scholarly journals Complexity in Linear Systems: A Chaotic Linear Operator on the Space of Odd2π-Periodic Functions

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Tamás Kalmár-Nagy ◽  
Márton Kiss
Keyword(s):  
Nonlinear Systems ◽  
Linear Operator ◽  
Linear Systems ◽  
Chaotic Dynamics ◽  
Periodic Points ◽  
Complex Behavior ◽  
Periodic Functions ◽  
Infinite Dimensional ◽  
Backward Shift ◽  
Principal Value

Not just nonlinear systems but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequencesl2displays chaotic dynamics. Here we construct the corresponding operatorCon the space of2π-periodic odd functions and provide its representation involving a Principal Value Integral. We explicitly calculate the eigenfunction of this operator, as well as its periodic points. We also provide examples of chaotic and unbounded trajectories ofC.

Chaotic backward shift operator on Chebyshev polynomials

2018 ◽  
Vol 30 (5) ◽  
pp. 1025-1037
Author(s):  
MÁRTON KISS ◽  
TAMÁS KALMÁR-NAGY
Keyword(s):  
Explicit Form ◽  
Chebyshev Polynomials ◽  
Chaotic Dynamics ◽  
Shift Operator ◽  
Periodic Points ◽  
Recurrence Plots ◽  
Chaotic Operator ◽  
Fractal Measures ◽  
Backward Shift ◽  
Principal Value

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.

scholarly journals Recurrent flows: the clockwork behind turbulence

2013 ◽  
Vol 726 ◽  
pp. 1-4 ◽  
Author(s):  
Predrag Cvitanović
Keyword(s):  
Fluid Dynamics ◽  
Reynolds Number ◽  
Chaotic Dynamics ◽  
High Dimensional ◽  
Governing Equations ◽  
Infinite Dimensional ◽  
Long Run ◽  
Moderate Reynolds Number ◽  
Dimensional State Space ◽  
Exhaustive Study

AbstractThe understanding of chaotic dynamics in high-dimensional systems that has emerged in the last decade offers a promising dynamical framework to study turbulence. Here turbulence is viewed as a walk through a forest of exact solutions in the infinite-dimensional state space of the governing equations. Recently, Chandler & Kerswell (J. Fluid Mech., vol. 722, 2013, pp. 554–595) carry out the most exhaustive study of this programme undertaken so far in fluid dynamics, a feat that requires every tool in the dynamicist’s toolbox: numerical searches for recurrent flows, computation of their stability, their symmetry classification, and estimating from these solutions statistical averages over the turbulent flow. In the long run this research promises to develop a quantitative, predictive description of moderate-Reynolds-number turbulence, and to use this description to control flows and explain their statistics.

Sign in / Sign up

Export Citation Format

Share Document