The cusp–Hopf bifurcation

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold-Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold-heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems

Coming of the third wave: A move toward best practice, user defined tools and mainstream integration for virtual worlds in education

The Gartner Hype Cycle has placed virtual worlds on the climb up the Slope of Enlightenment. While some authors in the past have made much of the educational use of virtual worlds languishing in the Trough of Disillusionment, there has been a community of authors, designers and educators working to further understanding of the limitations and affordances of such technologies. It is time to pool this knowledge, experience, tools and practice to solidify best practice, focus research on development of specific elements and forge ahead to shape the third wave of educational virtual worlds. This paper attempts to outline this information and practice while offering solutions for further development

Evaluating Data Assimilation Algorithms

Data assimilation leads naturally to a Bayesian formulation in which the posterior probability distribution of the system state, given the observations, plays a central conceptual role. The aim of this paper is to use this Bayesian posterior probability distribution as a gold standard against which to evaluate various commonly used data assimilation algorithms. A key aspect of geophysical data assimilation is the high dimensionality and low predictability of the computational model. With this in mind, yet with the goal of allowing an explicit and accurate computation of the posterior distribution, we study the 2D Navier-Stokes equations in a periodic geometry. We compute the posterior probability distribution by state-of-the-art statistical sampling techniques. The commonly used algorithms that we evaluate against this accurate gold standard, as quantified by comparing the relative error in reproducing its moments, are 4DVAR and a variety of sequential filtering approximations based on 3DVAR and on extended and ensemble Kalman filters. The primary conclusions are that: (i) with appropriate parameter choices, approximate filters can perform well in reproducing the mean of the desired probability distribution; (ii) however they typically perform poorly when attempting to reproduce the covariance; (iii) this poor performance is compounded by the need to modify the covariance, in order to induce stability. Thus, whilst filters can be a useful tool in predicting mean behavior, they should be viewed with caution as predictors of uncertainty. These conclusions are intrinsic to the algorithms and will not change if the model complexity is increased, for example by employing a smaller viscosity, or by using a detailed NWP model

Dimension reduction for systems with slow relaxation

We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model reduction, and build a mathematical framework for analyzing the reduced models. We introduce the notions of universal and asymptotic filters to characterize `optimal' model reductions for sloppy linear models. We illustrate our methods by applying them to the practically important problem of modeling evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof

Stochastic climate theory and modeling

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models

Learning TRIZ: Impact on confidence when facing problems

This paper explores the impact of learning problem solving tools such as Theory of Inventive Problem Solving (TRIZ) on self-efficacy. The paper utilises survey responses from 94 students who were enrolled in an RMIT-wide elective which taught students tools of TRIZ between 2006 and 2010. It was found that there were correlations between questions of self-efficacy and questions of attitude when facing future problems. A stronger correlation was observed between self-efficacy judged on enactive mastery experience (past performance) compared to self-efficacy judged on vicarious experience (peer comparison). Learning TRIZ was found to have a stronger influence on selfefficacy judged on past performance. The findings in this study suggest that learning specific tools of problem solving together with effective implementation can assist with the development of self-efficacy. Self-efficacy is vital as it impacts the willingness to face future problems which has a role in the improvement of problem solving ability. We propose that learning TRIZ leads to the development of problem solving skills. This paper is part of an ongoing PhD research that addresses the issue of the measure and transferability of innovative problem solving skills within the engineering field

Educating a reflective engineer: learning from engineering experts

Engineers are expected to engage in the process of reflection for their learning and when resolving problems. This is evident as reflection is one of the focuses in engineering education. Reflection can enhance the learning of students. However, recent literature also found that the process of reflection is not practiced by young engineers. To understand this phenomenon, the authors investigated the perceptions of expert and novice engineers on the process of reflection. This study is part of an overall research into how problem solving performance can be enhanced through formal instructions. PURPOSE The purpose of this paper is to investigate the following: • What can be learned about the process of reflection by comparing responses of expert and novice engineers? • If there are any differences, what caused these differences? DESIGN/METHOD The research design consists of two phases including qualitative and quantitative methods. Initial interview data was collected and analysed using Grounded Theory methodology, involving 22 engineers. The results of the interview data is then verified through a questionnaire, involving responses from 221 engineers. RESULTS The study found that novice and expert engineers perceived the process of reflection differently. It was discovered that novice engineers are likely to reflect only when mistakes are made. On the other hand, expert engineers are more likely to reflect continuously when they resolve problems.The outcome of this study has implications for engineering curricula. The insights gained from why expert engineers resolve problems the way they do, highlight the misconceptions that novice engineers have on the process of reflection which need to be addressed. The outcome of the research also revealed what is really required to achieve effective reflection for learning