PERIODIC PERTURBATION of QUADRATIC SYSTEMS WITH TWO INFINITE HETEROCLINIC CYCLES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On the existence of infinite heteroclinic cycles in polynomial systems and its dynamic consequences

In this work we consider the dynamic consequences of the existence of infinite heteroclinic cycle in planar polynomial vector fields, which is a trajectory connecting two saddle points at infinity. It is stated that, although the saddles which form the cycle belong to infinity, for certain types of nonautonomous perturbations the perturbed system may present a complex dynamic behavior of the solutions in a finite part of the phase plane, due to the existence of tangencies and transversal intersections of their stable and unstable manifolds. This phenomenon might be called the chaos arising from infinity. The global study at infinity is made via the Poincare Compactification and the argument used to prove the statement is the Birkhoff-Smale Theorem. (c) 2004 WILEY-NCH Verlag GmbH & Co. KGaA, Weinheim.

Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Resonances and subharmonic bifurcations of large amplitude periodic orbits of planar polynomial vector fields

In this work are studied periodic perturbations, depending on two parameters, of planar polynomial vector fields having an annulus of large amplitude periodic orbits, which accumulate on a symmetric infinite heteroclinic cycle. Such periodic orbits and the heteroclinic trajectory can be seen only by the global consideration of the polynomial vector fields on the whole plane, and not by their restriction to any compact set. The global study involving infinity is performed via the Poincare Compactification. It is shown that, for certain types of periodic perturbations, one can seek, in a neighborhood of the origin in the parameter plane, curves C-(m) of subharmonic bifurcations, for which the periodically perturbed system has subharmonics of order m, for any integer m.

Time-periodic perturbation of a Lienard equation with an unbounded homoclinic loop

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Periodic perturbations of quadratic planar polynomial vector fields

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A Statistical Model of Current Loops and Magnetic Monopoles

We formulate a natural model of loops and isolated vertices for arbitrary planar graphs, which we call the monopole-dimer model. We show that the partition function of this model can be expressed as a determinant. We then extend the method of Kasteleyn and Temperley-Fisher to calculate the partition function exactly in the case of rectangular grids. This partition function turns out to be a square of a polynomial with positive integer coefficients when the grid lengths are even. Finally, we analyse this formula in the infinite volume limit and show that the local monopole density, free energy and entropy can be expressed in terms of well-known elliptic functions. Our technique is a novel determinantal formula for the partition function of a model of isolated vertices and loops for arbitrary graphs.

Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system

In this paper, by using the Poincare compactification in R(3) we make a global analysis of the Lorenz system, including the complete description of its dynamic behavior on the sphere at infinity. Combining analytical and numerical techniques we show that for the parameter value b = 0 the system presents an infinite set of singularly degenerate heteroclinic cycles, which consist of invariant sets formed by a line of equilibria together with heteroclinic orbits connecting two of the equilibria. The dynamical consequences related to the existence of such cycles are discussed. In particular a possibly new mechanism behind the creation of Lorenz-like chaotic attractors, consisting of the change in the stability index of the saddle at the origin as the parameter b crosses the null value, is proposed. Based on the knowledge of this mechanism we have numerically found chaotic attractors for the Lorenz system in the case of small b > 0, so nearby the singularly degenerate heteroclinic cycles.

Infinite-horizon policy-gradient estimation

Gradient-based approaches to direct policy search in reinforcement learning have received much recent attention as a means to solve problems of partial observability and to avoid some of the problems associated with policy degradation in value-function methods. In this paper we introduce GPOMDP, a simulation-based algorithm for generating a biased estimate of the gradient of the average reward in Partially Observable Markov Decision Processes (POMDPs) controlled by parameterized stochastic policies. A similar algorithm was proposed by Kimura, Yamamura, and Kobayashi (1995). The algorithm's chief advantages are that it requires storage of only twice the number of policy parameters, uses one free parameter β ∈ [0,1) (which has a natural interpretation in terms of bias-variance trade-off), and requires no knowledge of the underlying state. We prove convergence of GPOMDP, and show how the correct choice of the parameter β is related to the mixing time of the controlled POMDP. We briefly describe extensions of GPOMDP to controlled Markov chains, continuous state, observation and control spaces, multiple-agents, higher-order derivatives, and a version for training stochastic policies with internal states. In a companion paper (Baxter, Bartlett, & Weaver, 2001) we show how the gradient estimates generated by GPOMDP can be used in both a traditional stochastic gradient algorithm and a conjugate-gradient procedure to find local optima of the average reward. ©2001 AI Access Foundation and Morgan Kaufmann Publishers. All rights reserved.

