For the given function x ˙ =5-re -x 2 the value of r at whichbifurcation occurs is to be determined. The bifurcation is to be classified as saddle-node, trans-critical, supercritical pitchfork, or subcritical pitchfork. The bifurcation diagram of fixed points is to be sketched. Concept Introduction: Bifurcation theory is used to study the stability of dynamical systems. The phenomenon in which fixed points are created and destroyed by varying the control parameter is termed as saddle-node bifurcation. Fixed points are the points where, x ˙ = 0 . A pitchfork bifurcation occurs where the system transitions from one fixed point to three fixed points. A subcritical pitchfork bifurcation occurs when there is a single unstable fixed point present, which after the change of parameters becomes unstable, and two new symmetric unstable fixed points appear.

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Chapter 3.4, Problem 7E

Interpretation Introduction

Interpretation:

For the given function x˙=5-re-x2 the value of r at whichbifurcation occurs is to be determined. The bifurcation is to be classified as saddle-node, trans-critical, supercritical pitchfork, or subcritical pitchfork. The bifurcation diagram of fixed points is to be sketched.

Concept Introduction:

Bifurcation theory is used to study the stability of dynamical systems.

The phenomenon in which fixed points are created and destroyed by varying the control parameter is termed as saddle-node bifurcation.

Fixed points are the points where, x˙ = 0.

A pitchfork bifurcation occurs where the system transitions from one fixed point to three fixed points.

A subcritical pitchfork bifurcation occurs when there is a single unstable fixed point present, which after the change of parameters becomes unstable, and two new symmetric unstable fixed points appear.

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