Detectability of deterministic non-linear processes in Earth rotation time-series-I. Embedding

Frède, Valérie; Mazzéga, Pierre

doi:10.1046/j.1365-246x.1999.00821.x

Cited by 15 publications

(16 citation statements)

References 35 publications

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“…On the other hand, KANTZ and SCHREIBER (1996) note that the estimation of s using AMI is reliable only for two-dimensional embeddings. In most practical applications both methods are employed and it is either found that the estimated delay times are similar (FREDE and MAZZEGA, 1999) or that the two methods yield significantly different values of s, in which case additional constraints may be needed (like the construction of twodimensional phase portraits for different values of s) in order to make a final decision (KONSTANTINOU, 2002).…”

Section: Determination Of the Delay Timementioning

confidence: 99%

Nonlinear Time Series Analysis of Volcanic Tremor Events Recorded at Sangay Volcano, Ecuador

Konstantinou

Lin

2004

Pure and Applied Geophysics

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The assumption that volcanic tremor may be generated by deterministic nonlinear source processes is now supported by a number of studies at different volcanoes worldwide that clearly demonstrate the low-dimensional nature of the phenomenon. We applied methods based on the theory of nonlinear dynamics to volcanic tremor events recorded at Sangay volcano, Ecuador in order to obtain more information regarding the physics of their source mechanism. The data were acquired during 21-26 April 1998 and were recorded using a sampling interval of 125 samples s À1 by two broadband seismometers installed near the active vent of the volcano. In a previous study JOHNSON and LEES (2000) classified the signals into three groups: (1) short duration (<1 min) impulses generated by degassing explosions at the vent; (2) extended degassing 'chugging' events with a duration 2-5 min containing well-defined integer overtones (1-5 Hz) and variable higher frequency content; (3) extended degassing events that contain significant energy above 5 Hz. We selected 12 events from groups 2 and 3 for our analysis that had a duration of at least 90 s and high signal-to-noise ratios. The phase space, which describes the evolution of the behavior of a nonlinear system, was reconstructed using the delay embedding theorem suggested by Takens. The delay time used for the reconstruction was chosen after examining the first zero crossing of the autocorrelation function and the first minimum of the Average Mutual Information (AMI) of the data. In most cases it was found that both methods yielded a delay time of 14-18 samples (0.112-0.144 s) for group 2 and 5 samples (0.04 s) for group 3 events. The sufficient embedding dimension was estimated using the false nearest neighbors method which had a value of 4 for events in group 2 and was in the range 5-7 for events in group 3. Based on these embedding parameters it was possible to calculate the correlation dimension of the resulting attractor, as well as the average divergence rate of nearby orbits given by the largest Lyapunov exponent. Events in group 2 exhibited lower values of both the correlation dimension (1.8-2.6) and largest Lyapunov exponent (0.013-0.022) in comparison with the events in group 3 where the values of these quantities were in the range 2.4-3.5 and 0.029-0.043, respectively. Theoretically, a nonlinear oscillation described by the equation € x x þ b_ x x þ cgðxÞ ¼ f cos xt can generate deterministic signals with characteristics similar to those observed in groups 2 and 3 as the values of the parameters b; c; f ; x are drifting, causing instability of orbits in the phase space.

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Section: Determination Of the Delay Timementioning

confidence: 99%

Nonlinear Time Series Analysis of Volcanic Tremor Events Recorded at Sangay Volcano, Ecuador

Konstantinou

Lin

2004

Pure and Applied Geophysics

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“…The positive local Lyapunov exponent that [25,26,27] find in the Chandler wobble data may reflect, not chaos (an aperiodic and strongly non-linear phenomenon), but rather the linear anti-damping of Eq. (20) (which makes the wobbleless equilibrium unstable).…”

Section: Discussionmentioning

confidence: 97%

Chandler Wobble: Stochastic and Deterministic Dynamics

Jenkins

2016

Dynamical Systems: Theoretical and Experimental Analysis

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We propose a model of the Earth's torqueless precession, the "Chandler wobble", as a self-oscillation driven by positive feedback between the wobble and the centrifugal deformation of the portion of the Earth's mass contained in circulating fluids. The wobble may thus run like a heat engine, extracting energy from heat-powered geophysical circulations whose natural periods would otherwise be unrelated to the wobble's observed period of about fourteen months. This can explain, more plausibly than previous models based on stochastic perturbations or forced resonance, how the wobble is maintained against viscous dissipation. The self-oscillation is a deterministic process, but stochastic variations in the magnitude and distribution of the circulations may turn off the positive feedback (a Hopf bifurcation), accounting for the occasional extinctions, followed by random phase jumps, seen in the data. This model may have implications for broader questions about the relation between stochastic and deterministic dynamics in complex systems, and the statistical analysis thereof.

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“…[7][8][9][10][11] The basic idea of that method is that the evolution of any single variable of a system is determined by the other variables that it interacts with. [7][8][9][10][11] The basic idea of that method is that the evolution of any single variable of a system is determined by the other variables that it interacts with.…”

Section: A Reconstruction Of the Phase Spacementioning

confidence: 99%

Chaotic operation by a single transistor circuit in the reverse active region

Hanias

Giannis

Tombras

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, we present an externally triggered experimental chaotic circuit with a bipolar junction transistor operating in its reverse active region in order to investigate for possible control features in its output phase portraits. Nonlinear time series modeling techniques are applied to analyze the circuit's output voltage oscillations and reveal the presence of chaos, while the chaos itself is achieved by controlling the amplitude of the applied input signal. The phase space, which describes the behavior evolution of a nonlinear system, is reconstructed using the delay embedding theorem suggested by Takens. The time delay used for this reconstruction is chosen after examining the first minimum of the collected data average mutual information, while the sufficient embedding dimension is estimated using the false nearest-neighbor algorithm which has a value of 5. Also the largest Lyapunov exponent is estimated and found equal to 0.020 48. Finally, the phase space embedding based weight predictor algorithm is employed to make a short-term prediction of the chaotic time series for which the system's governing equations may be unknown.

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Detectability of deterministic non-linear processes in Earth rotation time-series-I. Embedding

Cited by 15 publications

References 35 publications

Nonlinear Time Series Analysis of Volcanic Tremor Events Recorded at Sangay Volcano, Ecuador

Nonlinear Time Series Analysis of Volcanic Tremor Events Recorded at Sangay Volcano, Ecuador

Chandler Wobble: Stochastic and Deterministic Dynamics

Chaotic operation by a single transistor circuit in the reverse active region

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