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Chapter 7: Q9E (page 380)

Use eigenvalues to determine the stability of a dynamical system. Analyse the dynamical system, x鈫�(t+1)=Ax鈫�(t)where A is a real2脳2matrix with eigenvaluesforthe matrices A in Exercises 1 through 10, determine whether the zero state is a stable equilibrium of the dynamicalsystem.x鈫�(t+1)=Ax鈫�(t)

A=[0.80-0.600.700.600.8]

Short Answer

Expert verified

The stability of the dynamical system using eigenvalues is unstable.

Step by step solution

01

solve the given matrix by eigenvalues:

Eigenvalues位ofA are roots of the characteristic equation. Associated eigenvectors of A are nonzero solutions of the equation.(A鈭�位I)x=0

Using characteristic equation,

det(A鈭�位I)=0|0.8鈭�位00.600.7鈭�位00.600.8鈭�位|=0(0.8鈭�位)2(0.7鈭�位)鈭�0.36(0.7鈭�位)=0(0.7鈭�位)(位2鈭�1.6位+0.28)=0位1=0.7,位2=1.6卤2.56鈭�1.122=0.8卤0.6i

Since,|位2|=|位3|=1then the equilibrium is unstable.

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Most popular questions from this chapter

Find all2脳2matrices for whiche鈫�1=(10)is an eigenvector with associated eigenvalue 5.

consider the dynamical system

x鈫�(t+1)=[1.100位]x鈫�(t).

Sketch a phase portrait of this system for the given values of位:

位=1.2

Consider the linear space Vof alln脳nmatrices for which all the vectorse鈬赌1,.....,e鈬赌nare eigenvectors. Describe the spaceV(the matrices inV"have a name"), and determine the dimension ofV.

Show that similar matrices have the same eigenvalues. Hint:v鈫� Ifis an eigenvector ofS-1AS, thenrole="math" localid="1659529994406" Sv鈫�is an eigenvector of A.

Is v鈬赌an eigenvector of 7 A? If so, what is the eigenvalue?
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