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Chapter 7: Q7.6-39E (page 382)

Consider the dynamical system

x1(t+1)=0.1x1(t)+0.2x2(t)+1x2(t+1)=0.4x1(t)+0.3x2(t)+2.

See Exercise 7.4.35. Find the equilibrium state of this system and determine its stability. See Exercise 38. Sketch a phase portrait.

Short Answer

Expert verified

The equilibrium state of this systemv=24

Step by step solution

01

Define stability:

Stability is one of the most basic requirements for numerical models, which extends to mostly linear problems. If every solution that was initially close to it is always close to it. It is said to be stable and asymptotically stable although initially every solution close to it meets t鈫�鈭�.

02

Find the equilibrium state:

As given in question we could get the matrix as,

A=0.10.20.40.3;b=12

From pervious exercise we got,

v=(I-A)-1b

By substituting we get,

=0.9-0.2-0.40.7-112

The equilibrium state of this systemv=24

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Most popular questions from this chapter

In all parts of this problem, let V be the linear space of all 2 脳 2 matrices for which [12]is an eigenvector.

(a) Find a basis of V and thus determine the dimension of V.

(b) Consider the linear transformation T (A) = A[12] from V to R2. Find a basis of the image of Tand a basis of the kernel of T. Determine the rank of T .

(c) Consider the linear transformation L(A) = A[13] from V to R2. Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.

find an eigenbasis for the given matrice and diagonalize:

A=19[82-2255-244]

Representing the orthogonal projection onto the planex-2y+2z=0

Suppose v鈨�Supposeis an eigenvector of the n脳nmatrix A, with eigenvalue 4 . Explain why v鈨�is an eigenvector of A2+2A+3In.What is the associated eigenvalue?

consider an eigenvalue 位0of ann脳nmatrix A. we are told that the algebraic multiplicity of exceeds 1.Show thatf'(位0)=0(i.e.., the derivative of the characteristic polynomial of A vanishes are位0).

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[2-2001-100003-4002-3]
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