Computer-aided Computation of Abelian integrals and Robust Normal Forms

• This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied. In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements. Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees. In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Ordinary differential equations

parameter estimation

planar Hamiltonian systems

bifurcation theory

Abelian integrals

limit cycles

normal forms

hyperbolic fixed points

numerical integration

invariant manifolds

interval analysis. 2000 Mathematics Subject Classification. 34A60

34C07

34C20

37D10

37G15

37M20

37M99

65G20

65L09

65L70.

MATHEMATICS

MATEMATIK

matematik

Mathematics

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