An infinite-horizon robust filter for uncertain hidden Markov Models with conditional relative entropy constraints

We consider a robust filtering problem for uncertain discrete-time, homogeneous, first-order, finite-state hidden Markov models (HMMs). The class of uncertain HMMs considered is described by a conditional relative entropy constraint on measures perturbed from a nominal regular conditional probability distribution given the previous posterior state distribution and the latest measurement. Under this class of perturbations, a robust infinite horizon filtering problem is first formulated as a constrained optimization problem before being transformed via variational results into an unconstrained optimization problem; the latter can be elegantly solved using a risk-sensitive information-state based filtering.

Modelling bluetooth and fusing with loops

The invited presentation was delivered at Queensland Department of Main Roads, Brisbane Australia, 17th June 2013

Proteolytic activation of Chlamydia trachomatis HTRA is mediated by PDZ1 domain interactions with protease domain loops L3 and LC and beta strand β5

Chlamydia trachomatis is a bacterial pathogen responsible for one of the most prevalent sexually transmitted infections worldwide. Its unique development cycle has limited our understanding of its pathogenic mechanisms. However, CtHtrA has recently been identified as a potential C. trachomatis virulence factor. CtHtrA is a tightly regulated quality control protein with a monomeric structural unit comprised of a chymotrypsin-like protease domain and two PDZ domains. Activation of proteolytic activity relies on the C-terminus of the substrate allosterically binding to the PDZ1 domain, which triggers subsequent conformational change and oligomerization of the protein into 24-mers enabling proteolysis. This activation is mediated by a cascade of precise structural arrangements, but the specific CtHtrA residues and structural elements required to facilitate activation are unknown. Using in vitro analysis guided by homology modeling, we show that the mutation of residues Arg362 and Arg224, predicted to disrupt the interaction between the CtHtrA PDZ1 domain and loop L3, and between loop L3 and loop LD, respectively, are critical for the activation of proteolytic activity. We also demonstrate that mutation to residues Arg299 and Lys160, predicted to disrupt PDZ1 domain interactions with protease loop LC and strand β5, are also able to influence proteolysis, implying their involvement in the CtHtrA mechanism of activation. This is the first investigation of protease loop LC and strand β5 with respect to their potential interactions with the PDZ1 domain. Given their high level of conservation in bacterial HtrA, these structural elements may be equally significant in the activation mechanism of DegP and other HtrA family members.

Joint identification of infinite-frequency added mass and fluid-memory models of marine structures

This paper addresses the problem of joint identification of infinite-frequency added mass and fluid memory models of marine structures from finite frequency data. This problem is relevant for cases where the code used to compute the hydrodynamic coefficients of the marine structure does not give the infinite-frequency added mass. This case is typical of codes based on 2D-potential theory since most 3D-potential-theory codes solve the boundary value associated with the infinite frequency. The method proposed in this paper presents a simpler alternative approach to other methods previously presented in the literature. The advantage of the proposed method is that the same identification procedure can be used to identify the fluid-memory models with or without having access to the infinite-frequency added mass coefficient. Therefore, it provides an extension that puts the two identification problems into the same framework. The method also exploits the constraints related to relative degree and low-frequency asymptotic values of the hydrodynamic coefficients derived from the physics of the problem, which are used as prior information to refine the obtained models.

Open access to infinite content (or "in praise of law reviews")

This Article is about legal scholarly publication in a time of plenitude. It is an attempt to explain why the most pressing questions in legal scholarly publishing are about how we ensure access to an infinity of content. It explains why standard assumptions about resource scarcity in publication are wrong in general, and how the changes in the modality of publication affect legal scholarship. It talks about the economics of open access to legal material, and how this connects to a future where there is infinite content. And because student-edited law reviews fit this future better than their commercially-produced, peer-refereed cousins, this Article is, in part, a defense of the crazy-beautiful institution that is the American law review.