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Visualization of system dynamics using phasegrams.

Herbst CT, Herzel H, Svec JG, Wyman MT, Fitch WT - J R Soc Interface (2013)

Bottom Line: A phasegram can be interpreted as a bifurcation diagram in time.In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known.Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

View Article: PubMed Central - PubMed

Affiliation: Department of Cognitive Biology, Laboratory of Bioacoustics, University of Vienna, Althanstrasse 14, 1090 Vienna, Austria. christian.herbst@univie.ac.at

ABSTRACT
A new tool for visualization and analysis of system dynamics is introduced: the phasegram. Its application is illustrated with both classical nonlinear systems (logistic map and Lorenz system) and with biological voice signals. Phasegrams combine the advantages of sliding-window analysis (such as the spectrogram) with well-established visualization techniques from the domain of nonlinear dynamics. In a phasegram, time is mapped onto the x-axis, and various vibratory regimes, such as periodic oscillation, subharmonics or chaos, are identified within the generated graph by the number and stability of horizontal lines. A phasegram can be interpreted as a bifurcation diagram in time. In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known. Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

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Effect of the Poincaré section angle on phasegram generation, illustrated with a Lorenz system. (a) Parameter B of the Lorenz system, varied from 250 to 28 (see text). (b) Spectrogram of generated time-domain signal (output variable x of the Lorenz system). (c) Time-domain signal. (d) A 200 ms portion of the time-domain signal, centred around t = 9 s. (e) Phase portraits from the above signals, created by attractor reconstruction. A Poincaré section was created along the x-axis (left panel) and at an angle of 0.35 π radians, as determined by the algorithm described in the electronic supplementary material (right panel), yielding intersection points with the trajectory (red dots). (f) Histograms of the trajectory intersection points on Poincaré sections in the two conditions. (g) ‘Trajectory strips’: colour-coded histograms with Poincaré sections through phase portraits. (h) Phasegrams from the signal displayed in panel (c) for both Poincaré section angles, respectively. The markers at t = 9 s represent the position of the trajectory strips from (g) within the graph. See text for the note on the period doubling sequence at t = 3.4–3.8 s in the right panel.
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RSIF20130288F3: Effect of the Poincaré section angle on phasegram generation, illustrated with a Lorenz system. (a) Parameter B of the Lorenz system, varied from 250 to 28 (see text). (b) Spectrogram of generated time-domain signal (output variable x of the Lorenz system). (c) Time-domain signal. (d) A 200 ms portion of the time-domain signal, centred around t = 9 s. (e) Phase portraits from the above signals, created by attractor reconstruction. A Poincaré section was created along the x-axis (left panel) and at an angle of 0.35 π radians, as determined by the algorithm described in the electronic supplementary material (right panel), yielding intersection points with the trajectory (red dots). (f) Histograms of the trajectory intersection points on Poincaré sections in the two conditions. (g) ‘Trajectory strips’: colour-coded histograms with Poincaré sections through phase portraits. (h) Phasegrams from the signal displayed in panel (c) for both Poincaré section angles, respectively. The markers at t = 9 s represent the position of the trajectory strips from (g) within the graph. See text for the note on the period doubling sequence at t = 3.4–3.8 s in the right panel.

Mentions: The effect of the Poincaré section angle in phasegram generation is exemplified in figure 3 by a signal generated with a Lorenz system [16, equations 25–27]—for details see below. Poincaré sections were created at 0° (left panel) and 45° (right panel). The effect of the angle variation is seen in the histograms, the trajectory strips and the resulting phasegrams (figure 3f–h). Note that the phasegram created with the algorithmic angle selection (right panel in figure 3h) reveals a subharmonic sequence around t = 3.4–3.8 s, which is not apparent in the phasegram created with an arbitrary angle of 0 radians (left panel in figure 3h)—see also electronic supplementary material, movie S2.Figure 3.


