Question

Consider a dynamical system $\overrightarrow{\mathbf{x}}\left(t+1\right)\mathbf{=}\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$with two components. The accompanying sketch shows the initial state vector ${\overrightarrow{\mathbf{x}}}_{\mathbf{0}}$and two eigenvectors $\overrightarrow{{\mathbf{\upsilon }}_{\mathbf{1}}}\mathbf{ }\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathbf{ }\overrightarrow{{\mathbf{\upsilon }}_{\mathbf{2}}}$of A (with eigen values $\overrightarrow{{\mathbf{\lambda }}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\overrightarrow{{\mathbf{\lambda }}_{\mathbf{2}}}$ respectively). For the given values of $\overrightarrow{{\mathbf{\lambda }}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\overrightarrow{{\mathbf{\lambda }}_{\mathbf{2}}}$, draw a rough trajectory. Consider the future and the past of the system.

$\overrightarrow{{\mathbf{\lambda }}_{\mathbf{1}}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{,}\overrightarrow{{\mathbf{\lambda }}_{\mathbf{2}}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{1}$

Short answer

So, the required solution is $A^t{x}_0=1.2^t\alpha {\upsilon }_1+1.1^t\beta {\upsilon }_2$.

Step-by-step solution

Define the eigenvector

Eigenvector:An eigenvector of$\mathit{A}$is a nonzero vector$\mathit{v}$in${\mathit{R}}_{\mathbf{n}}$such that$\mathit{A}\mathit{v}\mathbf{=}\mathit{\lambda }\mathit{v}\mathbf{}$, for some scalar$\mathit{\lambda }$.

Note the given data

It is given that:

$$\begin{gathered} \overrightarrow{{\lambda }_1}=1.2, \\ \overrightarrow{{\lambda }_2}=1.1 \end{gathered}$$

Given graph is:

Calculate the required matrix

We have:

$$\begin{gathered} A{\upsilon }_1=1.2{\upsilon }_1 \\ A{\upsilon }_2=1.1{\upsilon }_2 \end{gathered}$$

For $x_0=\alpha {\upsilon }_1+\beta {\upsilon }_2$,We have:

$$\begin{gathered} A{x}_0=A(\alpha {\upsilon }_1+\beta {\upsilon }_2) \\ =\alpha A{\upsilon }_1+\beta A{\upsilon }_2 \\ =1.2\alpha {\upsilon }_1+1.1\beta {\upsilon }_2 \end{gathered}$$

Therefore, localid="1668085548202" $A^t{x}_0=1.2^t\alpha {\upsilon }_1+1.1^t\beta {\upsilon }_2$.

Hence, the solution is $A^t{x}_0=1.2^t\alpha {\upsilon }_1+1.1^t\beta {\upsilon }_2$..