Bifurcation

Saddle node

at bifurcation node, 1 created 1 destroyed

bifurcation point/node

point (parameter value) WHERE the nature changes

sudden qualitative change

transcritical

2 POSSIBLE nodes, 1 constant, 1 varies

Pitchfork

stable until node, split into 3

SUPERcrit pitchfork

from node, 2 STABLE symmetrical (diverge), 1 unstable straight

SUBcrit pitchfork

from node, 3 unstable, 2 backwards, 1 forward

dx/dt = 0

point where system stops changing/ at EQ

Hopf Bifurcation

fixed point affected by >1 altering variables, results in oscillation.

Subcritical Hopf bifurcation

Unstable oscillation born from USS, bound accordingly to USS limits

Supercritical Hopf Bifurcation

Stable oscillation, bound by SS limits (outer fork)

To analyse stability

differentiate

eigenvalue

scale value for linear ODE, can be used for non-linear applications

Supercritical Hopf oscillation

expanding to the limit (USS>SS)

Subcritical Hopf bifurcation

Shrinks to “zero” - stable (USS>SS)

Polar coordinates relative to multi ODEs

when r=0, ODE displays the usual nature of the trend
with ODE + radial coordinate = displays oscillation

Nature of oscillation

dictate by stability of ODE (r=0)

Augmentation

introduce “real-imperfection” to the system
“more dynamic”

Hysteresis/ Switching - Augmentation

addition of parameter resulting in irreversibility (trap system towards one of SS)