To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map. Concept Introduction: The 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 . Countable and uncountable sets, where the notion created by a cantor will be followed. Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

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Answer 1

Question

Chapter 12.1, Problem 2E

Interpretation Introduction

Interpretation:

To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map.

Concept Introduction:

The 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.

Countable and uncountable sets, where the notion created by a cantor will be followed.

Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

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Answer 2

Question

Chapter 12.1, Problem 2E

Interpretation Introduction

Interpretation:

To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map.

Concept Introduction:

The 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.

Countable and uncountable sets, where the notion created by a cantor will be followed.

Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

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Answer 3

Question

Chapter 12.1, Problem 2E

Interpretation Introduction

Interpretation:

To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map.

Concept Introduction:

The 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.

Countable and uncountable sets, where the notion created by a cantor will be followed.

Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

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Answer 4

Question

Chapter 12.1, Problem 2E

Interpretation Introduction

Interpretation:

To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map.

Concept Introduction:

The 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.

Countable and uncountable sets, where the notion created by a cantor will be followed.

Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

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Answer 5

Chapter 12.1, Problem 2E

Interpretation Introduction

Interpretation:

To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map.

Concept Introduction:

The 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.

Countable and uncountable sets, where the notion created by a cantor will be followed.

Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

Answer 6

Chapter 12.1, Problem 2E

Interpretation Introduction

Interpretation:

To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map.

Concept Introduction:

The 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.

Countable and uncountable sets, where the notion created by a cantor will be followed.

Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

Answer 7

Chapter 12.1, Problem 2E

Interpretation Introduction

Interpretation:

To find the condition on the parameter which ensures that the origin is globally asymptotically stable, where the considered map is the linear map.

Concept Introduction:

The 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.

Countable and uncountable sets, where the notion created by a cantor will be followed.

Different infinite sets can be compared as followed, considering the basis set N as the set of natural numbers; if set A has one - to – one correspondence, i.e., every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.

Answer 8

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Answer 10

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Answer 11

Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Prob. 6E Ch. 12.1 - Prob. 7E Ch. 12.1 - Prob. 8E Ch. 12.1 - Prob. 9E Ch. 12.2 - Prob. 1E

Solution Summary: The author explains how to find the condition parameters which ensure that the origin is globally asymptotically stable, where the considered map is the linear map.

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