Question

Question: Examine your phase portraits in Problem 51. Under what conditions will the phase portrait of a 2x2 homogeneous linear system with complex eigenvalues consist of a family of closed curves? Consist of a family of spirals? Under what conditions is the origin (0,0) a repeller? An attractor?

Short answer

The phase portraits for $\lambda =\alpha +\beta i$closed curves if $\alpha =0$, spirals if .$\alpha \ne 0$ A spiral is an attractor if $\alpha <0$, and the spiral is a repellor if $\alpha >0$.

Step-by-step solution

Determine the Phase Portrait

• A phase portrait is a geometric depiction of a dynamical system's paths in the phase plane.
• A separate curve, or point, represents each set of beginning circumstances.
• In the study of dynamical systems, phase pictures are a useful tool.

Find the Phase Portrait of the System

• For $\lambda =\alpha +\beta i$, if $\alpha =0$, then the phase portrait of the system will consist of closed curves.
• If $\alpha \ne 0$then the phase portrait will consist of spirals.
• The spiral is an attractor if $\alpha <0$and a repellor if $\alpha >0$.
• Therefore, the phase portraits for $\lambda =\alpha +\beta i$closed curves if $\alpha =0$, spirals if $\alpha \ne 0$.
• A spiral is an attractor if $\alpha <0$and a repellor if $\alpha >0$.