To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ . Concept Introduction: A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x ˙ is used and x&*#x00A0;is a fixed point Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit. Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

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Answer 1

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Chapter 8.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ.

Concept Introduction:

A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

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Answer 2

Question

Chapter 8.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ.

Concept Introduction:

A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

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Answer 3

Question

Chapter 8.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ.

Concept Introduction:

A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

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Answer 4

Question

Chapter 8.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ.

Concept Introduction:

A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

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Answer 5

Chapter 8.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ.

Concept Introduction:

A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

Answer 6

Chapter 8.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ.

Concept Introduction:

A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

Answer 7

Chapter 8.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the value of μ by using numerical integration at which homoclinic bifurcation occurs, and to sketch the phase portrait of the system just above and below that μ.

Concept Introduction:

A fixed point of a differential equation is a point where f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Holonomic bifurcation is the infinite period bifurcation in which limit cycle moves closer and closer to the saddle point, and at the bifurcation, cycle touches the saddle point and becomes a holonomic orbit.

Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about the evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

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 explains that the value of isobtained by numerical integration at which homoclinic bifurcation occurs. Phase portraits represent the trajectories of the system with respect to the

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