Characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series is one of the most intensively studied contemporary problems in nonlinear sciences. In this work, we address this problem by combining two established concepts—empirical mode decomposition (EMD) and generalized fractal dimensions—into a unified analysis framework. Specifically, we demonstrate that the intrinsic mode functions derived by EMD can be used as a source of local (in terms of scales) information about the properties of the phase-space trajectory of the system under study, allowing us to derive multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three well-known low-dimensional deterministic dynamical systems (the Hénon map, the Lorenz ’63 system, and the standard map), three realizations of fractional Brownian motion with different Hurst exponents, and two somewhat higher-dimensional deterministic dynamical systems (the Lorenz ’96 model and the on–off intermittency model). These examples allow us to assess the performance of our formalism with respect to practically relevant aspects like additive noise, different initial conditions, the length of the time series under study, low- vs high-dimensional dynamics, and bursting effects. Finally, by taking advantage of two real-world systems whose multiscale features have been widely investigated (a marine stack record providing a proxy of the global ice volume variability of the past years and the SYM-H geomagnetic index), we also illustrate the applicability of this formalism to real-world time series.
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December 2020
Research Article|
December 03 2020
Multiscale measures of phase-space trajectories
Tommaso Alberti
;
Tommaso Alberti
a)
1
INAF-Istituto di Astrofisica e Planetologia Spaziali
, Via del Fosso del Cavaliere 100, I-00133 Roma, Italy
a)Author to whom correspondence should be addressed: [email protected]
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Giuseppe Consolini
;
Giuseppe Consolini
b)
1
INAF-Istituto di Astrofisica e Planetologia Spaziali
, Via del Fosso del Cavaliere 100, I-00133 Roma, Italy
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Peter D. Ditlevsen
;
Peter D. Ditlevsen
c)
2
Centre for Ice and Climate, Niels Bohr Institute, University of Copenhagen
, Copenhagen 2200, Denmark
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Reik V. Donner
;
Reik V. Donner
d)
3
Department of Water, Environment, Construction and Safety, Magdeburg-Stendal University of Applied Sciences
, Breitscheidstraße 2, 39114 Magdeburg, Germany
4
Potsdam Institute for Climate Impact Research (PIK)—Member of the Leibniz Association
, Telegrafenberg A31, 14473 Potsdam, Germany
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Virgilio Quattrociocchi
Virgilio Quattrociocchi
e)
1
INAF-Istituto di Astrofisica e Planetologia Spaziali
, Via del Fosso del Cavaliere 100, I-00133 Roma, Italy
5
Dip. Scienze Fisiche e Chimiche, Università degli Studi dell’Aquila
, Via Vetoio, I-67100 L’Aquila, Italy
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a)Author to whom correspondence should be addressed: [email protected]
Chaos 30, 123116 (2020)
Article history
Received:
March 26 2020
Accepted:
November 12 2020
Citation
Tommaso Alberti, Giuseppe Consolini, Peter D. Ditlevsen, Reik V. Donner, Virgilio Quattrociocchi; Multiscale measures of phase-space trajectories. Chaos 1 December 2020; 30 (12): 123116. https://doi.org/10.1063/5.0008916
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