Browsing by Subject "attractors"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Attractors and Attracting Neighborhoods for Multiflows
    (2019-05) Negaard-Paper, Shannon
    We already know a great deal about dynamical systems with uniqueness in forward time. Indeed, flows, semiflows, and maps (both invertible and not) have been studied at length. A view that has proven particularly fruitful is topological: consider invariant sets (attractors, repellers, periodic orbits, etc.) as topological objects, and the connecting sets between them form gradient like flows. In the case of systems with uniqueness in forward time, an attractor in one system is related to nearby attractors in a family of other, "close enough" systems. One way of seeing that connection is through the Conley decomposition (and the Conley index) [2], [13]. This approach requires focusing on isolated invariant sets - that is, invariant sets with isolating neighborhoods. If there is an invariant set I, which has an isolating neighborhood N, we say that I is the invariant set associated to N, and N is an isolating neighborhood associated to I. When the invariant set in question is an attractor or a repeller, then the isolating neighborhood is called an attracting neighborhood or a repelling neighborhood, respectively. A more specialized case may be called an attractor block or a repeller block. This approach was expanded to discrete time systems which lack uniqueness in forward time, using relations, in [7] and [11]. Relations do not rely on uniqueness in forward time, but the graph of any map is a relation; thus they serve to generalize maps. Some of this is reviewed in the next few sections. In addition, I expanded on work done in [7] to show that in compact metric spaces, attractors for closed relations continue (see Section 6.1). On the continuous time side, more work needs to be done. This paper is a step toward a more systematic approach for continuous time systems which lack uniqueness in forward time. This work applies to Filippov systems [4] and in control theory [12]. In the following pages, we establish a tool (multiflows) for discussing the continuous time case and develop a framework for understanding attractors (and therefore stability) in these systems. A crucial part of this work was establishing attractor / attracting neighborhood pairs, which happens in Section 5.5.
  • Thumbnail Image
    Item
    Metric Properties of Attractors for Vector Fields via Bounded, Nonautonomous Control
    (2019-05) Meyer, Katherine
    Attractors organize long-term behavior in dynamical systems, and their robustness against perturbations reflects both the resilience of a model system and the likelihood that the inexact model captures essential features of reality. Quantifying attractor strength requires metric information beyond the topological setting of classical dynamics. We extend the theory of “intensity of attraction,” first developed by McGehee for maps on compact metric spaces, to the context of continuous-time dynamics determined from a vector field on Euclidean space. Intensity measures the strength of transient dynamics within a domain of attraction, and can be computed by tracking the sets reachable from the attractor under bounded, nonautonomous control. We connect bounded control systems to multiflows, a nascent framework of multivalued dynamics. A connection between reachable sets and isolating blocks implies that an attractor’s intensity not only reflects its capacity to retain solutions under time-varying perturbations, but also gives a lower bound on the distance the attractor continues in the space of vector fields.
  • Thumbnail Image
    Item
    Oral History Interview with Ernest Alan Edmonds
    (Charles Babbage Institute, 2022) Edmonds, Ernest Alan
    This interview is part of a series on Human-Computer Interaction (HCI) conducted by the Charles Babbage Institute for ACM SIGCHI (Association for Computing Machinery Special Interest Group on Computer-Human Interaction). In this interview, Professor Edmonds reflects on early interests in drawing and art, his development of interactive art and his research on computer-human interaction support for creativity. He describes himself as an “artist by inclination, a logician by training, and a computer scientist by accident.” The bulk of the interview explores these areas, his pioneering in work in computer based/algorithmic (from his influential Nineteen in the late 1960s forward) and interactive art (DataPack, in 1970 with Stroud Cornock, forward), as well as important contributions to computer science. This includes relating of influences of university mentors, the Constructivist school, and collaborations and friendships within the Systems Group of UK artists. He discusses his computer science contributions—at Leicester Polytechnic, Loughborough University, and University of Technology in Sydney—including his “adaptive approach” in 1970 software development challenging established “waterfall” techniques and anticipating and helping provide foundation to what later became termed as “agile.” He also relates his contributions to fostering intellectual community with computer scientists and artists, including and especially with his and Linda Candy’s impactful “Creativity and Cognition” an annual event launched 1993, becoming a SIGCHI Conference in 1997, and thriving to this day. In his career, thinking a step beyond current technology, and drawing on his concepts of “attractors, sustainers, and relators,” he has creatively advanced interaction between human and machine, and human interaction through machines.