345 research outputs found
Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation
We present an algorithm for the rigorous integration of Delay Differential
Equations (DDEs) of the form . As an application, we
give a computer assisted proof of the existence of two attracting periodic
orbits (before and after the first period-doubling bifurcation) in the
Mackey-Glass equation
An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs
We propose a new algorithm for computing validated bounds for the solutions
to the first order variational equations associated to ODEs. These validated
solutions are the kernel of numerics computer-assisted proofs in dynamical
systems literature. The method uses a high-order Taylor method as a predictor
step and an implicit method based on the Hermite-Obreshkov interpolation as a
corrector step. The proposed algorithm is an improvement of the -Lohner
algorithm proposed by Zgliczy\'nski and it provides sharper bounds.
As an application of the algorithm, we give a computer-assisted proof of the
existence of an attractor set in the R\"ossler system, and we show that the
attractor contains an invariant and uniformly hyperbolic subset on which the
dynamics is chaotic, that is, conjugated to subshift of finite type with
positive topological entropy.Comment: 33 pages, 11 figure
Lagrangian Reachabililty
We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that
conservatively approximates the set of reachable states of a nonlinear
dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF),
which is derived from an over-approximation of the gradient of the solution
flows. The SF measures the discrepancy between two states propagated by the
system solution from two initial states lying in a well-defined region, thereby
allowing LRT to compute a reachtube with a ball-overestimate in a metric where
the computed enclosure is as tight as possible. To evaluate its performance, we
implemented a prototype of LRT in C++/Matlab, and ran it on a set of
well-established benchmarks. Our results show that LRT compares very favorably
with respect to the CAPD and Flow* tools.Comment: Accepted to CAV 201
Hysteresis in Adiabatic Dynamical Systems: an Introduction
We give a nontechnical description of the behaviour of dynamical systems
governed by two distinct time scales. We discuss in particular memory effects,
such as bifurcation delay and hysteresis, and comment the scaling behaviour of
hysteresis cycles. These properties are illustrated on a few simple examples.Comment: 28 pages, 10 ps figures, AMS-LaTeX. This is the introduction of my
Ph.D. dissertation, available at
http://dpwww.epfl.ch/instituts/ipt/berglund/these.htm
Fractal Properties of Robust Strange Nonchaotic Attractors in Maps of Two or More Dimensions
We consider the existence of robust strange nonchaotic attractors (SNA's) in
a simple class of quasiperiodically forced systems. Rigorous results are
presented demonstrating that the resulting attractors are strange in the sense
that their box-counting dimension is N+1 while their information dimension is
N. We also show how these properties are manifested in numerical experiments.Comment: 9 pages, 14 figure
The Reach-and-Evolve Algorithm for Reachability Analysis of Nonlinear Dynamical Systems
This paper introduces a new algorithm dedicated to the rigorous reachability analysis of nonlinear dynamical systems. The algorithm is initially presented in the context of discrete time dynamical systems, and then extended to continuous time dynamical systems driven by ODEs. In continuous time, this algorithm is called the Reach and Evolve algorithm. The Reach and Evolve algorithm is based on interval analysis and a rigorous discretization of space and time. Promising numerical experiments are presented
High-order Lohner-type algorithm for rigorous computation of Poincar\'e maps in systems of Delay Differential Equations with several delays
We present a Lohner-type algorithm for rigorous integration of systems of
Delay Differential Equations (DDEs) with multiple delays and its application in
computation of Poincar\'e maps to study the dynamics of some bounded, eternal
solutions. The algorithm is based on a piecewise Taylor representation of the
solutions in the phase-space and it exploits the smoothing of solutions
occurring in DDEs to produces enclosures of solutions of a high order. We apply
the topological techniques to prove various kinds of dynamical behavior, for
example, existence of (apparently) unstable periodic orbits in Mackey-Glass
Equation (in the regime of parameters where chaos is numerically observed) and
persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the
R\"ossler system)
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
- …