example 1. differential equations (sode) stability...

Stability of critical points in Linear Systems of Ordinary

Differential Equations (SODE)

In this chapter Mathematica will be used to study the stability of the critical points of twin equationlinear SODE.

If the SODE is linear and homogeneous X' = AX with constant coefficients and the matrix A is nonsingular, then the only critical point is the origin and its stability can be studied by examining the eigenval-ues of A.

Example 1.

Compare the following SODE orbits and justify the different behaviour, analyzing the stability at thecritical point.a) x' = 3x-y, y' = 2x + y.b) x' = x +5y, y' = -x - y. c) x' = x -5y, y' = -x - y.

Solution

REMARK: The three SODE are autonomous, linear and with constant coefficients, with a single criticalpoint at the origin.Some trajectories are represented by phase planes.

� Section a.

The SODE is defined and it is solved with the DSolve command

sis31a = 8x'@tD == 3 x@tD - y@tD, y'@tD == 2 x@tD + y@tD<

8x¢@tD � 3 x@tD - y@tD, y¢@tD � 2 x@tD + y@tD<

solgen31a = DSolve@sis31a, 8x@tD, y@tD<, tD

99x@tD ® -ã2 t C@2D Sin@tD + ã

2 t C@1D HCos@tD + Sin@tDL,y@tD ® ã

2 t C@2D HCos@tD - Sin@tDL + 2 ã2 t C@1D Sin@tD==

The solutions x(t) and y(t) are defined:

x@t_D = solgen31a@@1, 1, 2DD

-ã2 t C@2D Sin@tD + ã

2 t C@1D HCos@tD + Sin@tDL

y@t_D = solgen31a@@1, 2, 2DD

ã2 t C@2D HCos@tD - Sin@tDL + 2 ã

2 t C@1D Sin@tD

Some solutions are generated with the instruction Table

solpar31a = Table@8x@tD, y@tD< �. 8C@1D -> i, C@2D -> j<, 8i, -6, 6, 4<, 8j, -6, 6, 4<D

9996 ã2 t Sin@tD - 6 ã

2 t HCos@tD + Sin@tDL, -6 ã2 t HCos@tD - Sin@tDL - 12 ã

2 t Sin@tD=,92 ã

2 t Sin@tD - 6 ã2 t HCos@tD + Sin@tDL, -2 ã

2 t HCos@tD - Sin@tDL - 12 ã2 t Sin@tD=,

9-2 ã2 t Sin@tD - 6 ã

2 t HCos@tD + Sin@tDL, 2 ã2 t HCos@tD - Sin@tDL - 12 ã

2 t Sin@tD=,9-6 ã

2 t Sin@tD - 6 ã2 t HCos@tD + Sin@tDL, 6 ã

2 t HCos@tD - Sin@tDL - 12 ã2 t Sin@tD==,

996 ã2 t Sin@tD - 2 ã


2 t Sin@tD=,92 ã



9-2 ã2 t Sin@tD - 2 ã


2 t Sin@tD=,9-6 ã


2 t HCos@tD - Sin@tDL - 4 ã2 t Sin@tD==,

996 ã2 t Sin@tD + 2 ã

2 t HCos@tD + Sin@tDL, -6 ã2 t HCos@tD - Sin@tDL + 4 ã

2 t Sin@tD=,92 ã

2 t Sin@tD + 2 ã2 t HCos@tD + Sin@tDL, -2 ã

2 t HCos@tD - Sin@tDL + 4 ã2 t Sin@tD=,

9-2 ã2 t Sin@tD + 2 ã

2 t HCos@tD + Sin@tDL, 2 ã2 t HCos@tD - Sin@tDL + 4 ã

2 t Sin@tD=,9-6 ã

2 t Sin@tD + 2 ã2 t HCos@tD + Sin@tDL, 6 ã

2 t HCos@tD - Sin@tDL + 4 ã2 t Sin@tD==,

996 ã2 t Sin@tD + 6 ã


2 t Sin@tD=,92 ã



9-2 ã2 t Sin@tD + 6 ã


2 t Sin@tD=,9-6 ã


2 t HCos@tD - Sin@tDL + 12 ã2 t Sin@tD===

The previous table is modified with the command Flatten to make a functions list that is graphicallyrepresented.

