For the system, bifurcation occurs at the origin when μ = 0 . The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined. Concept Introduction: Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ . If the decay becomes slower and slower, and finally changes to growth at a critical value μ c , the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation. A subcritical Hopf bifurcation occurs at μ = 0 , where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0 , the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Question

Want to see the full answer?

Answer 1

Question

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

D7. Determine the oscillation of x²-x²+x + x = 0.

What is the phase shift of the following graph? y = -4 sin( 7(x + 1)) O A. T ОВ. -4 ОС. 2 O D. -1

Enter the equation of the function cos(x) that has a vertical shift of 4 units up, and a phase shift of π6 units right.

Answer 2

Question

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Answer 6

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

Answer 7

Chapter 8.2, Problem 6E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when μ = 0. The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower, and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

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 describes Hopf bifurcation as the origination of a limit cycle from equilibrium in dynamical systems generated by ordinary differential equations.

Videos

Want to see the full answer?

Chapter 8 Solutions