Visualization of system dynamics using phasegrams.

Herbst CT, Herzel H, Svec JG, Wyman MT, Fitch WT - J R Soc Interface (2013)

Effect of the Poincaré section angle on phasegram generation, illustrated with a Lorenz system. (a) Parameter B of the Lorenz system, varied from 250 to 28 (see text). (b) Spectrogram of generated time-domain signal (output variable x of the Lorenz system). (c) Time-domain signal. (d) A 200 ms portion of the time-domain signal, centred around t = 9 s. (e) Phase portraits from the above signals, created by attractor reconstruction. A Poincaré section was created along the x-axis (left panel) and at an angle of 0.35 π radians, as determined by the algorithm described in the electronic supplementary material (right panel), yielding intersection points with the trajectory (red dots). (f) Histograms of the trajectory intersection points on Poincaré sections in the two conditions. (g) ‘Trajectory strips’: colour-coded histograms with Poincaré sections through phase portraits. (h) Phasegrams from the signal displayed in panel (c) for both Poincaré section angles, respectively. The markers at t = 9 s represent the position of the trajectory strips from (g) within the graph. See text for the note on the period doubling sequence at t = 3.4–3.8 s in the right panel.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
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RSIF20130288F3: Effect of the Poincaré section angle on phasegram generation, illustrated with a Lorenz system. (a) Parameter B of the Lorenz system, varied from 250 to 28 (see text). (b) Spectrogram of generated time-domain signal (output variable x of the Lorenz system). (c) Time-domain signal. (d) A 200 ms portion of the time-domain signal, centred around t = 9 s. (e) Phase portraits from the above signals, created by attractor reconstruction. A Poincaré section was created along the x-axis (left panel) and at an angle of 0.35 π radians, as determined by the algorithm described in the electronic supplementary material (right panel), yielding intersection points with the trajectory (red dots). (f) Histograms of the trajectory intersection points on Poincaré sections in the two conditions. (g) ‘Trajectory strips’: colour-coded histograms with Poincaré sections through phase portraits. (h) Phasegrams from the signal displayed in panel (c) for both Poincaré section angles, respectively. The markers at t = 9 s represent the position of the trajectory strips from (g) within the graph. See text for the note on the period doubling sequence at t = 3.4–3.8 s in the right panel.
Mentions: The effect of the Poincaré section angle in phasegram generation is exemplified in figure 3 by a signal generated with a Lorenz system [16, equations 25–27]—for details see below. Poincaré sections were created at 0° (left panel) and 45° (right panel). The effect of the angle variation is seen in the histograms, the trajectory strips and the resulting phasegrams (figure 3f–h). Note that the phasegram created with the algorithmic angle selection (right panel in figure 3h) reveals a subharmonic sequence around t = 3.4–3.8 s, which is not apparent in the phasegram created with an arbitrary angle of 0 radians (left panel in figure 3h)—see also electronic supplementary material, movie S2.Figure 3.

Bottom Line: A phasegram can be interpreted as a bifurcation diagram in time.In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known.Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

View Article: PubMed Central - PubMed

Affiliation: Department of Cognitive Biology, Laboratory of Bioacoustics, University of Vienna, Althanstrasse 14, 1090 Vienna, Austria. christian.herbst@univie.ac.at

ABSTRACT
A new tool for visualization and analysis of system dynamics is introduced: the phasegram. Its application is illustrated with both classical nonlinear systems (logistic map and Lorenz system) and with biological voice signals. Phasegrams combine the advantages of sliding-window analysis (such as the spectrogram) with well-established visualization techniques from the domain of nonlinear dynamics. In a phasegram, time is mapped onto the x-axis, and various vibratory regimes, such as periodic oscillation, subharmonics or chaos, are identified within the generated graph by the number and stability of horizontal lines. A phasegram can be interpreted as a bifurcation diagram in time. In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known. Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

Show MeSH
Related in: MedlinePlus