For the system, bifurcation occurs at the origin when μ = 0 . The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined. Concept Introduction: Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ . If the decay becomes slower and slower, and finally changes to growth at a critical value μ c , the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation. A subcritical Hopf bifurcation occurs at μ = 0 , where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0 , the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

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Answer 1

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Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

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Answer 2

Question

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

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Answer 3

Question

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

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Answer 4

Question

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

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Answer 5

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Answer 6

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Answer 7

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Answer 8

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Answer 11

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

Solution Summary: The author describes Hopf bifurcation as the origination of a limit cycle from equilibrium in dynamical systems generated by ordinary differential equations.

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