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Phase Plane for Two-Dimensional Autonomous System

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INSITTUTO DE ESTUDIOS SUPERIORES DE TAMAPULIPAS 

RED DE UNIVERSIDADES AN?HUAC 

 

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M?XICO MMVI 

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Phase Plane for Two-Dimensional Autonomous System  

 

Maple 10 Document
Prepared by
Prof. David Macias Ferrer
E-mail: david.macias@iest.edu.mx
Madero City, Mexico
 

URL: http://www.geocities.com/dmacias_iest/MyPage.html 

 

 

Goals 

  • To make a classification of the equilibrium points for a two-dimensional autonomous system.
 

  • To show the powerful Maple 10 graphics tools to visualize the phase portraits in all cases
 

 

2-D Autonomous System 

 

Let us consider a two-dimensional autonomous system: 

 

Diff(x(t), t) = a*x(t)+b*y(t) 

Diff(y(t), t) = c*x(t)+d*y(t) 

It can be written in the matrix form:  

Diff(X, t) = A*X 

where A is the associated matrix. It is evident that the autonomous system admits the trivial solution x = 0 and y = 0 . Geometrically this represent the origin on phase plane. The origin is also known critical point or equilibrium point. A non null solution of the system is a smooth curve called trajectory. The set of all trajectories is called phase portrait. The geometric properties of the phase portrait are closely related to the algebraic characteristics of eigenvalues of the matrix A. The expression: 

 

`+`(k^2, -(a+d)*k, ad-bc) = 0 

 

is called characteristic polynomial. So, the nature of equilibrium point is determined by the roots of this polynomial. 

 

The following cases are possible: 

 

restart; -1 

with(DEtools), with(linalg); -1 

 

Case I. Real and Distincts Roots 

 

a).- k[1] < 0, k[2] < 0. The roots of characteristic polynomial are real and negative. 

 

sys := {diff(x(t), t) = x(t)-4*y(t), diff(y(t), t) = 2*x(t)-5*y(t)} 

{diff(x(t), t) = x(t)-4*y(t), diff(y(t), t) = 2*x(t)-5*y(t)} (2.1.1)
 

The matrix associated is:  

A := Matrix(%id = 150421904) 

Matrix(%id = 150421904) (2.1.2)
 

The characteristic polynomial is: 

charpoly(A, k) 

k^2+4*k+3 (2.1.3)
 

The roots are: 

eigenvalues(A) 

-1, -3 (2.1.4)
 

In this case the origin is an asymptotically stable equilbrium point, also known as a sink. 

The general solution of the system is: 

sol := dsolve(sys) 

{y(t) = _C1*exp(-t)+_C2*exp(-3*t), x(t) = 2*_C1*exp(-t)+_C2*exp(-3*t)} (2.1.5)
 

The phase portrait is: 

DEplot(sys, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

b).- 0 < k[1], 0. < k[2] The roots of characteristic polynomial are real and positive 

sys1 := {diff(x(t), t) = x(t)-3*y(t), diff(y(t), t) = 2*y(t)} 

{diff(x(t), t) = x(t)-3*y(t), diff(y(t), t) = 2*y(t)} (2.1.6)
 

The matrix associated is: 

A1 := Matrix(%id = 147238480) 

Matrix(%id = 147238480) (2.1.7)
 

The characteristic polynomial is: 

charpoly(A1, k) 

(k-1)*(k-2) (2.1.8)
 

The roots are: 

eigenvalues(A1) 

1, 2 (2.1.9)
 

In this case the origin is an unstable equilbrium point, also known as a source. 

The general solution of the system is: 

sol1 := dsolve(sys1) 

{y(t) = _C2*exp(2*t), x(t) = -3*_C2*exp(2*t)+exp(t)*_C1} (2.1.10)
 

The phase portrait is: 

DEplot(sys1, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys1, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys1, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

c).- k[1] < 0, 0. < k[2] The roots of characteristic polynomial are real and of opposed signs. 

sys2 := {diff(x(t), t) = 7*x(t)-3*y(t), diff(y(t), t) = 6*x(t)-4*y(t)} 

{diff(x(t), t) = 7*x(t)-3*y(t), diff(y(t), t) = 6*x(t)-4*y(t)} (2.1.11)
 

The matrix associated is: 

A2 := Matrix(%id = 149916900) 

Matrix(%id = 149916900) (2.1.12)
 

The characteristic polynomial is: 

charpoly(A2, k) 

k^2-3*k-10 (2.1.13)
 

The roots are: 

eigenvalues(A2) 

5, -2 (2.1.14)
 

When one eigenvalue is positve, and the other is negative the origin is an unstable equilibrium point, also know as a saddle point, which is also unstable. 

