Phase Plane Plots

The Phase Plane refers to the (Subscript[x, 1])(Subscript[x, 2])- plane that solutions exist in.

Much of this is discussed in Section 10.4, 10.5, 10.6 of the text under the sections Geometric Properties.

Asymptotic means in the long time limit, as t->∞.

The only solution which passes through the origin at finite time is the zero solution.

The solutions do not cross each other (each point in the plane has exactly one solution passing through it).

Example 1

field=VectorPlot[{3x1-2x2,2x1-2x2},{x1,-2,2},{x2,-2,2},VectorScale->{Tiny,Small,None},VectorPoints->20,BaseStyle->20,FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"}];

eigenvectordirection=Plot[{2x1,1/2x1},{x1,-2,2},PlotStyle->{{Thick,Red},{Thick,Blue}},Exclusions->{{0,0}}];

origin=Graphics[Disk[{0,0},0.08]];

Show[field,eigenvectordirection,origin]

Example 2

field=VectorPlot[{2x1-x2,3x1-2x2},{x1,-3,3},{x2,-3,3},VectorScale->{Tiny,Small,None},VectorPoints->20,BaseStyle->20,FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"}];

eigenvectordirection=Plot[{x1,3x1},{x1,-3,3},PlotStyle->{{Thick,Red},{Thick,Blue}},Exclusions->{{0,0}}];

origin=Graphics[Disk[{0,0},0.08]];

Show[field,eigenvectordirection,origin]

Example 3

To get the eigenvalue “direction”, you can evaluate the derivative at a point to determine the direction of rotation. The direction field verifies what we found in class.

field=VectorPlot[{1x1+2x2,-5x1-x2},{x1,-2,2},{x2,-2,2},VectorScale->{Tiny,Small,None},VectorPoints->20,BaseStyle->20,FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"}];

Show[field,origin]