Question: The system = x?x3 (6.31) has one stable ?xed point for ? 0 which becomes unstable and splits o? two stable ?xed points at =

The system The system = x?x3 (6.31) has one stable ?xed point for ? = x?x3 (6.31)

has one stable ?xed point for ? 0 which becomes unstable and splits o? two stable ?xed points at = 0. This is known as pitchfork bifurcation.

The system

0 which becomes unstable and splits o? two stable ?xed points at = x?x2 (6.32)

has two stable ?xed points for 6= 0 which collide and exchange stability at = 0. This is known as transcritical bifurcation.

The system = 0. This is known as pitchfork bifurcation. The system = x?x2 = + x2 (6.33)

has one stable and one unstable ?xed point for

give me exact code to Draw phase plots as a function of for the three systems from above and prove all statements made above.

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