To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ = 0 . Concept Introduction: If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs. For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ . If by varying a parameter μ , the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μ c , then the system is said to have gone through supercritical Hopf bifurcation.

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Answer 1

Question

Chapter 8.2, Problem 3E

Interpretation Introduction

Interpretation:

To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ=0.

Concept Introduction:

If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs.

For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ. If by varying a parameter μ, the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μc, then the system is said to have gone through supercritical Hopf bifurcation.

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Answer 2

Question

Chapter 8.2, Problem 3E

Interpretation Introduction

Interpretation:

To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ=0.

Concept Introduction:

If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs.

For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ. If by varying a parameter μ, the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μc, then the system is said to have gone through supercritical Hopf bifurcation.

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Answer 3

Question

Chapter 8.2, Problem 3E

Interpretation Introduction

Interpretation:

To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ=0.

Concept Introduction:

If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs.

For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ. If by varying a parameter μ, the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μc, then the system is said to have gone through supercritical Hopf bifurcation.

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Answer 4

Question

Chapter 8.2, Problem 3E

Interpretation Introduction

Interpretation:

To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ=0.

Concept Introduction:

If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs.

For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ. If by varying a parameter μ, the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μc, then the system is said to have gone through supercritical Hopf bifurcation.

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Answer 5

Chapter 8.2, Problem 3E

Interpretation Introduction

Interpretation:

To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ=0.

Concept Introduction:

If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs.

For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ. If by varying a parameter μ, the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μc, then the system is said to have gone through supercritical Hopf bifurcation.

Answer 6

Chapter 8.2, Problem 3E

Interpretation Introduction

Interpretation:

To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ=0.

Concept Introduction:

If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs.

For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ. If by varying a parameter μ, the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μc, then the system is said to have gone through supercritical Hopf bifurcation.

Answer 7

Chapter 8.2, Problem 3E

Interpretation Introduction

Interpretation:

To plot the phase portrait for the system using computer and show that the Hopf bifurcation occurs at a μ=0.

Concept Introduction:

If topological structure of the phase portrait changes by varying a parameter, then it is said that the bifurcation occurs.

For a physical system which is coming to equilibrium through exponential decay, the decay rate depends on the value of control parameter μ. If by varying a parameter μ, the system is coming to equilibrium slowly and becomes unstable at a critical value of control parameter μc, then the system is said to have gone through supercritical Hopf bifurcation.

Answer 8

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Answer 9

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Answer 10

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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 plots the phase portrait for the system using computer and shows that the supercritical Hopf bifurcation occurs at a mu =0

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