Bifurcations

Normal forms

remember by heart for exam ^

• $\dot{x} = r + x^2$
• control parameter = $r$
• bifurcation $r = 0$

1. stacked vector fields
2. bifurcation
3. conventional bifurcation diagram - flip x and r - splitting into two branches

Example II

• split function into 2 parts
• $y = r - x$
• $y = e^{-x}$
• b) = bifurcation point = where the functions are tangential

Transcritical bifurcation

• $\dot{x} = rx - x^2$
• $rx$ makes difference to saddle-node bifurcation

Pitchfork bifurcation

• symmetry

1. Supercritical (related to 2nd order phase transitions)
2. Subcritical (related to 1st order phase transitions)

Supercritical normal form

• $\dot{x} = rx - x^3$

• left-right symmetry $x \to -x$

• $r = 0$ -> critical slowing down (linear team vanishes)

• control paramter = other stuff for example $\beta = \frac{x^}{\tanh x^}$

Subcritical pitchfork bifurcation

• $\dot{x} = rx + x^3$
• $+$ - destabilising term
• if reaches state behind unstable there is no turning back - bridge collapses
• $r > 0$ - blow-up
• usually comes with stabilising term ($\dot{x} = rx + x^3 - x^5$)
• hysteresis bring $r$ below value $r_s$ - lower value than we have been before (characteristics for first order system?)

Imperfect bifurcations

• $h$ - imperfectino parameter
• $\dot{x} = h + rx - x^3$
• for $h \neq 0$ the left-right symmetry is broken
• saddle-node bifurcation is when

Catastrophe with bifurcations

• cusp catastrophe