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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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Simulation of vocal fold vibration with a simplified two-mass model (Steinecke/Herzel) during a pressure sweep (0–2.5–0 kPa; Q = 0.51). (a) Narrow-band spectrogram of the simulated airflow; (b) and (c) displacement of left (b) and right (c) lower mass of the model as a function of time, extracted at t = 2, t = 2.76, t = 3.1 and t = 5 s. (d) Phase portraits from the above signals, created by plotting the time-varying position of the left lower vocal fold mass against that of the right lower mass. A Poincaré section was created at an angle of 0.375 π radians, yielding intersection points with the trajectory (red dots). (e) Phasegram: the vertical markers at t = 0.67; 2.7; 2.91; 3.51; 6.72; 7.27; 7.6 and 9.7 s represent bifurcations, i.e. changes from one distinct oscillatory regime to another (see text). (f) Trace of the time-varying simulated lung pressure used to drive the model. The vertical markers indicate bifurcation events (see above and text). (g) Bifurcation diagram, showing the distinct vibratory patterns in relation to lung pressure. Note the hysteresis caused by the direction of lung pressure change (increasing versus decreasing—see text for details).
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RSIF20130288F5: Simulation of vocal fold vibration with a simplified two-mass model (Steinecke/Herzel) during a pressure sweep (0–2.5–0 kPa; Q = 0.51). (a) Narrow-band spectrogram of the simulated airflow; (b) and (c) displacement of left (b) and right (c) lower mass of the model as a function of time, extracted at t = 2, t = 2.76, t = 3.1 and t = 5 s. (d) Phase portraits from the above signals, created by plotting the time-varying position of the left lower vocal fold mass against that of the right lower mass. A Poincaré section was created at an angle of 0.375 π radians, yielding intersection points with the trajectory (red dots). (e) Phasegram: the vertical markers at t = 0.67; 2.7; 2.91; 3.51; 6.72; 7.27; 7.6 and 9.7 s represent bifurcations, i.e. changes from one distinct oscillatory regime to another (see text). (f) Trace of the time-varying simulated lung pressure used to drive the model. The vertical markers indicate bifurcation events (see above and text). (g) Bifurcation diagram, showing the distinct vibratory patterns in relation to lung pressure. Note the hysteresis caused by the direction of lung pressure change (increasing versus decreasing—see text for details).

Mentions: Following Steinecke & Herzel [28], the time-varying displacements of the model's two lower masses were plotted against each other (figure 5). Four distinct vibratory regimes were seen in the right mass and in the glottal flow (not shown here) for the chosen model parameters: periodic, period doubling and other subharmonic regimes (see also electronic supplementary material, movie S3). Owing to the user-defined asymmetry in the model geometry and mechanical properties, the left mass had a more complex vibratory pattern than the right mass. The right and the left masses are synchronized by ratios of 1 : 1, 5 : 3, 5 : 7 and 4 : 2, respectively, for the four examples shown in figure 5b–d.Figure 5.


Visualization of system dynamics using phasegrams.

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

Simulation of vocal fold vibration with a simplified two-mass model (Steinecke/Herzel) during a pressure sweep (0–2.5–0 kPa; Q = 0.51). (a) Narrow-band spectrogram of the simulated airflow; (b) and (c) displacement of left (b) and right (c) lower mass of the model as a function of time, extracted at t = 2, t = 2.76, t = 3.1 and t = 5 s. (d) Phase portraits from the above signals, created by plotting the time-varying position of the left lower vocal fold mass against that of the right lower mass. A Poincaré section was created at an angle of 0.375 π radians, yielding intersection points with the trajectory (red dots). (e) Phasegram: the vertical markers at t = 0.67; 2.7; 2.91; 3.51; 6.72; 7.27; 7.6 and 9.7 s represent bifurcations, i.e. changes from one distinct oscillatory regime to another (see text). (f) Trace of the time-varying simulated lung pressure used to drive the model. The vertical markers indicate bifurcation events (see above and text). (g) Bifurcation diagram, showing the distinct vibratory patterns in relation to lung pressure. Note the hysteresis caused by the direction of lung pressure change (increasing versus decreasing—see text for details).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4043161&req=5

RSIF20130288F5: Simulation of vocal fold vibration with a simplified two-mass model (Steinecke/Herzel) during a pressure sweep (0–2.5–0 kPa; Q = 0.51). (a) Narrow-band spectrogram of the simulated airflow; (b) and (c) displacement of left (b) and right (c) lower mass of the model as a function of time, extracted at t = 2, t = 2.76, t = 3.1 and t = 5 s. (d) Phase portraits from the above signals, created by plotting the time-varying position of the left lower vocal fold mass against that of the right lower mass. A Poincaré section was created at an angle of 0.375 π radians, yielding intersection points with the trajectory (red dots). (e) Phasegram: the vertical markers at t = 0.67; 2.7; 2.91; 3.51; 6.72; 7.27; 7.6 and 9.7 s represent bifurcations, i.e. changes from one distinct oscillatory regime to another (see text). (f) Trace of the time-varying simulated lung pressure used to drive the model. The vertical markers indicate bifurcation events (see above and text). (g) Bifurcation diagram, showing the distinct vibratory patterns in relation to lung pressure. Note the hysteresis caused by the direction of lung pressure change (increasing versus decreasing—see text for details).
Mentions: Following Steinecke & Herzel [28], the time-varying displacements of the model's two lower masses were plotted against each other (figure 5). Four distinct vibratory regimes were seen in the right mass and in the glottal flow (not shown here) for the chosen model parameters: periodic, period doubling and other subharmonic regimes (see also electronic supplementary material, movie S3). Owing to the user-defined asymmetry in the model geometry and mechanical properties, the left mass had a more complex vibratory pattern than the right mass. The right and the left masses are synchronized by ratios of 1 : 1, 5 : 3, 5 : 7 and 4 : 2, respectively, for the four examples shown in figure 5b–d.Figure 5.

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