2 StabilitySG.nb


solpar31abis = Flatten@solpar31a, 1D

996 ã2 t Sin@tD - 6 ã


2 t Sin@tD=,92 ã



9-2 ã2 t Sin@tD - 6 ã


2 t Sin@tD=,9-6 ã



96 ã2 t Sin@tD - 2 ã


2 t Sin@tD=,92 ã



9-2 ã2 t Sin@tD - 2 ã


2 t Sin@tD=,9-6 ã



96 ã2 t Sin@tD + 2 ã


2 t Sin@tD=,92 ã



9-2 ã2 t Sin@tD + 2 ã


2 t Sin@tD=,9-6 ã



96 ã2 t Sin@tD + 6 ã


2 t Sin@tD=,92 ã



9-2 ã2 t Sin@tD + 6 ã


2 t Sin@tD=,9-6 ã


2 t HCos@tD - Sin@tDL + 12 ã2 t Sin@tD==

graf131a = ParametricPlot@Evaluate@solpar31abisD,8t, -2, 0.5`<, PlotStyle ® 88RGBColor@1, 0, 0D, [email protected]`D<<D

-4 -2 2 4

-5

5

The solutions and the directions field are represented together:

StabilitySG.nb 3

The solutions and the directions field are represented together:

graf231a = HNeeds@"VectorFieldPlots`"D;VectorFieldPlots`VectorFieldPlot@83 x - y, 2 x + y<, 8x, -15, 15<, 8y, -20, 20<,PlotPoints ® 50, DisplayFunction ® Identity, BaseStyle ® [email protected]`D<DL;

Show@graf131a, graf231aD

-4 -2 2 4

-5

5

The trajectories move further from the origin (they spin like a spiral) as t increases and they are closeto it when t is less than 0. This behaviour can be explained by looking at the eigenvalues of the coefficientmatrix.

The matrix associated with the SODE is defined as:

a =3 -1

2 1

883, -1<, 82, 1<<

4 StabilitySG.nb

Its eigenvalues are given by:

Eigenvalues@aD

82 + ä, 2 - ä<

The origin is an unstable spiral focus because the eigenvalues are complex conjugated and they havea positive real part.

� Section b.

Any possible assignments are cleared and the SODE is defined:

Clear@x, yD

sis31b = 8x'@tD == x@tD + 5 y@tD, y'@tD � -x@tD - y@tD<

8x¢@tD � x@tD + 5 y@tD, y¢@tD � -x@tD - y@tD<

It is solved using DSolve command:

solgen31b = DSolve@sis31b, 8x@tD, y@tD<, tD

::x@tD ®5

2C@2D Sin@2 tD +

1

2C@1D H2 Cos@2 tD + Sin@2 tDL,

y@tD ®1

2C@2D H2 Cos@2 tD - Sin@2 tDL -

1

2C@1D Sin@2 tD>>


StabilitySG.nb 5

x@t_D = solgen31b@@1, 1, 2DD

5

2C@2D Sin@2 tD +

1

2C@1D H2 Cos@2 tD + Sin@2 tDL

y@t_D = solgen31b@@1, 2, 2DD

1

2C@2D H2 Cos@2 tD - Sin@2 tDL -

1

2C@1D Sin@2 tD

Some solutions are generated with the instruction Table:

solpar31b = Table@8x@tD, y@tD< �. 8C@1D -> i, C@2D -> j<, 8i, 2, 8, 3<, 8j, 2, 8, 3<D

