Abstract : The aim of dynamical systems theory is to understand the long term behavior of large sets of initial conditions, often
for highly nonlinear models coming from realistic application problems. Due to the importance of nonlinear models, numerical calculations have long played an important role. In this session we bring together experts to discuss problems at the frontiers of our understanding. Some talks will focus on new computational methods, some on attempts to understand more and more realistic models, and some on theoretical issues which inform our approach to computational dynamics.
[02779] Polynomial discretisations of transfer and Koopman operators in chaotic dynamics
Author(s) :
Caroline Wormell (The Australian National University)
Abstract : Many long-term statistical properties of chaotic systems are encoded by transfer or Koopman operators. Orthogonal polynomial-based operator discretisations are generally very efficient, and I will show in expanding dynamics these operators are no exception: a Chebyshev discretisation allows fast, very accurate, rigorous estimates of expanding dynamics, even with parabolic fixed points. Furthermore, a theoretical extension to general orthogonal polynomials proves fast convergence of Extended Dynamical Mode Decomposition, a data-driven algorithm commonly used across physical sciences.
[03719] Estimating the spectra for annealed transfer operators of random dynamical systems
Author(s) :
Alex Blumenthal (Georgia Tech)
Isaia Nisoli (Universidade Federal de Rio de Janeiro)
Toby Taylor-Crush (University of Loughborough)
Abstract : I will describe some recent efforts with my collaborators Toby Taylor-Crush and Isaia Nisoli towards the computer-validated estimation of spectra for annealed transfer operators of random dynamical systems. Applications include the study of various stochastic bifurcations associated to the explosion of the support of a stationary measure as some underlying parameter is varied.
[03734] Energy growth in Hamiltonian systems with small dissipation
Author(s) :
Marian Gidea (Yeshiva Univesity)
Abstract : We consider a model for an energy harvesting device consisting of a rotator and a pendulum subject to a small perturbation given by a time-periodic Hamiltonian vector field plus a conformally symplectic vector field. In general, the system has energy dissipation. We provide explicit conditions so that the system exhibits energy growth. In theory, this shows Arnold diffusion in Hamiltonian systems with small dissipation. In practice, this translates into continuous generation of electricity.
[03874] A dynamical systems approach to low-damage seismic design
Author(s) :
Hinke M Osinga (University of Auckland)
Abstract : An example of low-damage seismic design is the post-tensioned moment-resisting frame, which exhibits geometric nonlinearity under large deformations. Whether the tilt angle of the frame exceeds a prescribed maximum depends on the forcing properties. We show that this failure boundary is organised by so-called grazing orbits, which reach but do not move beyond the design limit of the frame. We consider both harmonic and aperiodic waves with a broader frequency content.
[03886] Understanding how blenders emerge: weaving a carpet from global manifolds
Author(s) :
Dana C'Julio (University of Auckland)
Bernd Krauskopf (University of Auckland)
Hinke M Osinga (University of Auckland)
Abstract : A blender is a tool for constructing `wild' robust chaotic dynamics in partially hyperbolic systems. We make precise statements about how a blender emerges in a family of 3D Hénon-like maps as parameters are changed. To this end, we employ advanced numerical techniques to determine when the one-dimensional stable manifolds of two saddle points weave through phase space to form an impenetrable carpet, which is the characterising property of a blender.
[03948] A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries
Author(s) :
Emmanuel Fleurantin (George Mason University, University of North Carolina at Chapel Hill)
Katherine Slyman (University of North Carolina at Chapel Hill)
Blake Barker (Brigham Young University)
Christopher K.R.T. Jones (George Mason University, University of North Carolina at Chapel Hill)
Abstract : Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler-Lagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The Onsager-Machlup (OM) functional, which is treated as a perturbation of the Friedlin-Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of this talk will be focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.
[04096] On the connectedness and disconnectedness of the Julia set for the Hénon map.
Author(s) :
Zin Arai (Tokyo Institute of Technology)
Abstract : We discuss the connectedness and disconnectedness of the Julia set for the complex Hénon map. To prove the disconnectedness of the Julia set, we develop a topological method that uses the plurisubharmonic nature of the Green function. We also construct a hyperbolic complex Hénon map with a connected Julia set. Consequently, we obtain a certain topological property of the connectedness locus of the map. This is a joint work with Yutaka Ishii (Kyushu Univerisity).
[04135] Standard piecewise smooth symplectic maps
Author(s) :
Vered Rom-Kedar (The Weizmann Institute)
Michal Pnueli (The Weizmann Institute)
Alexandra Zobova (The Weizmann Institute)
Abstract : Return maps of near integrable/near quasi-integrable Hamiltonian impact systems are shown to produce piecewise smooth symplectic maps. In the integrable/quasi-integrable limit, these maps reduce to piecewise smooth families of rotations/interval exchange maps. As for the standard map, we introduce simplified models for these return maps and study their dynamics numerically. Regular and singular resonances emerge, as well as transient behavior, leading to conjectures regarding the non-existence of dividing circles in the singularity bands.
[04300] Optimal linear response for expanding circle maps
Author(s) :
Gary Froyland (UNSW Sydney)
Stefano Galatolo (University of Pisa)
Abstract : We consider the problem of optimal linear response for deterministic expanding maps of the circle. To each infinitesimal perturbation of a circle map we consider the response of the expectation of an observation function, and the response of isolated spectral points of the transfer operator. Under mild conditions on the set of feasible perturbations we show there is a unique optimal perturbation. We derive expressions for the unique optimum, and devise a Fourier-based computational scheme.
[04937] Finite element approximated manifolds for PDEs by the parameterization method
Author(s) :
Jorge Gonzalez (Georgia Tech)
Jason Desmond Mireles James (Florida Atlantic University)
Necibe Tuncer (Florida Atlantic University)
Abstract : The computation of invariant manifolds for PDEs is significantly challenging over irregular high dimensional domains where the classical Fourier methods are not applicable. This work presents a new framework of interest for practical applications that combines the parameterization method with the classical finite element method. We implement the method for a variety of examples having both polynomial and non-polynomial nonlinearities, on non-convex and not necessarily simply connected polygonal domains.
[05240] Dynamics of a Hill four-body problem with oblate bodies
Author(s) :
Wai Ting Lam (Florida Atlantic University)
Abstract : Consider a restricted four body problem with three oblate massive bodies, which are assumed to move in a plane under their mutual gravity, and an infinitestimal fourth body to move in the 3-dimensional space under the gravitational influence of the three heavy bodies, but without affecting them. By performing Hill approximation, we study the dynamics and properties of the infinitesimal body in a neighborhood of the smaller body.
[05259] Parametrisation method for large finite element models of engineering structures
Author(s) :
Alessandra Vizzaccaro (University of Exeter)
Andrea Opreni (Politecnico di Milano)
Giorgio Gobat (Politecnico di Milano)
Attilio Alberto Frangi (Politecnico di Milano)
Cyril Touze' (ENSTA Paris )
Abstract : In this contribution we present a method to directly compute asymptotic expansion of invariant manifolds of large finite element models from physical coordinates and their reduced order dynamics on the manifold. The focus of hte talk is on engineering structures, whose spectrum around the fixed point is usually composed of complex conjugate pair of eigenvalues with always negative but small real part. This gives rise to rich dynamical behaviour such as internal resonances, parametric resonances, and superharmonic resonances. The accuracy of the reduction on the slow invariant manifold will be shown on selected examples.
