An open set of maps for which every point is absolutely nonshadowable

We consider a class of nonhyperbolic systems, for which there are two fixed points in an attractor having a dense trajectory; the unstable manifold of one has dimension one and the other's is two dimensional. Under the condition that there exists a direction which is more expanding than other directions, we show that such attractors are nonshadowable. Using this theorem, we prove that there is an open set of diffeomorphisms (in the -topology, ) for which every point is absolutely nonshadowable, i.e., there exists such that, for every , almost every -pseudo trajectory starting from this point is -nonshadowable.

     

Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model

The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbersr somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence uponr is studied both numerically and (very close to the criticalr) analytically.This work was supported in part by NASA grant NSG 5209; partial support of computer costs was provided by the University of Maryland-Baltimore County Computer Center.     

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.     

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.     

Crossing bifurcations and unstable dimension variability

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this Letter, we describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3. We present evidence of unstable dimension variability as a result of the crisis. We then derive a new scaling law describing the density of the new portion of the attractor formed in the crisis. We illustrate this new type of bifurcation with a specific example of a three-dimensional chaotic attractor undergoing a crisis.     

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.     

Period-doubling cascades for large perturbations of Hénon families

The Hénon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends a general theory explaining why cascades occur [5].     

Topological horseshoes

When does a continuous map have chaotic dynamics in a set ? More specifically, when does it factor over a shift on symbols? This paper is an attempt to clarify some of the issues when there is no hyperbolicity assumed. We find that the key is to define a ``crossing number' for that set . If that number is and 1$">, then contains a compact invariant set which factors over a shift on symbols.