Find the fixed points and classify them, sketch the neighboring trajectories Concept Introduction: The parametric curves traced by solutions of a differential equation are known as trajectories. The geometrical representation of collection of trajectories in a phase plane is called as phase portrait. The point which satisfies the condition f(x * )=0 is known as fixed point, it correspond to steady state of the system (or equilibria of the system). Closed Orbit corresponds to periodic solution of the system i.e. x(t+T)= x(t) for all t. If nearby trajectories moving away from the fixed point then the point is said to be saddle point. If the trajectories swirling around the fixed point then it is a unstable fixed point. If nearby trajectories moving away from the fixed point then the point is said to be unstable fixed point. If nearby trajectories moving towards the fixed point then the point is said to be stable fixed point. To check the stability of fixed point use Jacobian matrix ( ∂ x ˙ ∂ x ∂ x ˙ ∂ y ∂ y ˙ ∂ x ∂ y ˙ ∂ y ) The point ( x * , y * ) is said to be a stable fixed point if eigenvalues of Jacobian matrix evaluated at this point having negative real parts and point is said to be unstable if one of its eigenvalue has positive real part. If the both eigenvalues are purely real then the fixed point is saddle point.

Question

Want to see the full answer?

Answer 1

Question

Chapter 6.3, Problem 6E

Interpretation Introduction

Interpretation:

Find the fixed points and classify them, sketch the neighboring trajectories

Concept Introduction:

The parametric curves traced by solutions of a differential equation are known as trajectories.

The geometrical representation of collection of trajectories in a phase plane is called as phase portrait.

The point which satisfies the condition f(x)=0 is known as fixed point, it correspond to steady state of the system (or equilibria of the system).

Closed Orbit corresponds to periodic solution of the system i.e. x(t+T)= x(t) for all t.

If nearby trajectories moving away from the fixed point then the point is said to be saddle point.

If the trajectories swirling around the fixed point then it is a unstable fixed point.

If nearby trajectories moving away from the fixed point then the point is said to be unstable fixed point.

If nearby trajectories moving towards the fixed point then the point is said to be stable fixed point.

To check the stability of fixed point use Jacobian matrix

(∂x˙∂x∂x˙∂y∂y˙∂x∂y˙∂y)

The point (x,y*) is said to be a stable fixed point if eigenvalues of Jacobian matrix evaluated at this point having negative real parts and point is said to be unstable if one of its eigenvalue has positive real part. If the both eigenvalues are purely real then the fixed point is saddle point.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

Use the method of undetermined coefficients to solve the given nonhomogeneous system. dx dt = 2x + 3y − 8 dy dt = −x − 2y + 6 X(t) =

As discussed in Section 8.3, the Markowitz model uses the variance of the portfolio as the measure of risk. However, variance includes deviations both below and above the mean return. Semivariance includes only deviations below the mean and is considered by many to be a better measure of risk. (a) Develop a model that minimizes semivariance for the Hauck Financial data given in the file HauckData with a required return of 10%. Assume that the five planning scenarios in the Hauck Financial Services model are equally likely to occur. Hint: Modify model (8.10)–(8.19). Define a variable ds for each scenario and let ds ≥ R − Rs with ds ≥ 0. Then make the objective function: Min 1 5 5 s = 1 ds2. Let FS = proportion of portfolio invested in the foreign stock mutual fund IB = proportion of portfolio invested in the intermediate-term bond fund LG = proportion of portfolio invested in the large-cap growth fund LV = proportion of portfolio invested in the large-cap value fund…

For each month of the year, Taylor collected the average high temperatures in Jackson, Mississippi. He used the data to create the histogram shown. Which set of data did he use to create the histogram? A 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92\ 55,\ 60,\ 64,\ 72,\ 73,\ 75,\ 77,\ 81,\ 83,\ 91,\ 91,\ 92 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92 B 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91\ 55,\ 57,\ 60,\ 65,\ 70,\ 71,\ 78,\ 79,\ 85,\ 86,\ 88,\ 91 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91 C 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93\ 55,\ 60,\ 63,\ 64,\ 65,\ 71,\ 83,\ 87,\ 88,\ 88,\ 89,\ 93 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93 D 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91\ 55,\ 58,\ 60,\ 66,\ 68,\ 75,\ 77,\ 82,\ 86,\ 89,\ 91,\ 91 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91