different components, and each component contains a period-doubling cascade. These components are one-manifolds of orbits, meaning that we can reasonably say that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ ₂) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in \({J \times \mathfrak{M}}\).As corollaries of our main results, we give several examples, we prove that the map in each example has infinitely many cascades, and we count the cascades.     

Fractal basin boundaries

Basin boundaries for dynamical systems can be either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structures are given, and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.     

When all solutions of x′ = −∑ qi(t) x(t − Ti(t)) oscillate

In this paper the long-term behavior of solutions to the equation in the title are examined, where q_i(t) and T_i(t) are positive. In particular, it is shown that if lim inf_{t → ∝} ∑_{i = 1}ⁿT_i(t) q_i(t) > 1/e, all solutions oscillate about 0 infinitely often.     

Cranberry changes the physicochemical surface properties of E. coli and adhesion with uroepithelial cells

Cranberries have been suggested to decrease the attachment of bacteria to uroepithelial cells (UC), thus preventing urinary tract infections, although the mechanisms are not well understood. A thermodynamic approach was used to calculate the Gibbs free energy of adhesion changes (DeltaG(adh)) for bacteria-UC interactions, based on measuring contact angles with three probe liquids. Interfacial tensions and DeltaG(adh) values were calculated for Escherichia coli HB101pDC1 (P-fimbriated) and HB101 (non-fimbriated) exposed to cranberry juice (0-27 wt.%). HB101pDC1 can form strong bonds with the Gal-Gal disaccharide receptor on uroepithelial cells, while HB101-UC interactions are only non-specific. For HB101 interacting with UC, DeltaG(adh) was always negative, suggesting favorable adhesion, and the values were insensitive to cranberry juice concentration. For the HB101pDC1-UC system, DeltaG(adh) became positive at 27wt.% cranberry juice, suggesting that adhesion was unfavorable. Acid-base (AB) interactions dominated the interfacial tensions, compared to Lifshitz-van der Waals (LW) interactions. Exposure to cranberry juice increased the AB component of the interfacial tension of HB101pDC1. LW interactions were small and insensitive to cranberry juice concentration. The number of bacteria attached to UC was quantified in batch adhesion assays and quantitatively correlated with DeltaG(adh). Since the thermodynamic approach should not agree with the experimental results when specific interactions are present, such as HB101pDC-UC ligand-receptor bonds, our results may suggest that cranberry juice disrupts bacterial ligand-UC receptor binding. These results help form the mechanistic explanation of how cranberry products can be used to prevent bacterial attachment to host tissue, and may lead to the development of better therapies based on natural products.     

Structure, electronic properties and catalytic behaviour of an activity-enhancing CYP102A1 (P450(BM3)) variant

The substrate-free crystal structure of a five-mutation directed evolution variant of CYP102A1 (P450(BM3)) with generic activity-enhancing properties ("KT2") has been determined to 1.9-? resolution. There is a close resemblance to substrate-bound structures of the wild-type enzyme (WT). The disruption of two salt bridges that link the G- and I-helices in WT causes conformational changes that break several hydrogen bonds and reduce the angle of the kink in the I-helix where dioxygen activation is thought to take place. The side-chain of a key active site residue, Phe87, is rotated in one molecule of the asymmetric unit, and the side-chains of Phe158 and Phe261 cascade into the orientations found in fatty-acid-bound forms of the enzyme. The iron is out of the porphyrin plane, towards the proximal cysteine. Unusually, the axial water ligand to the haem iron is not hydrogen-bonded to Ala264. The first electron transfer from the reductase domain to the haem domain of substrate-free KT2 is almost as fast as in palmitate-bound WT even though the reduction potential of the haem domain is only slightly more oxidising than that of substrate-free WT. However, NADPH is turned over slowly in the absence of substrate, so the catalytic cycle is gated by a step subsequent to the first electron transfer-a contrast to WT. Propylbenzene binding slightly raises the first electron transfer rate in WT but not in KT2. It is proposed that the generic rate accelerating properties of KT2 arise from the substrate-free form being in a catalytically ready conformation, such that substrate-induced changes to the structure play a less significant role in promoting the first electron transfer than in WT.     

“Structure and mechanical properties of blown films of ionomer-compatibilized LDPE nanocomposites”

Blown films based on low density polyethylene (LDPE) organoclay nanocomposites (NCs) were obtained by melt extrusion followed by film blowing, using a zinc ionomer of poly(ethylene-co-methacrylic acid) (Pema-Zn) as a compatibilizer. The parameters studied were the compatibilizer and the montmorillonite (MMT) contents that ranged from 0 to 20% and from 0 to 5%, respectively. The presence of clay hindered Pema-Zn crystallization indicating the existence of interaction between the Pema-Zn and the clay. Analysis of the nanostructure showed that the MMT was found inside microscopic domains of Pema-Zn distributed throughout the LDPE matrix. The addition of Pema-Zn improved the dispersion of the clay in LDPE films resulting in synergistic improvements in the mechanical properties. These improvements occur both in the machine and transverse directions. Thus, the presence of Pema-Zn is a determining factor in biaxiality and can clearly be attributed to the bidimensional laminar structure of clays such as MMT.