The general solution of the system is: 

sol2 := dsolve(sys2) 

{y(t) = 3*_C1*exp(-2*t)+2/3*_C2*exp(5*t), x(t) = _C1*exp(-2*t)+_C2*exp(5*t)} (2.1.15)
 

The phase portrait is: 

DEplot(sys2, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = .3, y(0) = .6]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys2, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = .3, y(0) = .6]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys2, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = .3, y(0) = .6]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys2, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = .3, y(0) = .6]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

 

 

 

Case II. Complex Roots 

 

a).- k[1, 2] = p+`&+-`(qi), p < 0, q <> 0.The roots of the characteristic polynomial are complex, but the real part is negative. 

sys3 := {diff(x(t), t) = x(t)+2*y(t), diff(y(t), t) = -9*x(t)-5*y(t)} 

{diff(x(t), t) = x(t)+2*y(t), diff(y(t), t) = -9*x(t)-5*y(t)} (2.2.1)
 

The matrix associated is: 

A3 := Matrix(%id = 147288752) 

Matrix(%id = 147288752) (2.2.2)
 

The characteristic polynomial is: 

charpoly(A3, k) 

k^2+4*k+13 (2.2.3)
 

The roots are: 

eigenvalues(A3) 

-2+3*I, -2-3*I (2.2.4)
 

When the eigenvalues are complex and the real part is negative, the origin is an asymptotically stable focus, also known as a spiral sink. 

The general solution of the system is: 

sol3 := dsolve(sys3) 

{y(t) = -3/2*exp(-2*t)*(_C1*sin(3*t)-_C1*cos(3*t)+_C2*cos(3*t)+_C2*sin(3*t)), x(t) = exp(-2*t)*(_C1*sin(3*t)+_C2*cos(3*t))} (2.2.5)
 

The phase portrait is: 

DEplot(sys3, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys3, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys3, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys3, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

b).- k[1, 2] = p+`&+-`(qi), 0 < p, q <> 0.The roots of the characteristic polynomial are complex, but the real part is positive. 

sys4 := {diff(x(t), t) = 2*x(t)+3*y(t), diff(y(t), t) = -3*x(t)+2*y(t)} 

{diff(x(t), t) = 2*x(t)+3*y(t), diff(y(t), t) = -3*x(t)+2*y(t)} (2.2.6)
 

The matrix associated is: 

A4 := Matrix(%id = 154605824) 

Matrix(%id = 154605824) (2.2.7)
 

The characteristic polynomial is: 

charpoly(A4, k) 

k^2-4*k+13 (2.2.8)
 

The roots are: 

eigenvalues(A4) 

2+3*I, 2-3*I (2.2.9)
 

When the eigenvalues are complex and the real part is positive, the origin is an unstable focus, also known as a spiral source. 

The general solution of the system is: 

sol4 := dsolve(sys4) 

{y(t) = exp(2*t)*(_C1*cos(3*t)-_C2*sin(3*t)), x(t) = exp(2*t)*(_C1*sin(3*t)+_C2*cos(3*t))} (2.2.10)
 

The phase portrait is: 

DEplot(sys4, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys4, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys4, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

c).- k[1, 2] = p+`&+-`(qi), p = 0, q <> 0.The roots of the characteristic polynomial are complex, but the real part is zero. 

sys5 := {diff(x(t), t) = 2*x(t)+2*y(t), diff(y(t), t) = -4*x(t)-2*y(t)} 

{diff(x(t), t) = 2*x(t)+2*y(t), diff(y(t), t) = -4*x(t)-2*y(t)} (2.2.11)
 

The matrix associated is: 

A5 := Matrix(%id = 149846560) 

Matrix(%id = 149846560) (2.2.12)
 

The characteristic polynomial is: 

charpoly(A5, k) 

k^2+4 (2.2.13)
 

The roots are: 

eigenvalues(A5) 

2*I, -2*I (2.2.14)
 

When the eigenvalues are complex and the real part is zero, the origin is a stable equilibrium point (but not asymptotically stable). It is also called a center. 

The general solution of the system is: 

sol5 := dsolve(sys5) 

{x(t) = _C1*sin(2*t)+_C2*cos(2*t), y(t) = _C1*cos(2*t)-_C2*sin(2*t)-_C1*sin(2*t)-_C2*cos(2*t)} (2.2.15)
 

The phase portrait is: 

DEplot(sys5, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys5, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys5, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

 

 

 

 

Case III. Multiple Roots 

 

a).- k[1] = k[2] and k[2] < 0.The roots of the characteristic polynomial are real, multiple and negative.  

sys6 := {diff(x(t), t) = 3*x(t)-5*y(t), diff(y(t), t) = 5*x(t)-7*y(t)} 

{diff(x(t), t) = 3*x(t)-5*y(t), diff(y(t), t) = 5*x(t)-7*y(t)} (2.3.1)
 

The matrix associated is: 

A6 := Matrix(%id = 146564000) 

Matrix(%id = 146564000) (2.3.2)
 

The characteristic polynomial is: 

charpoly(A6, k) 

k^2+4*k+4 (2.3.3)
 

The roots are: 

eigenvalues(A6) 

-2, -2 (2.3.4)
 

In this case, the origin is an asymptotically stable equilbrium, and is a degenerate sink. 

