WOLFRAM NOTEBOOK

Flip Bifurcation in Dynamical Systems

0.3234

A flip bifurcation occurs when increasing the parameter

causes the graph of the function

f(x)=4αx(1-x)

(x)

to intersect the line

y=x

. See Example 2.32 of[1]. In a flip bifurcation, an eigenvalue leaves the unit circle through the point

-1

. When this happens, the period two points become stable; thus, this is also known as a period-doubling bifurcation. Varying

, the zero solution becomes unstable for

α>0.25

; the period one blue branch becomes unstable for

α>0.75

; the period-doubling bifurcation occurs at

α=0.75

. At the period-doubling bifurcation, the fixed points of

(x)

become stable.

[1] A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, New York: Wiley, 1995.

Edmon Perkins, Ali Nayfeh, Balakumar Balachandran

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.

I understand and wish to continue anyway »