Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

The linking homomorphism of one-dimensional minimal sets

AbstractWe introduce a way of characterizing the linking of one-dimensional minimal sets in three-dimensional flows and carry out the characterization for some minimal sets within flows modelled by templates, with an emphasis on the linking of Denjoy continua. We also show that any aperiodic minimal subshift of minimal block growth has a suspension which is homeomorphic to a Denjoy continuum

Some elements for a history of the dynamical systems theory

Leon Glass would like to thank the Natural Sciences and Engineering Research Council (Canada) for its continuous support of curiosity-driven research for over 40 years starting with the events recounted here. He also thanks his colleagues and collaborators including Stuart Kauffman, Rafael Perez, Ronald Shymko, Michael Mackey for their wonderful insights and collaborations during the times recounted here. R.G. is endebted to the following friends and colleagues, listed in the order encountered on the road described: F. T. Arecchi, L. M. Narducci, J. R. Tredicce, H. G. Solari, E. Eschenazi, G. B. Mindlin, J. L. Birman, J. S. Birman, P. Glorieux, M. Lefranc, C. Letellier, V. Messager, O. E. Rössler, R. Williams. U.P. would like to thank the following friends and colleagues who accompanied his first steps into the world of nonlinear phenomena: U. Dressler, I. Eick, V. Englisch, K. Geist, J. Holzfuss, T. Klinker, W. Knop, A. Kramer, T. Kurz, W. Lauterborn, W. Meyer-Ilse, C. Scheffczyk, E. Suchla and M. Wisenfeldt. The work by L. Pecora and T. Carroll was supported directly by the Office of Naval Research (ONR) and by ONR through the Naval Research Laboratory’s Basic Research Program. C.L. would like to thank Jürgen Kurths for his support to this project.Peer reviewedPostprintPublisher PD

Stability and Diversity in Collective Adaptation

We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show that, although individual agents interact with their environment and other agents in a purely self-interested way, macroscopic behavior can be interpreted as game dynamics. Application to several familiar, explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors. The simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation. We also analyze the adaptation dynamics from an information-theoretic viewpoint and discuss self-organization induced by information flux between agents, giving a novel view of collective adaptation.Comment: 22 pages, 23 figures; updated references, corrected typos, changed conten

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry related to Hamiltonian dynamics