To analyze the nonlinear system graphically and to sketch the vector field graphically on the real line. All fixed points are to be determined; their stability should be classified, and graph of x(t) is to be sketched for different initial conditions. Also, if possible, the analytic solution for x(t) is to be obtained. Concept Introduction: The given equation is x ˙ =1-2cosx . Initially, we want to find the fixed point of the given equation. Fixed points of the given equation are values of x at which x ˙ =0 . The point at which the velocity is zero can be obtained by graphing the function x ˙ vs x (position). Stable points are points at which the local flow is toward them. They represent stable equilibria at which small disturbances damp out in time away from it. Unstable points are points at which the local flow is away from them. They represent unstable equilibria.

Question

Want to see the full answer?

Answer 1

Question

Chapter 2.2, Problem 6E

Interpretation Introduction

Interpretation:

To analyze the nonlinear system graphically and to sketch the vector field graphically on the real line. All fixed points are to be determined; their stability should be classified, and graph of x(t) is to be sketched for different initial conditions. Also, if possible, the analytic solution for x(t) is to be obtained.

Concept Introduction:

The given equation is x˙=1-2cosx. Initially, we want to find the fixed point of the given equation. Fixed points of the given equation are values of x at which x˙=0.

The point at which the velocity is zero can be obtained by graphing the function x˙ vs x(position).

Stable points are points at which the local flow is toward them. They represent stable equilibria at which small disturbances damp out in time away from it.

Unstable points are points at which the local flow is away from them. They represent unstable equilibria.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

Q4 3 Points 1 Let A = 2 3 7 5 11 Give one nontrivial solution X of the homogeneous system Ax = 0. (Your vector x should have explicit numbers as its entries, as opposed to variables/parameters). Show your work for how you found it. Please select file(s) Select file(s) Save Answer

4. Assume that a risk-free money market account is added to the market described in Q3. The continuously compounded rate of return on the money market account is log (1.1). (i) For each given μ, use Lagrange multipliers to determine the proportions (as a function of μ) of wealth invested in the three assets available for the minimum variance portfolio with expected return μ. (ii) Determine the market portfolio in this market and calculate its Sharp ratio.

3. A market consists of two risky assets with rates of return R₁ and R2 and no risk-free asset. From market data the following have been estimated: ER₁ = 0.25, ER2 = 0.05, Var R₁ = 0.01, Var R2 = 0.04 and the correlation between R1 and R2 is p = -0.75. (i) Given that an investor is targeting a total expected return of μ = 0.2. What portfolio weights should they choose to meet this goal with minimum portfolio variance? Correct all your calculations up to 4 decimal points. (ii) Determine the global minimum-variance portfolio and the expected return and variance of return of this portfolio (4 d.p.). (iii) Sketch the minimum-variance frontier in the μ-σ² plane and indicate the efficient frontier. (iv) Without further calculation, explain how the minimum variance of the investor's portfolio return will change if the two risky assets were independent.