Phase Portrait and Field Directions of Two-Dimensional Linear Systems of ODEs

Manipulate[
m={{a1,b1},{a2,b2}};Show[VectorPlot[m.{x,y},{x,-10,10},{y,-10,10},StreamPoints->{{pt1,pt2,pt3,pt4}},StreamStyle->{Red,Thick},ImageSize->{460,310}],Graphics[{Thick,Orange,Map[Line[{-100#,100#}]&,Select[Eigenvectors[m],(Im[#[[1]]]==0&&Im[#[[2]]]==0)&]]}]],
Style["\!\(\*OverscriptBox[\(\(x\)\(\\ \)\), \(.\)]\)= (Subscript[a, 1])x + (Subscript[b, 1])y",Bold],
{{a1,1,"(Subscript[a, 1])"},-10,10,Appearance->"Labeled"},
{{b1,1,"(Subscript[b, 1])"},-10,10,Appearance->"Labeled"},
Style["(OverDot[y]) = (Subscript[a, 2])x + (Subscript[b, 2])y",Bold],
{{a2,2,"(Subscript[a, 2])"},-10,10,Appearance->"Labeled"},{{b2,-2,"(Subscript[b, 2])"},-10,10,Appearance->"Labeled"},{{pt1,{-5.1,4.8}},{-10,-10},{10,10},Locator},
{{pt2,{4.9,5.1}},{-10,-10},{10,10},Locator},
{{pt3,{-4.9,-5}},{-10,-10},{10,10},Locator},
{{pt4,{4.9,-5.2}},{-10,-10},{10,10},Locator},TrackedSymbols->True,ControlPlacement->Left]

This Demonstration plots the phase portrait (or phase plane) and the vector field of directions around the fixed point (0,0) of the two-dimensional linear system of first-order ordinary differential equations

(OverDot[x])=(Subscript[a, 1])x+(Subscript[b, 1])y

(OverDot[y])=(Subscript[a, 2])x+(Subscript[b, 2])y.

Drag the four locators to see the trajectories of four solutions of the system that go through them. The position of these points can be chosen by clicking on wherever you like inside the graphics. Thick orange lines parallel to the eigenvectors are shown if the eigenvalues of the system are real.

For more information on phase portraits and types of fixed points for linear systems of ODEs, see, for example:

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Cambridge: Westview Press, 2000.

G. F. Simmons, Differential Equations with Applications and Historical Notes, New York: McGraw-Hill, 1991.

Contributed by: Santos Bravo Yuste