16 research outputs found
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order
polynomial expansions of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated truncation error
bounds, and maximizes the size of the image of the polynomial approximation
relative to some specified constraints. More precisely we use that the manifold
computations depend heavily on the scalings of the eigenvectors: indeed we
study the precise effects of these scalings on the estimates which determine
the validated error bounds. This relationship between the eigenvector scalings
and the error estimates plays a central role in our automatic procedures. In
order to illustrate the utility of these methods we present several
applications, including visualization of invariant manifolds in the Lorenz and
FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable
manifolds in a suspension bridge problem. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
Numerical computation of transverse homoclinic orbits for periodic solutions of delay differential equations
We present a computational method for studying transverse homoclinic orbits
for periodic solutions of delay differential equations, a phenomenon that we
refer to as the \emph{Poincar\'{e} scenario}. The strategy is geometric in
nature, and consists of viewing the connection as the zero of a nonlinear map,
such that the invertibility of its Fr\'{e}chet derivative implies the
transversality of the intersection. The map is defined by a projected boundary
value problem (BVP), with boundary conditions in the (finite dimensional)
unstable and (infinite dimensional) stable manifolds of the periodic orbit. The
parameterization method is used to compute the unstable manifold and the BVP is
solved using a discrete time dynamical system approach (defined via the
\emph{method of steps}) and Chebyshev series expansions. We illustrate this
technique by computing transverse homoclinic orbits in the cubic Ikeda and
Mackey-Glass systems
Analytic enclosure of the fundamental matrix solution
This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing the enclosure in the C s category
Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields
In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the Newton–Kantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained
Automatic differentiation for Fourier series and the radii polynomial approach
In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential
equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order
to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the
augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii
polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is
analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of
sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a
number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem
(PCRTBP
Stationary coexistence of hexagons and rolls via rigorous computations
In this work we introduce a rigorous computational method for finding heteroclinic solutions of a
system of two second order differential equations. These solutions correspond to standing waves
between rolls and hexagonal patterns of a two-dimensional pattern formation PDE model. After
reformulating the problem as a projected boundary value problem (BVP) with boundaries in the
stable/unstable manifolds, we compute the local manifolds using the parameterization method and
solve the BVP using Chebyshev series and the radii polynomial approach. Our results settle a
conjecture by Doelman et al. [European J. Appl. Math., 14 (2003), pp. 85–110] about the coexistence
of hexagons and rolls