Fixed Point of a 2x2 Linear System of ODEs

Manipulate[With[{expr=If[k==1,{(E^(3 t)) ((1+t) 2Cos[θ]+t 2Sin[θ]),(E^(3 t)) (2Sin[θ]-t (2Cos[θ]+2Sin[θ]))},{(1/(2 (Sqrt[1-k])))(E^(-(-3+(Sqrt[1-k])) t)) ((-1+(E^(2 (Sqrt[1-k]) t)) (1+(Sqrt[1-k]))+(Sqrt[1-k])) 2Cos[θ]+(-1+(E^(2 (Sqrt[1-k]) t))) k 2 Sin[θ]),(1/(2 (Sqrt[1-k])))(E^(-(-3+(Sqrt[1-k])) t)) (-(-1+(E^(2 (Sqrt[1-k]) t)))2Cos[θ]+(1+(E^(2 (Sqrt[1-k]) t)) (-1+(Sqrt[1-k]))+(Sqrt[1-k])) 2 Sin[θ])}]},ParametricPlot[Evaluate[Table[expr,{θ,0,2π-π/8,π/8}]],{t,-2,2},PlotRange->3,PlotStyle->ColorData["HTML","SlateBlue"],PlotLabel ->Row[{MatrixForm[{{(OverDot[x])}, {(OverDot[y])}}], "=", MatrixForm[{{4, k}, {-1, 2}}] , MatrixForm[{{x}, {y}}]}],Epilog->{ColorData["HTML","SlateBlue"],Table[Evaluate[Arrow[{expr/.t->-0.0001,expr/.t->0.0001}]],{θ,0,2π-π/8,π/8}],RGBColor[1,.47,0],Thick,Map[Line[{-5#,5#}]&,Select[Eigenvectors[{{4,k},{-1,2}}],(Im[#[[1]]]==0&&Im[#[[2]]]==0)&]]},ImageSize->{450,450}]],{{k,0},-20,20}]

This Demonstration shows trajectories in the linear autonomous system

(({{(OverDot[x])}, {(OverDot[y])}}))=(({{4, k}, {-1, 2}}))(({{x}, {y}}))

for various values of the parameter k. Different values of k are associated with the different types possible for the fixed point at the origin. For k>1, the fixed point is an unstable spiral; for k=1, it is an unstable improper node; for -8<k<1, it is an unstable node; for k<-8, it is a saddle.

When the eigenvalues of the matrix are real, orange lines are shown that are parallel to the eigenvectors.

For all values of k, the second-order linear autonomous system

(({{(OverDot[x])}, {(OverDot[y])}}))=(({{4, k}, {-1, 2}}))(({{x}, {y}}))

has a fixed (critical) point at the origin; for all values apart from -8, this fixed point is unique.

The nature of the fixed point depends on the eigenvalues of the stability matrix

(({{4, k}, {-1, 2}})),

which are 3±(Sqrt[1-k]).

For k>1, these eigenvalues are non-real and conjugate, with a positive real part; in these circumstances, all orbits spiral outward, and the fixed point is an unstable spiral. An example is shown in the first snapshot.

In the degenerate case k=1, the eigenvalues are real, positive, and equal, and there is only one eigenvector, to which all trajectories are tangential. The fixed point is an unstable improper node. This is shown in the second snapshot.

For -8<k<1, the eigenvalues are real, positive, and distinct; in these circumstances, all trajectories are tangential to the eigenvector associated with the smaller eigenvalue (except those directly along the other eigenvector), and the fixed point is an unstable node. This is shown in the third snapshot.

In the case k=-8, one eigenvector is positive and the other is zero, and all points on the eigenvector line 2y=x are fixed points. The fixed point at the origin is no longer isolated, and all trajectories are parallel to the other eigenvector. This is shown in the fourth (penultimate) snapshot.

For k<-8, the eigenvalues are real and of opposite sign. All trajectories are asymptotic to the eigenvector associated with the positive eigenvalue, except for those along the other eigenvector (which are the only ones that are bounded as t->∞). The fixed point is a saddle. This is shown in the final snapshot.

Contributed by: Phil Ramsden