::82 Cos@2 tD + 6 Sin@2 tD, 2 Cos@2 tD - 2 Sin@2 tD<,

:2 Cos@2 tD +27

2Sin@2 tD,

5

2H2 Cos@2 tD - Sin@2 tDL - Sin@2 tD>,

82 Cos@2 tD + 21 Sin@2 tD, 4 H2 Cos@2 tD - Sin@2 tDL - Sin@2 tD<>,

::5 Sin@2 tD +5

2H2 Cos@2 tD + Sin@2 tDL, 2 Cos@2 tD -

7

2Sin@2 tD>,

:25

2Sin@2 tD +

5

2H2 Cos@2 tD + Sin@2 tDL,

5

2H2 Cos@2 tD - Sin@2 tDL -

5

2Sin@2 tD>,

:20 Sin@2 tD +5

2H2 Cos@2 tD + Sin@2 tDL, 4 H2 Cos@2 tD - Sin@2 tDL -

5

2Sin@2 tD>>,

:85 Sin@2 tD + 4 H2 Cos@2 tD + Sin@2 tDL, 2 Cos@2 tD - 5 Sin@2 tD<,

:25

2Sin@2 tD + 4 H2 Cos@2 tD + Sin@2 tDL,

5

2H2 Cos@2 tD - Sin@2 tDL - 4 Sin@2 tD>,

820 Sin@2 tD + 4 H2 Cos@2 tD + Sin@2 tDL, 4 H2 Cos@2 tD - Sin@2 tDL - 4 Sin@2 tD<>>


6 StabilitySG.nb

solpar31bbis = Flatten@solpar31b, 1D

:82 Cos@2 tD + 6 Sin@2 tD, 2 Cos@2 tD - 2 Sin@2 tD<,

:2 Cos@2 tD +27

2Sin@2 tD,

5

2H2 Cos@2 tD - Sin@2 tDL - Sin@2 tD>,

82 Cos@2 tD + 21 Sin@2 tD, 4 H2 Cos@2 tD - Sin@2 tDL - Sin@2 tD<,

:5 Sin@2 tD +5

2H2 Cos@2 tD + Sin@2 tDL, 2 Cos@2 tD -

7

2Sin@2 tD>,

:25

2Sin@2 tD +

5

2H2 Cos@2 tD + Sin@2 tDL,

5

2H2 Cos@2 tD - Sin@2 tDL -

5

2Sin@2 tD>,

:20 Sin@2 tD +5

2H2 Cos@2 tD + Sin@2 tDL, 4 H2 Cos@2 tD - Sin@2 tDL -

5

2Sin@2 tD>,

85 Sin@2 tD + 4 H2 Cos@2 tD + Sin@2 tDL, 2 Cos@2 tD - 5 Sin@2 tD<,

:25

2Sin@2 tD + 4 H2 Cos@2 tD + Sin@2 tDL,

5

2H2 Cos@2 tD - Sin@2 tDL - 4 Sin@2 tD>,

820 Sin@2 tD + 4 H2 Cos@2 tD + Sin@2 tDL, 4 H2 Cos@2 tD - Sin@2 tDL - 4 Sin@2 tD<>

graf131b = ParametricPlot@Evaluate@solpar31bbisD,8t, 0, 2 Π<, PlotStyle ® 88RGBColor@1, 0, 0D, [email protected]`D<<D

-20 -10 10 20

-10

-5

5

10

The solutions and the direction fields are represented together:

graf231b = HNeeds@"VectorFieldPlots`"D;VectorFieldPlots`VectorFieldPlot@8x + 5 y, -x - y<, 8x, -25, 25<, 8y, -12, 12<,PlotPoints ® 20, DisplayFunction ® Identity, BaseStyle ® [email protected]`D<DL;

StabilitySG.nb 7

Show@graf131b, graf231bD

-20 -10 10 20

-10

-5

5

10

The trajectories are closed curves, in this case ellipses, that are centered in the origin. This behav-iour can be explained by looking at the eigenvalues of the coefficient matrix.

The matrix associated with the SODE is defined as:

a =1 5

-1 -1

881, 5<, 8-1, -1<<

Its eigenvalues are given by the command:

Eigenvalues@aD

82 ä, -2 ä<

The origin is a stable centre because the eigenvalues are pure imaginary.

� Section c.

Any possible assignments are cleared and the SODE is defined:

Clear@x, yD

8 StabilitySG.nb

sis31c = 8x'@tD == x@tD - 5 y@tD, y'@tD � -x@tD - y@tD<

8x¢@tD � x@tD - 5 y@tD, y¢@tD � -x@tD - y@tD<

It is solved using the command:

solgen31c = DSolve@sis31c, 8x@tD, y@tD<, tD

::x@tD ®1

12ã

- 6 t J6 - 6 + 6 ã2 6 t

+ 6 ã2 6 tN C@1D -

5 ã- 6 t J-1 + ã2 6 tN C@2D

2 6,

y@tD ® -

ã- 6 t J-1 + ã2 6 tN C@1D

2 6-

1

12ã

- 6 t J-6 - 6 - 6 ã2 6 t

+ 6 ã2 6 tN C@2D>>


x@t_D = solgen31c@@1, 1, 2DD

1

12ã

- 6 t J6 - 6 + 6 ã2 6 t

+ 6 ã2 6 tN C@1D -

5 ã- 6 t J-1 + ã2 6 tN C@2D

2 6

y@t_D = solgen31c@@1, 2, 2DD

-

ã- 6 t J-1 + ã2 6 tN C@1D

2 6-

1

12ã

- 6 t J-6 - 6 - 6 ã2 6 t

+ 6 ã2 6 tN C@2D

Some solutions are generated with the instruction Table:

StabilitySG.nb 9

solpar31c =

Simplify@Table@8x@tD, y@tD< �. 8C@1D -> i, C@2D -> j<, 8i, 1, 5, 2<, 8j, 1, 5, 2<DD

