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2D Homogeneous Linear Systems with Constant Coefficients: a zero eigenvalue a line of equilibria Xu-Yan Chen

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Page 1: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Homogeneous Linear Systems withConstant Coefficients:

a zero eigenvalue ⇒ a line of equilibria

Xu-Yan Chen

Page 2: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Systems of Diff Eqs:d~x

dt= A~x

where ~x(t) =

[x1(t)x2(t)

], A is a 2× 2 real constant matrix

Things to explore:

I General solutions

I Initial value problems

I Geometric figures

I Solutions graphs x1 vs t & x2 vs tI Direction fields in the (x1, x2) planeI Phase portraits in the (x1, x2) plane

I Stability/instability of equilibrium (x1, x2) = (0, 0)

Page 3: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Systems:d~x

dt= A~x

What if we have a zero eigenvalue?Assume that the eigenvalues of A are: λ1 = 0, λ2 6= 0.

Solution Method: (nothing new)

I Find an eigenvector ~u1 for λ1, by solving

(A− λ1I)~x = 0

⇒ A~x = 0

I Find an eigenvector ~u2 for λ2, by solving

(A− λ2I)~x = 0.

I General solutions are

~x(t) = C1eλ1t~u1 + C2e

λ2t~u2

⇒ ~x(t) = C1~u1 + C2eλ2t~u2

Page 4: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Systems:d~x

dt= A~x

What if we have a zero eigenvalue?Assume that the eigenvalues of A are: λ1 = 0, λ2 6= 0.

Solution Method: (nothing new)

I Find an eigenvector ~u1 for λ1, by solving

(A− λ1I)~x = 0 ⇒ A~x = 0

I Find an eigenvector ~u2 for λ2, by solving

(A− λ2I)~x = 0.

I General solutions are

~x(t) = C1eλ1t~u1 + C2e

λ2t~u2

⇒ ~x(t) = C1~u1 + C2eλ2t~u2

Page 5: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Systems:d~x

dt= A~x

What if we have a zero eigenvalue?Assume that the eigenvalues of A are: λ1 = 0, λ2 6= 0.

Solution Method: (nothing new)

I Find an eigenvector ~u1 for λ1, by solving

(A− λ1I)~x = 0 ⇒ A~x = 0

I Find an eigenvector ~u2 for λ2, by solving

(A− λ2I)~x = 0.

I General solutions are

~x(t) = C1eλ1t~u1 + C2e

λ2t~u2 ⇒ ~x(t) = C1~u1 + C2eλ2t~u2

Page 6: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Systems:d~x

dt= A~x

Phase portraits & stabilitywhen the eigenvalues of A are: λ1 = 0, λ2 6= 0.

General solutions: ~x(t) = C1~u1 + C2eλ2t~u2

On the eigenspace of the zero eigenvalue λ1, the solutions are timeindependent; i.e., the eigenspace of the zero eigenvalue is a line ofequilibria.

I λ1 = 0, λ2 < 0⇒ Attractive line of equilibria,

stable,but not asymptotically stable

I λ1 = 0, λ2 > 0⇒ Repulsive line of equilibria,

unstable

Page 7: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Systems:d~x

dt= A~x

Phase portraits & stabilitywhen the eigenvalues of A are: λ1 = 0, λ2 6= 0.

General solutions: ~x(t) = C1~u1 + C2eλ2t~u2

On the eigenspace of the zero eigenvalue λ1, the solutions are timeindependent; i.e., the eigenspace of the zero eigenvalue is a line ofequilibria.

I λ1 = 0, λ2 < 0⇒ Attractive line of equilibria,

stable,but not asymptotically stable

I λ1 = 0, λ2 > 0⇒ Repulsive line of equilibria,

unstable

Page 8: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Systems:d~x

dt= A~x

Phase portraits & stabilitywhen the eigenvalues of A are: λ1 = 0, λ2 6= 0.

General solutions: ~x(t) = C1~u1 + C2eλ2t~u2

On the eigenspace of the zero eigenvalue λ1, the solutions are timeindependent; i.e., the eigenspace of the zero eigenvalue is a line ofequilibria.

I λ1 = 0, λ2 < 0⇒ Attractive line of equilibria,

stable,but not asymptotically stable

I λ1 = 0, λ2 > 0⇒ Repulsive line of equilibria,

unstable

Page 9: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

2D Systems:d~x

dt= A~x

Phase portraits & stabilitywhen the eigenvalues of A are: λ1 = 0, λ2 6= 0.

General solutions: ~x(t) = C1~u1 + C2eλ2t~u2

On the eigenspace of the zero eigenvalue λ1, the solutions are timeindependent; i.e., the eigenspace of the zero eigenvalue is a line ofequilibria.

