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Chaotic backward shift operator on Chebyshev polynomials

Part of: Ordinary differential operators General theory of linear operators General theory in ordinary differential equations Boundary value problems

Published online by Cambridge University Press:  21 December 2018

MÁRTON KISS Open the ORCID record for MÁRTON KISS [Opens in a new window]  and
TAMÁS KALMÁR-NAGY
MÁRTON KISS
Affiliation:
Department of Differential Equations, Institute of Mathematics, Budapest University of Technology and Economics, H 1111 Budapest, Müegyetem rkp. 3-9, Hungary email: mkiss@math.bme.hu
TAMÁS KALMÁR-NAGY
Affiliation:
Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, H 1111 Budapest, Müegyetem rkp. 3-9, Hungary email: ejam@kalmarnagy.com
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Abstract

We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.

Keywords

Backward shiftprincipal value integralChebyshev polynomials

MSC classification

Primary: 34A55: Inverse problems 34B20: Weyl theory and its generalizations 34B24: Sturm-Liouville theory 34B45: Boundary value problems on graphs and networks
Secondary: 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.) 47A75: Eigenvalue problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 1025 - 1037
Copyright
© Cambridge University Press 2018 

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