By using definitions of different types of stabilities, the stabilities of fixed points are to be proved. Concept Introduction: A fixed point x * = 0 is said to be globally attracting, if x * attracts all trajectories in the phase plane. Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x * and remains close to it for all positive time. When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

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Answer 1

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Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point x=0 is said to be globally attracting, if x attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x* and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

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Answer 2

Question

Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point x=0 is said to be globally attracting, if x attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x* and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

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Answer 3

Question

Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point x=0 is said to be globally attracting, if x attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x* and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

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Answer 4

Question

Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point x=0 is said to be globally attracting, if x attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x* and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

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Answer 5

Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point x=0 is said to be globally attracting, if x attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x* and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

Answer 6

Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point x=0 is said to be globally attracting, if x attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x* and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

Answer 7

Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point x=0 is said to be globally attracting, if x attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to x* and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.

Answer 8

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Answer 9

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Answer 11

Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E

Solution Summary: The author explains the stability of fixed points by using definitions of different types of stabilities.

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