Dynamical Systems Dynamical Systems

Dynamical Systems Dynamical Systems

First Order Systems First Order Systems

1-D flow1-D flow

Source(unstable)Sink(stable)

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

Bifurcations of 1-D systemsBifurcations of 1-D systems

• Sadle-Node Bifurcation • Transcritical Bifurcation • Pitchfork Bifurcation

The Saddle-Node BifurcationThe Saddle-Node Bifurcation

The Saddle-Node BifurcationThe Saddle-Node Bifurcation

Figure Bifurcation Diagram of a Saddle-Node Bifurcation

The Transcritical BifurcationThe Transcritical Bifurcation

The Transcritical BifurcationThe Transcritical Bifurcation

Figure Bifurcation Diagram of a Trascritical Bifurcation

The Pitchfork Bifurcation The Pitchfork Bifurcation

The Pitchfork BifurcationThe Pitchfork Bifurcation

Figure 5: Supercritical PitchforkBifurcation Diagram

Figure 5: Subcritical PitchforkBifurcation Diagram

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

Typical Power SpectraTypical Power Spectra

Figure 4 Power spectrum of a chaotic motion

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

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)

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)

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