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The main motivation to study combinatorial dynamics comes from data science. Since the publication in 1998 of the seminal work by Robin Forman on combinatorial Morse theory there has been growing interest in dynamical systems on finite spaces.
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Computer-assisted proofs, in particular for the existence of renormalisation fixed points and the spectral properties of the relevant linearised operators. His main research interests are in renormalisation for dynamical systems, with an emphasis on computer-assisted proofs. He also co-founded and chaired the Portsmouth Cafe Scientifique over 2006-2016. He has been involved in several successful projects on the interface between science and public understanding, including the Mathematics Posters on the London Underground project in 2000, with the Isaac Newton Institute for Mathematical Sciences in Cambridge (and later on the Boston Subway with the Clay Mathematics Institute), and the Dynamics of Spin exhibit at the Royal Society Summer Science Exhibition in London in 2007 and at Techfest (Asia's largest Science and Technology festival) held in Mumbai in 2008. Subsequently he went on to work at Hewlett-Packard's research laboratories in Bristol before taking up Postdoctoral research positions in the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge (studying the dynamics of nonexpansive maps, with applications to discrete event systems) and later in the School of Mathematics at the University of Bristol where he gained a permanent post as Scientific Programmer (studying transport phenomena in Hamiltonian systems, with applications to problems in Physics, Chemistry, and Celestial Mechanics). He also gave a course of lectures in Part III of the University of Cambridge Mathematics Tripos (Applied) on renormalisation in Dynamical Systems. He joined the Department of Mathematics at the University of Portsmouth in 2005, and is now a member of the Applied Mathematics Group. For golden mean rotation number, he used bounds on the fixed point of the corresponding (Fibonacci-type) renormalisation operator to verify the necklace hypotheses of Stirnemann and thereby to give a rigorous proof of conjectures of Widom explaining the universality observed by Manton and Nauenberg.
Wiedzmin 2 operator series#
Our computations use multi-precision interval arithmetic with rigorous directed rounding modes to bound tightly the coefficients of the relevant power series and their high-order terms, and the corresponding universal constants.Īndrew Burbanks is the Associate Head for Research and Innovation and a Principal Lecturer in the School of Mathematics and Physics at the University of Portsmouth in the UK. He completed his PhD at the University of Loughborough, using computer-assisted techniques to explain the universality observed in the breakup of quasiperiodicity (conjugacy to rigid rotation) on the boundary of Siegel discs in complex maps - a prototypical KAM-type scenario. We adapt the procedure to the eigenproblem for the scaling of added uncorrelated noise. In particular, we gain tight bounds on the eigenfunction corresponding to the essential expanding eigenvalue delta.
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We bound the spectrum of the Frechet derivative of the renormalisation operator at the fixed point, establishing the hyperbolic structure, in which the presence of a single essential expanding eigenvalue explains the universal asymptotically self-similar bifurcation structure observed in the iterations of families of maps lying in the relevant universality class.īy recasting the eigenproblem for the Frechet derivative in a particular nonlinear form, we again use the contraction mapping principle to gain rigorous bounds on eigenfunctions and their corresponding eigenvalues. Building on previous work, our proof uses rigorous computer-assisted means to bound operations in a space of analytic functions and hence to show that a quasi-Newton operator for the fixed-point problem is a contraction map on a suitable ball.
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We work with a modified operator that encodes the action of the renormalisation operator on even functions. We prove the existence of a fixed point to the renormalisation operator for period doubling in maps of even degree at the critical point.