The algebraic expressions for all the fixed points of a system x ˙ = rx + x 3 - x 5 exhibiting subcritical pitchfork bifurcation as r varies are to be determined. The vector fields as r varies are to be sketched including all fixed points and their stability and r s in saddle-node bifurcation should be calculated. Concept Introduction: The fixed points of the system occur at x ˙ = 0 . Bifurcation is used to study the stability of the dynamical systems. In pitchfork bifurcation, the fixed points appear and disappear in symmetrical pairs. There are two types of pitchfork bifurcation, one is supercritical and another is subcritical. In supercriticalbifurcation, a stable fixed point is present and after changing parameters it becomes unstable and two new symmetric unstable points generate. In subcriticalbifurcation, an unstable fixed point is present and after changing parameters it becomes stable and two new symmetric stable points generate. Saddle-node bifurcation is one of the bifurcation mechanism in which fixed points create, collide and destroy.

Question

Want to see the full answer?

Answer 1

Question

Chapter 3.4, Problem 14E

Interpretation Introduction

Interpretation:

The algebraic expressions for all the fixed points of a system x˙ = rx + x3- x5 exhibiting subcritical pitchfork bifurcation as r varies are to be determined. The vector fields as r varies are to be sketched including all fixed points and their stability and rs in saddle-node bifurcation should be calculated.

Concept Introduction:

The fixed points of the system occur at x˙ = 0.

Bifurcation is used to study the stability of the dynamical systems.

In pitchfork bifurcation, the fixed points appear and disappear in symmetrical pairs.

There are two types of pitchfork bifurcation, one is supercritical and another is subcritical.

In supercriticalbifurcation, a stable fixed point is present and after changing parameters it becomes unstable and two new symmetric unstable points generate.

In subcriticalbifurcation, an unstable fixed point is present and after changing parameters it becomes stable and two new symmetric stable points generate.

Saddle-node bifurcation is one of the bifurcation mechanism in which fixed points create, collide and destroy.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

5. The volume V of a given mass of monoatomic gas changes with temperat re T according to the relation V = KT2/3. The work done when temperature changes by 90 K will be xR. The value of x is (a) 60 (b)20 (c)30 S (d)90

Consider a matrix 3 -2 1 A = 0 5 4 -6 2 -1 Define matrix B as transpose of the inverse of matrix A. Find the determinant of matrix A + B.

5) State any theorems that you use in determining your solution. a) Suppose you are given a model with two explanatory variables such that: Yi = a +ẞ1x1 + ẞ2x2i + Ui, i = 1, 2, ... n Using partial differentiation derive expressions for the intercept and slope coefficients for the model above. [25 marks] b) A production function is specified as: Yi = α + B₁x1i + ẞ2x2i + Ui, i = 1, 2, ... n, u₁~N(0,σ²) where: y = log(output), x₁ = log(labor input), x2 = log(capital input) The results are as follows: x₁ = 10, x2 = 5, ỹ = 12, S11 = 12, S12= 8, S22 = 12, S₁y = 10, = 8, Syy = 10, S2y n = 23 (individual firms) i) Compute values for the intercept, the slope coefficients and σ². [20 marks] ii) Show that SE (B₁) = 0.102. [15 marks] iii) Test the hypotheses: ẞ1 = 1 and B2 = 0, separately at the 5% significance level. You may take without calculation that SE (a) = 0.78 and SE (B2) = 0.102 [20 marks] iv) Find a 95% confidence interval for the estimate ẞ2. [20 marks]