Student Solutions Manual for Nonlinear Dynamics and Chaos by Mitchal Dichter

Author:Mitchal Dichter
Language: eng
Format: epub
Publisher: CRC Press

8.1.7

First find the fixed points, which are the intersections of the nullclines.

Next find the linearization at the fixed points.

(x, y) = (0,0) ⇒ Δ = ab – 1 τ = −(a + b)

Looking more closely at this system, we see that the fixed point goes to infinity when either a or b is zero. Also notice that there is only one fixed point when ab = 1, and there are two fixed points when ab ≠ 1, a ≠ 0, and b ≠ 0. There are several cases for the stability of the two fixed points.

ab > 1 and a + b > 0 ⇒ The origin is a stable node and the other fixed point is a saddle point.

0 < ab < 1 and a + b > 0 ⇒ The origin is a saddle point and the other fixed point is a stable node.

ab > 1 and a + b < 0 ⇒ The origin is an unstable node and the other fixed point is a saddle point.

0 < ab < 1 and a + b < 0 ⇒ The origin is a saddle point and the other fixed point is an unstable node.

ab < 0 ⇒ The origin and the other fixed point are saddle points. This last case completes all possible cases.

From this we can conclude that transcritical bifurcations occur along the boundary in parameter space.

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