Question

For the values of ${\mathit{\lambda }}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{1}$and ${\mathit{\lambda }}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{2}$, sketch the trajectories for all nine initial values shown in the following figures. For each of the points, trace out both future and past of the system.

Short answer

The trajectories for all nine initial values is

Step-by-step solution

Draw the future and past of the system.

Consider the Eigen values ${\lambda }_1=-1$and ${\lambda }_2=-2$.

As 0 > ${\lambda }_1>{\lambda }_2$, the future and past of the system toward outside.

Draw the graph of $E_{-1}$and $E_{-2}$as follows:

Draw the trajectory of the system.

If the Eigen values of the system are ${\mathit{\lambda }}_{\mathbf{1}}$and ${\mathit{\lambda }}_{\mathbf{2}}$such that ${\mathit{\lambda }}_{\mathbf{1}}\mathbf{>}\mathbf{0}\mathbf{>}{\mathit{\lambda }}_{\mathbf{2}}$then the phase portraits of the system is

By the trajectory definition, draw the graph of the trajectory of the system as follows:

Hence the trajectories for all nine initial values and traces of the future and past of the system are sketched.