For the given system ϕ " +α ϕ ' + sin ϕ = I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead. Concept Introduction: The trajectories of the dynamic system are called phase portrait. Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Question

Want to see the full answer?

Answer 1

Question

Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ϕ"+αϕ'+sinϕ= I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

Find the general solution of the Cauchy-Euler equation: X^3y’’’ - x^2y’’ + 2xy’ - 2y = x^3

Find Particular Sol. of xsin2 ycosy y' + sin3y/ 1+lnx = 4 ; y(e)= π/2

Find all the second partial derivatives z=cos2(5x+2y)

Answer 2

Question

Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ϕ"+αϕ'+sinϕ= I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ϕ"+αϕ'+sinϕ= I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ϕ"+αϕ'+sinϕ= I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ϕ"+αϕ'+sinϕ= I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Answer 6

Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ϕ"+αϕ'+sinϕ= I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Answer 7

Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ϕ"+αϕ'+sinϕ= I to show that if α is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as I decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

Answer 8

Expert Solution & Answer

Answer 9

Expert Solution & Answer

Answer 10

Want to see the full answer?

Check out a sample textbook solution

See solution

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 phase portrait of the system equation is plotted as follows: if is fixed, the stable limit cycle is destroyed in a homoclinic bifurcation.

Videos

Want to see the full answer?

Chapter 8 Solutions