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= ??

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= ??

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= ??

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= ??

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= ??

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= ??

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= ??

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)

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?

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?

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?

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?

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?

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?

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?

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?