:::1

6ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN,

1

6ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN>,

:1

6ã

- 6 t J3 + 7 6 + J3 - 7 6 N ã2 6 tN,

1

6ã

- 6 t J9 + 2 6 + J9 - 2 6 N ã2 6 tN>,

:1

2ã

- 6 t J1 + 4 6 + J1 - 4 6 N ã2 6 tN,

1

2ã

- 6 t J5 + 6 - J-5 + 6 N ã2 6 tN>>,

::1

6ã

- 6 t J9 + 6 - J-9 + 6 N ã2 6 tN,

1

6ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN>,

:1

2ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN,

1

2ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN>,

:1

6ã

- 6 t J9 + 11 6 + J9 - 11 6 N ã2 6 tN,

1

6ã

- 6 t J15 + 4 6 + J15 - 4 6 N ã2 6 tN>>,

::5

2ã

- 6 t J1 + ã2 6 tN,

1

2ã

- 6 t J1 + 6 - J-1 + 6 N ã2 6 tN>,

:5

6ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN,

1

6ã

- 6 t J9 + 4 6 + J9 - 4 6 N ã2 6 tN>,

:5

6ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN,

5

6ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN>>>


solpar31cbis = Flatten@solpar31c, 1D

::1

6ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN,

1

6ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN>,

:1

6ã

- 6 t J3 + 7 6 + J3 - 7 6 N ã2 6 tN,

1

6ã

- 6 t J9 + 2 6 + J9 - 2 6 N ã2 6 tN>,

:1

2ã

- 6 t J1 + 4 6 + J1 - 4 6 N ã2 6 tN,

1

2ã

- 6 t J5 + 6 - J-5 + 6 N ã2 6 tN>,

:1

6ã

- 6 t J9 + 6 - J-9 + 6 N ã2 6 tN,

1

6ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN>,

:1

2ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN,

1

2ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN>,

:1

6ã

- 6 t J9 + 11 6 + J9 - 11 6 N ã2 6 tN,

1

6ã

- 6 t J15 + 4 6 + J15 - 4 6 N ã2 6 tN>,

:5

2ã

- 6 t J1 + ã2 6 tN,

1

2ã

- 6 t J1 + 6 - J-1 + 6 N ã2 6 tN>,

:5

6ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN,

1

6ã

- 6 t J9 + 4 6 + J9 - 4 6 N ã2 6 tN>,

:5

6ã

- 6 t J3 + 2 6 + J3 - 2 6 N ã2 6 tN,

5

6ã

- 6 t J3 + 6 - J-3 + 6 N ã2 6 tN>>

10 StabilitySG.nb

graf131c = ParametricPlot@Evaluate@solpar31cbisD,8t, -0.5`, 1<, PlotStyle ® 88RGBColor@1, 0, 0D, [email protected]`D<<D

-15 -10 -5 5 10 15

-7.5

-5

-2.5

2.5

5

7.5

10

The solutions and the direction fields are represented together:

graf231c = HNeeds@"VectorFieldPlots`"D;VectorFieldPlots`VectorFieldPlot@8x - 5 y, -x - y<, 8x, -50, 28<, 8y, -10, 15<,PlotPoints ® 20, DisplayFunction ® Identity, BaseStyle ® [email protected]`D<DL;

Show@graf131c, graf231cD

-40 -20 20

-10

-5

5

10

15

The eigenvalues of the matrix associated with the SODE are calculated to explain the nature of thepoint.

The matrix associated to the SODE is defined as:

StabilitySG.nb 11

a =1 -5

-1 -1

881, -5<, 8-1, -1<<

Its eigenvalues are given by the command:

Eigenvalues@aD

:- 6 , 6 >

The origin is a saddle point because the eigenvalues are real and with different sign.

The eigenvectors are calculated and the straight lines with this directions are represented.

Eigenvectors@aD

::-1 + 6 , 1>, :-1 - 6 , 1>>

The straight line of the eigenvector corresponding to the negative eigenvalue is

12 StabilitySG.nb

graf331c = ParametricPlotB:J-1 - 6 N t, t>,

8t, -8, 15<, PlotStyle ® 8RGBColor@0, 0, 1D, [email protected]`D<F

-40 -20 20

-5

5

10

15

The straight line of the eigenvector corresponding to the positive eigenvalue is

graf431c = ParametricPlotB:J-1 + 6 N t, t>,

8t, -8, 15<, PlotStyle ® 8RGBColor@0, 0, 1D, [email protected]`D<F

-10 -5 5 10 15 20

-5

5

10

15

And the four graphs together:

StabilitySG.nb 13

Show@graf131c, graf231c, graf331c, graf431cD

-40 -20 20

-10

-5

5

10

15

It is possible to say that any trajectory that starts at a point on the straight line corresponding to thenegative eigenvalue remains above it and tends to the origin. Any trajectory that starts at a point on thestraight line corresponding to the positive eigenvalue remains below it and tends to the origin.

14 StabilitySG.nb

example 1. differential equations (sode) stability...

Documents