For the given function x ˙ = rx - x 1+x 2 the value of r at which bifurcation occur 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 x * vs . r 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. The transcritical bifurcation occurs when the fixed points exchange their stabilities as the parameter is changed. 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 are stable. A supercritical pitchfork bifurcation occurs when there is a single stable fixed point present, which after the change of parameters becomes unstable, and two new symmetric fixed points appear are stable.

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Chapter 3.4, Problem 8E

Interpretation Introduction

Interpretation:

For the given function x˙= rx - x1+x2 the value of r at which bifurcation occur 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 x*vs. r 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.

The transcritical bifurcation occurs when the fixed points exchange their stabilities as the parameter is changed.

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 are stable.

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

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