I λ1 = 0, λ2 < 0⇒ Attractive line of equilibria,

stable,but not asymptotically stable

I λ1 = 0, λ2 > 0⇒ Repulsive line of equilibria,

unstable

Page 10: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (an attractive line of equilibria).

Consider ~x′ = A~x, where A =

[−3 −6−1 −2

].

(a) Find general solutions of ~x′ =

[−3 −6−1 −2

]~x.

(b) Solve the initial value problem ~x′ =

[−3 −6−1 −2

]~x, ~x(0) =

[23

].

(c) Sketch the phase portrait.

(d) Is the equilibrium (0, 0) stable, asymptotically stable, or unstable?

Page 11: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (a)d~x

dt=

[−3 −6−1 −2

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[−3− λ −6−1 −2− λ

]= λ2 + 5λ = 0 ⇒ λ1 = 0, λ2 = −5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[−3 −6−1 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−21

]⇒ An eigenvector ~u1 =

[−21

].

I Eigenvectors of A for λ2 = −5, by solving (A− λ2I)~x = 0:

(A+ 5I)~x = 0⇔[

2 −6−1 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[31

]⇒ An eigenvector ~u2 =

[31

].

I General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Page 12: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (a)d~x

dt=

[−3 −6−1 −2

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[−3− λ −6−1 −2− λ

]= λ2 + 5λ = 0 ⇒ λ1 = 0, λ2 = −5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[−3 −6−1 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−21

]⇒ An eigenvector ~u1 =

[−21

].

I Eigenvectors of A for λ2 = −5, by solving (A− λ2I)~x = 0:

(A+ 5I)~x = 0⇔[

2 −6−1 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[31

]⇒ An eigenvector ~u2 =

[31

].

I General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Page 13: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (a)d~x

dt=

[−3 −6−1 −2

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[−3− λ −6−1 −2− λ

]= λ2 + 5λ = 0 ⇒ λ1 = 0, λ2 = −5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[−3 −6−1 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−21

]⇒ An eigenvector ~u1 =

[−21

].

I Eigenvectors of A for λ2 = −5, by solving (A− λ2I)~x = 0:

(A+ 5I)~x = 0⇔[

2 −6−1 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[31

]⇒ An eigenvector ~u2 =

[31

].

I General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Page 14: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (a)d~x

dt=

[−3 −6−1 −2

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[−3− λ −6−1 −2− λ

]= λ2 + 5λ = 0 ⇒ λ1 = 0, λ2 = −5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[−3 −6−1 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−21

]⇒ An eigenvector ~u1 =

[−21

].

I Eigenvectors of A for λ2 = −5, by solving (A− λ2I)~x = 0:

(A+ 5I)~x = 0⇔[

2 −6−1 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[31

]⇒ An eigenvector ~u2 =

[31

].

I General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Page 15: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (a)d~x

dt=

[−3 −6−1 −2

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[−3− λ −6−1 −2− λ

]= λ2 + 5λ = 0 ⇒ λ1 = 0, λ2 = −5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[−3 −6−1 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−21

]⇒ An eigenvector ~u1 =

[−21

].

I Eigenvectors of A for λ2 = −5, by solving (A− λ2I)~x = 0:

(A+ 5I)~x = 0⇔[

2 −6−1 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[31

]⇒ An eigenvector ~u2 =

[31

].

I General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Page 16: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (b) Solved~x

dt=

[−3 −6−1 −2

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[−21

]+ C2e

−5t[31

]I Use the initial condition:

~x(0) =

[23

]⇒ C1

[−21

]+ C2

[31

]=

[23

]⇒

[−2 31 1

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[7/58/5

]I The solution to the initial value problem:

~x(t) = 75

[−21

]+ 8

5e−5t

[31

]

Page 17: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (b) Solved~x

dt=

[−3 −6−1 −2

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[−21

]+ C2e

−5t[31

]

I Use the initial condition:

~x(0) =

[23

]⇒ C1

[−21

]+ C2

[31

]=

[23

]⇒

[−2 31 1

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[7/58/5

]I The solution to the initial value problem:

~x(t) = 75

[−21

]+ 8

5e−5t

[31

]

Page 18: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (b) Solved~x

dt=

[−3 −6−1 −2

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[−21

]+ C2e

−5t[31

]I Use the initial condition:

~x(0) =

[23

]⇒ C1

[−21

]+ C2

[31

]=

[23

]⇒

[−2 31 1

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[7/58/5

]

I The solution to the initial value problem:

~x(t) = 75

[−21

]+ 8

5e−5t

[31

]

Page 19: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (b) Solved~x

dt=

[−3 −6−1 −2

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[−21

]+ C2e

−5t[31

]I Use the initial condition:

~x(0) =

[23

]⇒ C1

[−21

]+ C2

[31

]=

[23

]⇒

[−2 31 1

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[7/58/5

]I The solution to the initial value problem:

~x(t) = 75

[−21

]+ 8

5e−5t

[31

]

Page 20: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (c) Phase portrait ofd~x

dt=

[−3 −6−1 −2

]~x

General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

• C1

[−21

]are eigenvectors of λ1 = 0. These are equilibria.

