The Computational Complexity of Symbolic Dynamics at the Onset of Chaos

In a variety of studies of dynamical systems, the edge of order and chaos has been singled out as a region of complexity. It was suggested by Wolfram, on the basis of qualitative behaviour of cellular automata, that the computational basis for modelling this region is the Universal Turing Machine. In this paper, following a suggestion of Crutchfield, we try to show that the Turing machine model may often be too powerful as a computational model to describe the boundary of order and chaos. In particular we study the region of the first accumulation of period doubling in unimodal and bimodal maps of the interval, from the point of view of language theory. We show that in relation to the ``extended'' Chomsky hierarchy, the relevant computational model in the unimodal case is the nested stack automaton or the related indexed languages, while the bimodal case is modeled by the linear bounded automaton or the related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of manuscrip

Analysis and simulations for a phase‐field fracture model at finite strains based on modified invariants

Phase‐field models have already been proven to predict complex fracture patterns for brittle fracture at small strains. In this paper we discuss a model for phase‐field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. Here we present a phase‐field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split of the modified invariants of the right Cauchy‐Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the time‐discrete solutions converge in a weak sense to a solution of the time‐continuous formulation of the model. Numerical examples in two and three space dimensions illustrate the range of validity of the analytical results

Process intensification education contributes to sustainable development goals : part 1

In 2015 all the United Nations (UN) member states adopted 17 sustainable development goals (UN-SDG) as part of the 2030 Agenda, which is a 15-year plan to meet ambitious targets to eradicate poverty, protect the environment, and improve the quality of life around the world. Although the global community has progressed, the pace of implementation must accelerate to reach the UN-SDG time-line. For this to happen, professionals, institutions, companies, governments and the general public must become cognizant of the challenges that our world faces and the potential technological solutions at hand, including those provided by chemical engineering. Process intensification (PI) is a recent engineering approach with demonstrated potential to significantly improve process efficiency and safety while reducing cost. It offers opportunities for attaining the UN-SDG goals in a cost-effective and timely manner. However, the pedagogical tools to educate undergraduate, graduate students, and professionals active in the field of PI lack clarity and focus. This paper sets out the state-of-the-art, main discussion points and guidelines for enhanced PI teaching, deliberated by experts in PI with either an academic or industrial background, as well as representatives from government and specialists in pedagogy gathered at the Lorentz Center (Leiden, The Netherlands) in June 2019 with the aim of uniting the efforts on education in PI and produce guidelines. In this Part 1, we discuss the societal and industrial needs for an educational strategy in the framework of PI. The terminology and background information on PI, related to educational implementation in industry and academia, are provided as a preamble to Part 2, which presents practical examples that will help educating on Process Intensification