The general solution of system is: 

sol6 := dsolve(sys6) 

{y(t) = 1/5*exp(-2*t)*(5*_C1+5*_C2*t-_C2), x(t) = exp(-2*t)*(_C1+_C2*t)} (2.3.5)
 

The phase portrait is: 

DEplot(sys6, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys6, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys6, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys6, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

b).- k[1] = k[2] and 0. < k[2]The roots of the characteristic polynomial are real, multiple and positive. 

sys7 := {diff(x(t), t) = x(t), diff(y(t), t) = 3*x(t)+y(t)} 

{diff(x(t), t) = x(t), diff(y(t), t) = 3*x(t)+y(t)} (2.3.6)
 

The matrix associated is: 

A7 := Matrix(%id = 152499372) 

Matrix(%id = 152499372) (2.3.7)
 

The characteristic polynomial is: 

charpoly(A7, k) 

(k-1)^2 (2.3.8)
 

The roots are: 

eigenvalues(A7) 

1, 1 (2.3.9)
 

In this case, the origin is an unstable equilbrium point, and is a degenerate source. 

The general solution of system is: 

sol7 := dsolve(sys7) 

{y(t) = (3*_C2*t+_C1)*exp(t), x(t) = _C2*exp(t)} (2.3.10)
 

The phase portrait is: 

DEplot(sys7, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys7, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys7, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys7, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

 

 

Case IV. Null Real Root 

 

a).- k[1] = 0, k[2] < 0.The null real root is negative. 

sys8 := {diff(x(t), t) = x(t)+y(t), diff(y(t), t) = -5*x(t)-5*y(t)} 

{diff(x(t), t) = x(t)+y(t), diff(y(t), t) = -5*x(t)-5*y(t)} (2.4.1)
 

The matrix associated is: 

A8 := Matrix(%id = 154677920) 

Matrix(%id = 154677920) (2.4.2)
 

The characteristic polynomial is: 

charpoly(A8, k) 

k^2+4*k (2.4.3)
 

The roots are: 

eigenvalues(A8) 

0, -4 (2.4.4)
 

In this case, the zero eigenvalue determine a line of stable equilibria. 

The general solution is: 

sol8 := dsolve(sys8) 

{y(t) = _C1+_C2*exp(-4*t), x(t) = -1/5*_C2*exp(-4*t)-_C1} (2.4.5)
 

The phase portrait is: 

DEplot(sys8, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys8, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys8, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

b).- k[1] = 0, 0. < k[2]Now, the null real root is positive. 

sys9 := {diff(x(t), t) = -2*x(t)+y(t), diff(y(t), t) = 2*x(t)-y(t)} 

{diff(x(t), t) = -2*x(t)+y(t), diff(y(t), t) = 2*x(t)-y(t)} (2.4.6)
 

The matrix associated is: 

A9 := Matrix(%id = 148366768) 

Matrix(%id = 148366768) (2.4.7)
 

The characteristic polynomial is: 

charpoly(A9, k) 

k^2+3*k (2.4.8)
 

The roots are: 

eigenvalues(A9) 

0, -3 (2.4.9)
 

In this case, the zero eigenvalue determine a line of unstable equilibria. 

The general solution is: 

sol9 := dsolve(sys9) 

{y(t) = -_C2*exp(-3*t)+2*_C1, x(t) = _C1+_C2*exp(-3*t)} (2.4.10)
 

The phase portrait is: 

DEplot(sys9, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys9, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys9, [x(t), y(t)], t = -7 .. 7, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

c).- k[1] = k[2] and k[2] = 0.Boths eigenvalues are zero. 

sys10 := {diff(y(t), t) = -x(t)-y(t), diff(x(t), t) = x(t)+y(t)} 

{diff(y(t), t) = -x(t)-y(t), diff(x(t), t) = x(t)+y(t)} (2.4.11)
 

The matrix associated is: 

A10 := Matrix(%id = 151465852) 

Matrix(%id = 151465852) (2.4.12)
 

The characteristic polyinomial is: 

charpoly(A10, k) 

k^2 (2.4.13)
 

The roots are: 

eigenvalues(A10) 

0, 0 (2.4.14)
 

In this case, there is only one linearly independent eigenvector. This eigenvector determines a line of unstable equilibria and other solutions are parallel to this line. 

The general solution is: 

sol10 := dsolve(sys10) 

{x(t) = -_C1-_C1*t-_C2, y(t) = _C1*t+_C2} (2.4.15)
 

The phase portrait is: 

DEplot(sys10, [x(t), y(t)], t = -15 .. 10, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys10, [x(t), y(t)], t = -15 .. 10, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
DEplot(sys10, [x(t), y(t)], t = -15 .. 10, x = -2 .. 2, y = -2 .. 2, [[x(0) = 1, y(0) = -.8]], stepsize = .1, linecolor = blue, thickness = 2, arrows = medium, title =
 

Plot
 

 

 

Bibliography 

Elgoltz L., "Ecuaciones Diferenciales y C?lculo Variacional", Tercera Edici?n en Espa?ol, Editorial Mir Mosc?, Rusia, 1983. 

 


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