• C2e−5t

[31

]decays to the origin, along the eigenspace of λ2 = −5.

Page 21: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (c) Phase portrait ofd~x

dt=

[−3 −6−1 −2

]~x

General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

• C1

[−21

]are eigenvectors of λ1 = 0. These are equilibria.

• C2e−5t

[31

]decays to the origin, along the eigenspace of λ2 = −5.

Page 22: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (c) Phase portrait ofd~x

dt=

[−3 −6−1 −2

]~x

General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]• C1

[−21

]are eigenvectors of λ1 = 0. These are equilibria.

• C2e−5t

[31

]decays to the origin, along the eigenspace of λ2 = −5.

Page 23: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (c) Phase portrait ofd~x

dt=

[−3 −6−1 −2

]~x

General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]When C1 and C2 are both 6= 0, lim

t→∞~x(t) = C1

[−21

].

The solution trajectories are lines parallel to the eigenspace ofλ2 = −5.

Page 24: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (d) Is the equilibrium (0, 0) stable,asymptotically stable, or unstable?

General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Every equilibrium isstable,but not asymptotically stable.

We have anattractive line of equilibria,when λ1 = 0, λ2 < 0.

Page 25: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (d) Is the equilibrium (0, 0) stable,asymptotically stable, or unstable?

General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Every equilibrium isstable,but not asymptotically stable.

We have anattractive line of equilibria,when λ1 = 0, λ2 < 0.

Page 26: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 1 (d) Is the equilibrium (0, 0) stable,asymptotically stable, or unstable?

General solutions: ~x(t) = C1

[−21

]+ C2e

−5t[31

]

Every equilibrium isstable,but not asymptotically stable.

We have anattractive line of equilibria,when λ1 = 0, λ2 < 0.

Page 27: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (a repulsive line of equilibria).

Consider ~x′ = A~x, where A =

[2 16 3

].

(a) Find general solutions of ~x′ =

[2 16 3

]~x.

(b) Solve the initial value problem ~x′ =

[2 16 3

]~x, ~x(0) =

[23

].

(c) Sketch the phase portrait.

(d) Is the equilibrium (0, 0) stable, asymptotically stable, or unstable?

Page 28: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (a)d~x

dt=

[2 16 3

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[2− λ 1

6 3− λ

]= λ2 − 5λ = 0 ⇒ λ1 = 0, λ2 = 5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[2 16 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−1/2

1

]⇒ An eigenvector ~u1 =

[1−2

].

I Eigenvectors of A for λ2 = 5, by solving (A− λ2I)~x = 0:

(A− 5I)~x = 0 ⇔[−3 16 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[1/31

]⇒ An eigenvector ~u2 =

[13

].

I General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Page 29: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (a)d~x

dt=

[2 16 3

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[2− λ 1

6 3− λ

]= λ2 − 5λ = 0 ⇒ λ1 = 0, λ2 = 5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[2 16 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−1/2

1

]⇒ An eigenvector ~u1 =

[1−2

].

I Eigenvectors of A for λ2 = 5, by solving (A− λ2I)~x = 0:

(A− 5I)~x = 0 ⇔[−3 16 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[1/31

]⇒ An eigenvector ~u2 =

[13

].

I General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Page 30: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (a)d~x

dt=

[2 16 3

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[2− λ 1

6 3− λ

]= λ2 − 5λ = 0 ⇒ λ1 = 0, λ2 = 5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[2 16 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−1/2

1

]⇒ An eigenvector ~u1 =

[1−2

].

I Eigenvectors of A for λ2 = 5, by solving (A− λ2I)~x = 0:

(A− 5I)~x = 0 ⇔[−3 16 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[1/31

]⇒ An eigenvector ~u2 =

[13

].

I General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Page 31: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (a)d~x

dt=

[2 16 3

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[2− λ 1

6 3− λ

]= λ2 − 5λ = 0 ⇒ λ1 = 0, λ2 = 5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[2 16 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−1/2

1

]⇒ An eigenvector ~u1 =

[1−2

].

