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Chapter 9: Q37E (page 427)

Sketch rough phase portraits for the dynamical systems given in Exercise 32 through 39.

x→(t+1)=[10.3-0.21.7]x→(t)

Short Answer

Expert verified

Thus, the rough sketch of the system in the explanation.

Step by step solution

01

Given in the question.

The given systems are:

x→t+1=10.3-0.21.7x→t

02

Solve the given system.

Solve the given system as follows:

t1+1t2+1=10.3-0.21.7t1t2t1+1t2+1=t1+0.3t2-0.2t1+1.7t2

Now, equating both sides gives:

{t1+1=t1+0.3t2t2+1=-0.2t1+1.7t2{1=0.3t21=-0.2t1+1.7t2

Solve the above two equations gives:

t2=3.33t1=6.66

Thus the solutions is the span of12

03

Graph of the given system.

The graph is plotted below:

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Most popular questions from this chapter

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation dxdt=fxas dxfx=dtand integrate both sides.

9.dxdt=xk(with k≠1),x(0)=1

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Solve the differential equationf"(t)+2f'(t)+f(t)=0and find all the real solutions of the differential equation.

Consider a systemdx→dt=Ax→where A is a2×2matrix withtrA<0. We are told that A has no real eigenvalue. What can you say about the stability of the system

feedback Loops:Suppose some quantitieskix1(t),x2(t),...,xn(t)can be modelled by differential equations of the form localid="1662090443855">|\begingathereddx1dt=-k1x                 -bxn\hfilldx2dt=x1-k2                      \hfill.\hfill.\hfilldxndt =                           xn-1-knxn\hfill\endgathered|

Where b is positive and the localid="1662090454144">ki are positive.(The matrix of this system has negative numbers on the diagonal, localid="1662090460105" 1's directly below the diagonal and a negative number in the top right corner)We say that the quantities localid="1662090470062">x1,x2,...,xndescribe a (linear) negative feedback loop

  1. Describe the significance of the entries in this system inpractical terms.
  2. Is a negative feedback loop with two componentslocalid="1662090478095">(n=2) necessarily stable?
  3. Is a negative feedback loop with three components necessarily stable?
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