Abstract : Understanding the long-term behaviour of a dynamical system is an important challenge that requires many mathematical techniques and numerical tools. Dynamical systems are models with a natural time evolution $($discrete or continuous$)$. The invariant sets $($fixed points, periodic orbits, invariant manifolds, strange attractors, etc$)$ are all objects that persist beyond the transient phase, and thus carry important information. It is also necessary to understand the bifurcations that occur as model parameters are varied. In this minisymposium, we will learn of new results for analyzing dynamical systems. Examples include passive dynamic walking, Fitzhugh-Nagumo neurons, and self-propelled particle systems.
Organizer(s) : Warwick Tucker (University of Melbourne, Australia), Yoshitaka Saiki (Hitotsubashi University, Japan), Hiroe Oka (Ryukoku University, Japan), Hiroshi Kokubu (Kyoto University, Japan)
[03364] Connectedness of graphs of dynamical systems
Format : Talk at Waseda University
Author(s) :
James A Yorke (Univ. Of Maryland )
Roberto De Leo (Howard Univ.)
Abstract : We have been developing the general theory of graphs of maps or differential equations that works for maps, ordinary differential equations and parabolic partial differential equations. We have a complete theory for the logistic map.
We have created axioms for graphs that can be used to prove results about graphs, axioms that hold for all of the standard definitions. We prove that the graph is connected under mild hypotheses.
[03770] Changes in basin of attraction by homoclinic and heteroclinic tangencies in passive dynamic walking
Format : Talk at Waseda University
Author(s) :
Kota Okamoto (Kyoto University)
Ippei Obayashi (Okayama University)
Hiroshi Kokubu (Kyoto University)
Kei Senda (Kyoto University)
Kazuo Tsuchiya (Kyoto University)
Shinya Aoi (Osaka University)
Abstract : In the passive dynamic walking that walks down a shallow slope without any control or input, countless sudden changes appear in the basin of attraction depending on the slope angle. An infinite number of periodic solutions also appear. We investigated the mechanism of the sudden changes in the basin of attraction of the passive dynamic walking based on the homoclinic and heteroclinic tangencies of the manifolds of the periodic solutions.
[04997] A novel bifurcation in hybrid dynamical systems: a model of human locomotion and its generalization
Format : Talk at Waseda University
Author(s) :
Hidetoshi Morita (Shitennoji University)
Shinya Aoi (Osaka Univeristy)
Kazuo Tsuchiya (Kyoto University)
Hiroshi Kokubu (Kyoto University)
Abstract : Aiming to understand the coexistence of walk and run in human locomotion, we study a simpler vertical motion of inverted spring-mass model. We observe the coexistence of two limit cycles between which no unstable periodic orbit lies, unlike the usual coexistence of attractors. We analyze the mechanism of this novel type of coexistence. We further consider a generalized hybrid dynamical systems model, and analyze the corresponding, as well as yet other, behaviors.
[03723] Coupled Hopf bifurcations: interaction between Fitzhugh-Nagumo neurons
Format : Talk at Waseda University
Author(s) :
Fátima Drubi (University of Oviedo)
Santiago Ibáñez (University of Oviedo)
Diego Noriega (University of Oviedo)
Abstract : Coupled dynamical systems allow to model a wide range of phenomena. We are interested in models where identical pieces with simple dynamics are coupled through simple mechanisms, wondering about the complexity that interaction may imply. The coupling of identical families of vector fields exhibiting a Hopf bifurcation is paradigmatic. After discussing a general model, we will focus on the interaction between two Fitzhugh-Nagumo planar systems and study the appearance of Hopf-Pitchfork and Hopf-Hopf bifurcations.
[04209] A method of computing Morse decomposition via approximate ODE solvers and its application
Format : Talk at Waseda University
Author(s) :
Tomoyuki Miyaji (Kyoto University)
Abstract : Computing Conley--Morse graphs for dynamical systems defined by ordinary differential equations using rigorous ODE solvers such as Lohner’s method is suffered from wrapping effects and is computationally expensive. It is especially remarkable when applied to Poincaré maps. In this talk, we apply approximate solvers, such as the Runge--Kutta methods, to computing Morse decomposition. We will discuss its application to Poincaré maps for 4D ODEs arising in modeling the motion of a self-propelled particle.
[03407] Finite-resolution recurrence in dynamical models
Format : Talk at Waseda University
Author(s) :
Pawel Pilarczyk (Gdansk University of Technology, Faculty of Applied Physics and Mathematics)
Justyna Signerska-Rynkowska (Dioscuri Centre in Topological Data Analysis, Institute of Mathematics of the Polish Academy of Sciences)
Grzegorz Graff (Gdansk University of Technology, Faculty of Applied Physics and Mathematics)
Abstract : In order to improve the method for rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 (SIAM J. Appl. Dyn. Syst. 8, 757-789), we introduce the notion of finite-resolution recurrence, develop an effective algorithm for its computation, and apply it to a two-dimensional model of a neuron. We additionally use machine learning to distinguish between different types of dynamics observed.
[04891] Reconstruction of stationary measures from `cycles’ in random dynamical systems
Format : Talk at Waseda University
Author(s) :
Hiroki Takahasi (Keio Institute of Pure and Applied Sciences)
Abstract : One leading idea in the qualitative understanding of deterministic dynamical systems is to use collections of periodic orbits to structure the dynamics. This idea traces back to Poincar\'e, and has been supported by Bowen who proved that periodic orbits of topologically mixing Axiom A diffeomorphisms equidistribute with respect to the measure of maximal entropy. We consider an analogue of Bowen's equidistribution theorem of periodic orbits in the context of simple random dynamical systems.
[03823] Computer verifiable criteria for chaos in piecewise smooth dynamical systems
Format : Talk at Waseda University
Author(s) :
Paul Glendinning (University of Manchester)
Abstract : Many theoretical results on chaotic behaviour in piecewise smooth maps rely on relatively simple criteria that could in principle be checked by hand. I will describe some of these results and how they can be implemented numerically. This makes it possible to demonstrate the existence of phenomena such as robust chaos and unstable manifold variability in examples. Many of the results presented are from joint work with D.J.W. Simpson (Massey University, New Zealand).