I Eigenvectors of A for λ2 = 5, by solving (A− λ2I)~x = 0:

(A− 5I)~x = 0 ⇔[−3 16 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[1/31

]⇒ An eigenvector ~u2 =

[13

].

I General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Page 32: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (a)d~x

dt=

[2 16 3

]~x

I Eigenvalues of A, by solving det(A− λI) = 0:

det

[2− λ 1

6 3− λ

]= λ2 − 5λ = 0 ⇒ λ1 = 0, λ2 = 5

I Eigenvectors of A for λ1 = 0, by solving (A− λ1I)~x = 0:

A~x = 0 ⇔[2 16 3

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[−1/2

1

]⇒ An eigenvector ~u1 =

[1−2

].

I Eigenvectors of A for λ2 = 5, by solving (A− λ2I)~x = 0:

(A− 5I)~x = 0 ⇔[−3 16 −2

] [x1x2

]=

[00

]⇔

[x1x2

]= x2

[1/31

]⇒ An eigenvector ~u2 =

[13

].

I General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Page 33: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (b) Solved~x

dt=

[2 16 3

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[1−2

]+ C2e

5t

[13

]I Use the initial condition:

~x(0) =

[23

]⇒ C1

[1−2

]+ C2

[13

]=

[23

]⇒

[1 1−2 3

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[3/57/5

]I The solution to the initial value problem:

~x(t) = 35

[1−2

]+ 7

5e5t

[13

]

Page 34: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (b) Solved~x

dt=

[2 16 3

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[1−2

]+ C2e

5t

[13

]

I Use the initial condition:

~x(0) =

[23

]⇒ C1

[1−2

]+ C2

[13

]=

[23

]⇒

[1 1−2 3

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[3/57/5

]I The solution to the initial value problem:

~x(t) = 35

[1−2

]+ 7

5e5t

[13

]

Page 35: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (b) Solved~x

dt=

[2 16 3

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[1−2

]+ C2e

5t

[13

]I Use the initial condition:

~x(0) =

[23

]⇒ C1

[1−2

]+ C2

[13

]=

[23

]⇒

[1 1−2 3

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[3/57/5

]

I The solution to the initial value problem:

~x(t) = 35

[1−2

]+ 7

5e5t

[13

]

Page 36: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (b) Solved~x

dt=

[2 16 3

]~x, ~x(0) =

[23

]

I General solutions:

~x(t) = C1

[1−2

]+ C2e

5t

[13

]I Use the initial condition:

~x(0) =

[23

]⇒ C1

[1−2

]+ C2

[13

]=

[23

]⇒

[1 1−2 3

] [C1

C2

]=

[23

]⇒

[C1

C2

]=

[3/57/5

]I The solution to the initial value problem:

~x(t) = 35

[1−2

]+ 7

5e5t

[13

]

Page 37: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (c) Phase portrait ofd~x

dt=

[2 16 3

]~x

General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

• C1

[1−2

]are eigenvectors of λ1 = 0. These are equilibria.

• C2e5t

[13

]grows, along the eigenspace of λ2 = 5.

Page 38: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (c) Phase portrait ofd~x

dt=

[2 16 3

]~x

General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

• C1

[1−2

]are eigenvectors of λ1 = 0. These are equilibria.

• C2e5t

[13

]grows, along the eigenspace of λ2 = 5.

Page 39: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (c) Phase portrait ofd~x

dt=

[2 16 3

]~x

General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]• C1

[1−2

]are eigenvectors of λ1 = 0. These are equilibria.

• C2e5t

[13

]grows, along the eigenspace of λ2 = 5.

Page 40: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (c) Phase portrait ofd~x

dt=

[2 16 3

]~x

General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]When C1 and C2 are both 6= 0:

All these solutions grow to infinity.The trajectories are lines parallel to the eigenspace of λ2 = 5.

Page 41: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (d) Are the equilibria stable,asymptotically stable, or unstable?

General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Every equilibrium isunstable.

We have arepulsive line of equilibria,when λ1 = 0, λ2 > 0.

Page 42: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (d) Are the equilibria stable,asymptotically stable, or unstable?

General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Every equilibrium isunstable.

We have arepulsive line of equilibria,when λ1 = 0, λ2 > 0.

Page 43: 2D Homogeneous Linear Systems with Constant …people.math.gatech.edu/~xchen/teach/ode/2d-lin-sys-zero.pdf · 2D Homogeneous Linear Systems with Constant Coe cients: ... Is the equilibrium

Example 2 (d) Are the equilibria stable,asymptotically stable, or unstable?

General solutions: ~x(t) = C1

[1−2

]+ C2e

5t

[13

]

Every equilibrium isunstable.

We have arepulsive line of equilibria,when λ1 = 0, λ2 > 0.