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Dynamical Systems Dynamical Systems
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Dynamical Systems Dynamical Systems
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First Order Systems First Order Systems
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1-D flow1-D flow
Source(unstable)Sink(stable)
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BifurcationsBifurcations• Can be understood as a change in the
dynamics of the system as a control parameter is varied past a critical value
Figure 10 Bifurcations: a mechanical example
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Bifurcations of 1-D systemsBifurcations of 1-D systems
• Sadle-Node Bifurcation • Transcritical Bifurcation • Pitchfork Bifurcation
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The Saddle-Node BifurcationThe Saddle-Node Bifurcation
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The Saddle-Node BifurcationThe Saddle-Node Bifurcation
Figure Bifurcation Diagram of a Saddle-Node Bifurcation
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The Transcritical BifurcationThe Transcritical Bifurcation
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The Transcritical BifurcationThe Transcritical Bifurcation
Figure Bifurcation Diagram of a Trascritical Bifurcation
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The Pitchfork Bifurcation The Pitchfork Bifurcation
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The Pitchfork BifurcationThe Pitchfork Bifurcation
Figure 5: Supercritical PitchforkBifurcation Diagram
Figure 5: Subcritical PitchforkBifurcation Diagram
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Dynamics of a kite in a wind tunnelDynamics of a kite in a wind tunnel
• Investigated the dynamics of the kite by calculating the autospectral density function
• This function shows how much of the signal is at a frequency f
• Different waves have characteristic power spectra so the power spectrum can, for instance help the identification of a chaotic dynamical system
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Typical Power SpectraTypical Power Spectra
Figure 4 Power spectrum of a chaotic motion
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Dynamics of a kite in a wind tunnelDynamics of a kite in a wind tunnel
Wind speed (ms-1)
Number of Frequency Components Relationship between Peak Frequencies
x coordinate y coordinate x coordinate y coordinate
Tail
2.6 Varies (1 to 3) Varies (2,3,4) N/A N/A
3 1 Varies (2,3) N/A N/A
3.6 1 with fluctuations Varies (1,2) N/A N/A
3.9 1 2 N/A f2=2f1
No Tail
2.7 Varies (1 to 3) Varies (2,3,4) N/A N/A
2.9 Varies (2,3) Varies (2,3) N/A N/A
3.6 1 2 N/A f2=2f1
4 1 2 N/A f2=2f1
Table 1 Spectral Analysis Results
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Dynamics of a kite in a wind tunnelDynamics of a kite in a wind tunnel
Figure 3 Power Spectrum of no-tail kite at 2.7 ms-1
(x coordinate, 4th marker) Figure 3 Power Spectrum of no-tail kite at 2.7 ms-1 (y coordinate, 4th marker)
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Dynamics of a kite in a wind tunnelDynamics of a kite in a wind tunnel
Figure 3 Power Spectrum of tailed kite at 3.9 ms-1 (x coordinate, 4th marker)
Figure 3 Power Spectrum of tailed kite at 3.9 ms -1 (x coordinate, 4th marker)
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Dynamics of a kite in a wind tunnelDynamics of a kite in a wind tunnel
• At high wind speeds there the power spectra of the kite seem to follow the same pattern in both situations (with or without tail)
• As the wind speed is decreased, the spectra become more irregular and there is variation in the number of frequency components across the markers
• Dissimilarity in the behaviour across the four marker points may indicate that the body is not perfectly rigid