Bifurcation and Chaos in Coupled Ratchets exhibiting Synchronized Dynamics

The bifurcation and chaotic behaviour of unidirectionally coupled deterministic ratchets is studied as a function of the driving force amplitude ($a$) and frequency ($\omega$). A classification of the various types of bifurcations likely to be encountered in this system was done by examining the stability of the steady state in linear response as well as constructing a two-parameter phase diagram in the ($a -\omega$) plane. Numerical explorations revealed varieties of bifurcation sequences including quasiperiodic route to chaos. Besides, the familiar period-doubling and crises route to chaos exhibited by the one-dimensional ratchet were also found. In addition, the coupled ratchets display symmetry-breaking, saddle-nodes and bubbles of bifurcations. Chaotic behaviour is characterized by using the sensitivity to initial condition as well as the Lyapunov exponent spectrum; while a perusal of the phase space projected in the Poincar$\acute{e}$ cross-section confirms some of the striking features.Comment: 7 pages; 8 figure

Discrete embedded solitons

We address the existence and properties of discrete embedded solitons (ESs), i.e., localized waves existing inside the phonon band in a nonlinear dynamical-lattice model. The model describes a one-dimensional array of optical waveguides with both the quadratic (second-harmonic generation) and cubic nonlinearities. A rich family of ESs was previously known in the continuum limit of the model. First, a simple motivating problem is considered, in which the cubic nonlinearity acts in a single waveguide. An explicit solution is constructed asymptotically in the large-wavenumber limit. The general problem is then shown to be equivalent to the existence of a homoclinic orbit in a four-dimensional reversible map. From properties of such maps, it is shown that (unlike ordinary gap solitons), discrete ESs have the same codimension as their continuum counterparts. A specific numerical method is developed to compute homoclinic solutions of the map, that are symmetric under a specific reversing transformation. Existence is then studied in the full parameter space of the problem. Numerical results agree with the asymptotic results in the appropriate limit and suggest that the discrete ESs may be semi-stable as in the continuous case.Comment: A revtex4 text file and 51 eps figure files. To appear in Nonlinearit

Bubbling and bistability in two parameter discrete systems

We present a graphical analysis of the mechanisms underlying the occurrences of bubbling sequences and bistability regions in the bifurcation scenario of a special class of one dimensional two parameter maps. The main result of the analysis is that whether it is bubbling or bistability is decided by the sign of the third derivative at the inflection point of the map function.Comment: LaTeX v2.09, 14 pages with 4 PNG figure

Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map : Mechanisms and their characterizations

A simple quasiperiodically forced one-dimensional cubic map is shown to exhibit very many types of routes to chaos via strange nonchaotic attractors (SNAs) with reference to a two-parameter $(A-f)$ space. The routes include transitions to chaos via SNAs from both one frequency torus and period doubled torus. In the former case, we identify the fractalization and type I intermittency routes. In the latter case, we point out that atleast four distinct routes through which the truncation of torus doubling bifurcation and the birth of SNAs take place in this model. In particular, the formation of SNAs through Heagy-Hammel, fractalization and type--III intermittent mechanisms are described. In addition, it has been found that in this system there are some regions in the parameter space where a novel dynamics involving a sudden expansion of the attractor which tames the growth of period-doubling bifurcation takes place, giving birth to SNA. The SNAs created through different mechanisms are characterized by the behaviour of the Lyapunov exponents and their variance, by the estimation of phase sensitivity exponent as well as through the distribution of finite-time Lyapunov exponents.Comment: 27 pages, RevTeX 4, 16 EPS figures. Phys. Rev. E (2001) to appea

Increased Dickkopf-1 expression in breast cancer bone metastases

The aim of this study was to determine whether Dickkopf-1 (Dkk-1) expression in breast cancer was associated with bone metastases. We first analysed Dkk-1 expression by human breast cancer cell lines that induce osteolytic or osteoblastic lesions in animals. Dickkopf-1 levels were then measured in the bone marrow aspirates of hind limbs from eight NMRI mice inoculated with breast cancer cells that induced bone metastases and 11 age-matched non-inoculated control animals. Finally, Dkk-1 was measured in the serum of 17 women with breast cancer in complete remission, 19 women with breast cancer and bone metastases, 16 women with breast cancer and metastases at non-bone sites and 16 healthy women. Only breast cancer cells that induce osteolytic lesions in animals produced Dkk-1. There was a six-fold increase in Dkk-1 levels in the bone marrow from animals inoculated with MDA-B02 cells when compared with that of control non-inoculated animals (P=0.003). Median Dkk-1 levels in the serum of patients with breast cancer and bone metastases were significantly higher than levels of patients in complete remission (P=0.016), patients with breast cancer having metastases at non-bone sites (P<0.0001) and healthy women (P=0.047), although there was a large overlap in individual levels between the different groups. In conclusion, Dkk-1 is secreted by osteolytic human breast cancer cells lines and increased circulating levels are associated with the presence of bone metastases in patients with breast cancer. Measurements of circulating Dkk-1 levels may be useful for the clinical investigation of patients with breast cancer and bone metastases

The mixmaster universe: A chaotic Farey tale

When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer dependent answers. Here we resolve the conflict by using observer independent, fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the 6-dimensional minisuperspace of the full Einstein equations, and in a 2-dimensional discretisation of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy and multifractal dimensions. As all of these quantities are coordinate, or gauge independent, there is no longer any ambiguity--the mixmaster universe is indeed chaotic.Comment: 45 pages, RevTeX, 19 Figures included, submitted to PR

Efficacy of Wnt-1 monoclonal antibody in sarcoma cells

BACKGROUND: Sarcomas are one of the most refractory diseases among malignant tumors. More effective therapies based on an increased understanding of the molecular biology of sarcomas are needed as current forms of therapy remain inadequate. Recently, it has been reported that Wnt-1/β-catenin signaling inhibits apoptosis in several cancers. In this study, we investigated the efficacy of a monoclonal anti-Wnt-1 antibody in sarcoma cells. METHODS: We treated cell lines A-204, SJSA-1, and fresh primary cultures of lung metastasis of sarcoma with a monoclonal anti-Wnt-1 antibody. Wnt-1 siRNA treatment was carried out in A-204. We assessed cell death using Crystal Violet staining. Apoptosis induction was estimated by flow cytometry analysis (Annexin V and PI staining). Cell signaling changes were determined by western blotting analysis. RESULTS: We detected Wnt-1 expression in all tissue samples and cell lines. Significant apoptosis induction was found in monoclonal anti-Wnt-1 antibody treated cells compared to control monoclonal antibody treated cells (p < 0.02). Similarly, we observed increased apoptosis in Wnt-1 siRNA treated cells. Blockade of Wnt-1 signaling in both experiments was confirmed by analyzing intracellular levels of Dishevelled-3 and of cytosolic β-catenin. Furthermore, the monoclonal anti-Wnt-1 antibody also induced cell death in fresh primary cultures of metastatic sarcoma in which Wnt-1 signaling was active. CONCLUSION: Our results indicate that Wnt-1 blockade by either monoclonal antibody or siRNA induces cell death in sarcoma cells. These data suggest that Wnt-1 may be a novel therapeutic target for the treatment of a subset of sarcoma cells in which Wnt-1/β-catenin signaling is active