For the system x ˙ = rx - sin x , when r = 0 , all the fixed points are to be found and classified, and the vector field is to be sketched. When r > 1 , it is to be shown that there is only one fixed point. As r decreases from ∞ to 0 , all the bifurcations that occur are to be classified. For 0 < r << 1 , an approximate formula for values of r at which bifurcations occur is to be found. All the bifurcations that occur as r decreases from 0 to - ∞ are to be classified. The bifurcation diagram for - ∞ < r < ∞ is to be plotted and the stability of the various branches of fixed points is to be indicated. Concept Introduction: Fixed points are the points where x ˙ = 0 . The graph of the vector field for different values of r can be plotted. The saddle-node bifurcation is the basic mechanism by which fixed points are created and destroyed. As a parameter is varied, two fixed points move toward each other, collide, andmutually annihilate. 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 11E

Interpretation Introduction

Interpretation:

For the system x˙ = rx - sin x, when r = 0, all the fixed points are to be found and classified, and the vector field is to be sketched. When r > 1, it is to be shown that there is only one fixed point. As r decreases from ∞ to 0, all the bifurcations that occur are to be classified. For 0 < r << 1, an approximate formula for values of r at which bifurcations occur is to be found.

All the bifurcations that occur as r decreases from 0 to - ∞ are to be classified. The bifurcation diagram for - ∞ < r < ∞ is to be plotted and the stability of the various branches of fixed points is to be indicated.

Concept Introduction:

Fixed points are the points where x˙ = 0.

The graph of the vector field for different values of r can be plotted.

The saddle-node bifurcation is the basic mechanism by which fixed points are created and destroyed. As a parameter is varied, two fixed points move toward each other, collide, andmutually annihilate.

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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