transcritical bifurcation
TRANSCRIPT
![Page 1: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/1.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= ??
![Page 2: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/2.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= ??
![Page 3: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/3.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= ??
![Page 4: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/4.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= ??
![Page 5: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/5.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= ??
![Page 6: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/6.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= ??
![Page 7: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/7.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= ??
![Page 8: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/8.jpg)
A transcritical bifurcation
I A transcritical bifurcation occurs when, as you vary aparameter value, a stable ss and an unstable ss collide, crossthrough each other and exchange stability.
x
f(x)
I Example:dx
dt= x(a− x)
![Page 9: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/9.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x)ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?
![Page 10: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/10.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
a
xss
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x)ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?
![Page 11: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/11.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
a
xss
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x) + ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?
![Page 12: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/12.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
a
xss
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x) + ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?
![Page 13: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/13.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
a
xss
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x)− ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?
![Page 14: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/14.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
a
xss
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x)− ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?
![Page 15: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/15.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
a
xss
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x)ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?
![Page 16: Transcritical Bifurcation](https://reader031.vdocuments.mx/reader031/viewer/2022021500/577cc9f31a28aba711a50867/html5/thumbnails/16.jpg)
Bifurcation diagram
I The bifurcation diagram for a transcritical bifurcation:
x
f(x)
a
xss
I If we perturb the equation a little bit, the diagram changesqualitatively :
dx
dt= x(a− x)ε
I This is called unfolding a degenerate bifurcation.
I Is a fold (saddle-node) bifurcation degenerate? Can it beunfolded?