Spiralling dynamics near heteroclinic networks

There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. We construct explicitly a {two parameter family of vector fields} on the three-dimensional sphere \EU^3, whose flow has a spiralling attractor containing the following: two hyperbolic equilibria, heteroclinic trajectories connecting them {transversely} and a non-trivial hyperbolic, invariant and transitive set. The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The vector field is the restriction to \EU^3 of a polynomial vector field in \RR^4. In this article, we also identify global bifurcations that induce chaotic dynamics of different types.Comment: change in one figur

Reliability prediction in model driven development

Evaluating the implications of an architecture design early in the software development lifecycle is important in order to reduce costs of development. Reliability is an important concern with regard to the correct delivery of software system service. Recently, the UML Profile for Modeling Quality of Service has defined a set of UML extensions to represent dependability concerns (including reliability) and other non-functional requirements in early stages of the software development lifecycle. Our research has shown that these extensions are not comprehensive enough to support reliability analysis for model-driven software engineering, because the description of reliability characteristics in this profile lacks support for certain dynamic aspects that are essential in modeling reliability. In this work, we define a profile for reliability analysis by extending the UML 2.0 specification to support reliability prediction based on scenario specifications. A UML model specified using the profile is translated to a labelled transition system (LTS), which is used for automated reliability prediction and identification of implied scenarios; the results of this analysis are then fed back to the UML model. The result is a comprehensive framework for addressing software reliability modeling, including analysis and evolution of reliability predictions. We exemplify our approach using the Boiler System used in previous work and demonstrate how reliability analysis results can be integrated into UML models

Sensitivity Analysis for a Scenario-Based Reliability Prediction Model

As a popular means for capturing behavioural requirements, scenariosshow how components interact to provide system-level functionality.If component reliability information is available, scenarioscan be used to perform early system reliability assessment. Inprevious work we presented an automated approach for predictingsoftware system reliability that extends a scenario specificationto model (1) the probability of component failure, and (2) scenariotransition probabilities. Probabilistic behaviour models ofthe system are then synthesized from the extended scenario specification.From the system behaviour model, reliability predictioncan be computed. This paper complements our previous work andpresents a sensitivity analysis that supports reasoning about howcomponent reliability and usage profiles impact on the overall systemreliability. For this purpose, we present how the system reliabilityvaries as a function of the components reliabilities and thescenario transition probabilities. Taking into account the concurrentnature of component-based software systems, we also analysethe effect of implied scenarios prevention into the sensitivity analysisof our reliability prediction technique

A family of rotation numbers for discrete random dynamics on the circle

We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on the circle. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincar\'{e} lifts) approach does depend on the choice of lifts (e.g. continuously for nonatomic randomness). Furthermore, the winding orbit rotation number does not agree with the topological rotation number. Existence and conversion formulae between these distinct numbers are presented. Finally, we prove a sampling in time theorem which recover the rotation number of continuous Stratonovich stochastic dynamical systems on $S^1$ out of its time discretisation of the flow.Comment: 15 page

On Takens' Last Problem: tangencies and time averages near heteroclinic networks

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields $C^2$-close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. We illustrate our results with an explicit example where historic behaviour arises $C^2$-close of a $\textbf{SO(2)}$-equivariant vector field