The phase portrait of θ ˙ = μ sinθ - sin2θ a function of the control parameter μ is to be drawn, and bifurcation, which occurs as μ varies is to be classified. Also, all the bifurcation values of μ are to be obtained. Concept Introduction: The qualitative change in the dynamics of the flow with parameters is called bifurcation, and the points at which this occurs is called the bifurcation point. To study the stability of the dynamical systems, bifurcation is used. By changing the parameter, the fixed points move towards each other, collide, and mutually annihilate is known as Saddle Node Bifurcation. When the single stable fixed point is present, and it turns to an unstable point due to change in parameter and two new symmetric stable fixed points occurs is called Supercritical Pitchfork Bifurcation. When the single unstable fixed point is present, and it turns to a stable point due to change in parameter and two new symmetric unstable fixed points occur is called Subcritical Pitchfork Bifurcation.

Question

Want to see the full answer?

Answer 1

Question

Chapter 4.3, Problem 3E

Interpretation Introduction

Interpretation:

The phase portrait of θ˙ = μ sinθ - sin2θ a function of the control parameter μ is to be drawn, and bifurcation, which occurs as μ varies is to be classified. Also, all the bifurcation values of μ are to be obtained.

Concept Introduction:

The qualitative change in the dynamics of the flow with parameters is called bifurcation, and the points at which this occurs is called the bifurcation point.

To study the stability of the dynamical systems, bifurcation is used.

By changing the parameter, the fixed points move towards each other, collide, and mutually annihilate is known as Saddle Node Bifurcation.

When the single stable fixed point is present, and it turns to an unstable point due to change in parameter and two new symmetric stable fixed points occurs is called Supercritical Pitchfork Bifurcation.

When the single unstable fixed point is present, and it turns to a stable point due to change in parameter and two new symmetric unstable fixed points occur is called Subcritical Pitchfork Bifurcation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

Theorem: show that XCH) = M(E) M" (6) E + t Mcfic S a Solution of ODE -9CA)- x = ACE) x + g (t) + X (E) - E

5. (a) State the Residue Theorem. Your answer should include all the conditions required for the theorem to hold. (4 marks) (b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the anti-clockwise direction. Evaluate に dz. You must check all of the conditions of any results that you use. (5 marks) (c) Evaluate L You must check all of the conditions of any results that you use. ཙ x sin(Tx) x²+2x+5 da. (11 marks)

3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula for L(y). (1 mark) (b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a contour. Suppose there exists a finite real number M such that |f(z)| < M for all z in the image of y. Prove that < ||, f(z)dz| ≤ ML(y). (3 marks) (c) State and prove Liouville's theorem. You may use Cauchy's integral formula without proof. (d) Let R0. Let w € C. Let (10 marks) U = { z Є C : | z − w| < R} . Let f UC be a holomorphic function such that 0 < |ƒ(w)| < |f(z)| for all z Є U. Show, using the local maximum modulus principle, that f is constant. (6 marks)