The Phase Plane
Phase portraits; type and stability classifications of equilibrium solutions of
systems of differential equations
Phase Portraits of Linear Systems
Consider a systems of linear differential equations x′ = Ax. Its phase
portrait is a representative set of its solutions, plotted as parametric curves
(with t as the parameter) on the Cartesian plane tracing the path of each
particular solution (x, y) = (x1(t), 2 (t)), −∞ < t < ∞. Similar to a direction
field, a phase portrait is a graphical tool to visualize how the solutions of a
given system of differential equations would behave in the long run.
In this context, the Cartesian plane where the phase portrait resides is called
the phase plane. The parametric curves traced by the solutions are
sometimes also called their trajectories.
Remark: It is quite labor▯intensive, but it is possible to sketch the phase
portrait by hand without first having to solve the system of equations that it
represents. Just like a direction field, a phase portrait can be a tool to predict
the behaviors of a system’s solutions. To do so, we draw a grid on the phase
plane. Then, at each grid point x = (α, β), we can calculate the solution
trajectory’s instantaneous direction of motion at that point by using the given
system of equations to compute the tangent / velocity vector, x′. Namely
plug in x = (α, β) to compute x′ = Ax.
In the first section we will examine the phase portrait of linear system of
differential equations. We will classify the type and stability the equilibrium
solution of a given linear system by the shape and behavior of its phase
© 2008 Zachary S Tseng D▯2 ▯ 1 Equilibrium Solution (a.k.a. Critical Point, or Stationary Point)
An equilibrium solution of the system x′ = Ax is a point (x , x )1whe2e x′ = 0,
that is, where x 1 = 0 = x 2. An equilibrium solution is a constant solution of
the system, and is usually called a critical point. For a linear system
x′ = Ax, an equilibrium solution occurs at each solution of the system (of
homogeneous algebraic equations) Ax = 0. As we have seen, such a system
has exactly one solution, located at the origin, if det(A) ≠ 0. If det(A) = 0,
then there are infinitely many solutions.
For our purpose, and unless otherwise noted, we will only consider systems
of linear differential equations whose coefficient matrix A has nonzero
determinant. That is, we will only consider systems where the origin is the
only critical point.
Note: A matrix could only have zero as one of its eigenvalues if and only if
its determinant is also zero. Therefore, since we limit ourselves to consider
only those systems where det(A) ≠ 0, we will not encounter in this section
any matrix with zero as an eigenvalue.
© 2008 Zachary S Tseng D▯2 ▯ 2 Classification of Critical Points
Similar to the earlier discussion on the equilibrium solutions of a single first
order differential equation using the direction field, we will presently
classify the critical points of various systems of first order linear differential
equations by their stability. In addition, due to the truly two▯dimensional
nature of the parametric curves, we will also classify the type of those
critical points by their shapes (or, rather, by the shape formed by the
trajectories about each critical point).
Comment: The accurate tracing of the parametric curves of the solutions is
not an easy task without electronic aids. However, we can obtain very
reasonable approximation of a trajectory by using the very same idea behind
the direction field, namely the tangent line approximation. At each point
x = (x 1, 2 ) on the ty▯plane, the direction of motion of the solution curve the
passes through the point is determined by the direction vector (i.e. the
tangent vector) x′, the derivative of the solution vector x, evaluated at the
given point. The tangent vector at each given point can be calculated
directly from the given matrix▯vector equation x′ = Ax, using the position
vector x = (x 1 x 2. Like working with a direction field, there is no need to
find the solution first before performing this approximation.
© 2008 Zachary S Tseng D▯2 ▯ 3 Given x′ = Ax, where there is only one critical point, at (0,0):
Case I. Distinct real eigenvalues
1 t 2 t
The general solution is x= C k 1 1 + C k2e 2 .
1. When r a1d r ar2 both positive, or are both negative
The phase portrait shows trajectories either moving away from the
critical point to infinite▯distant away (when r > 0), or moving directly
toward, and converge to the critical point (when r < 0). The
trajectories that are the eigenvectors move in straight lines. The rest
of the trajectories move, initially when near the critical point, roughly
in the same direction as the eigenvector of the eigenvalue with the
smaller absolute value. Then, farther away, they would bend toward
the direction of the eigenvector of the eigenvalue with the larger
absolute value The trajectories either move away from the critical
point to infinite▯distant away (when r are both positive), or move
toward from infinite▯distant out and eventually converge to the critical
point (when r are both negative). This type of critical point is called a
node. It is asymptotically stable if r are both negative, unstable if r
are both positive.
© 2008 Zachary S Tseng D▯2 ▯ 4 Two distinct real eigenvalues, both of the same sign
Stability: It is unstable if both eigenvalues are positive;
asymptotically stable if they are both negative.
© 2008 Zachary S Tseng D▯2 ▯ 5 2. When r an1 r have2opposite signs (say r > 0 and r1< 0) 2
In this type of phase portrait, the trajectories given by the eigenvectors
of the negative eigenvalue initially start at infinite▯distant away, move
toward and eventually converge at the critical point. The trajectories
that represent the eigenvectors of the positive eigenvalue move in
exactly the opposite way: start at the critical point then diverge to
infinite▯distant out. Every other trajectory starts at infinite▯distant
away, moves toward but never converges to the critical point, before
changing direction and moves back to infinite▯distant away. All the
while it would roughly follow the 2 sets of eigenvectors. This type of
critical point is called a saddle point. It is always unstable.
© 2008 Zachary S Tseng D▯2 ▯ 6 Two distinct real eigenvalues, opposite signs
Type: Saddle Point
Stability: It is always unstable.
© 2008 Zachary S Tseng D▯2 ▯ 7 Case II. Repeated real eigenvalue
3. When there are two linearly independent eigenvectors k and k .
The general solution is x = C k e + C k e = e (C k + C k ) .
1 1 2 2 1 1 2 2
Every nonzero solution traces a straight▯line trajectory, in the
direction given by the vector C 1 1 C k 2 2 . The phase portrait thus
has a distinct star▯burst shape. The trajectories either move directly
away from the critical point to infinite▯distant away (when r > 0), or
move directly toward, and converge to the critical point (when r < 0).
This type of critical point is called a proper node (or a starl point). It
is asymptotically stable if r < 0, unstable if r > 0.
Note: For 2 × 2 systems of linear differential equations, this will
occur if, and only if, when the coefficient matrix A is a constant
multiple of the identity matrix:
1 0 α 0
A = α = , α = any nonzero constant . *
0 1 0 α
*In the case of α = 0, the solution is C 1 . Every solution is an equilibrium
x= C10 + C21 =C
solution. Therefore, every trajectory on its phase portrait consists of a single point, and every point on the
phase plane is a trajectory.
© 2008 Zachary S Tseng D▯2 ▯ 8 A repeated real eigenvalue, two linearly independent eigenvectors
Type: Proper Node (or Star Point)
Stability: It is unstable if the eigenvalue is positive; asymptotically
stable if the eigenvalue is negative.
© 2008 Zachary S Tseng D▯2 ▯ 9 4. When there is only one linearly independent eigenvector k.
rt rt rt
Then the general solution is x = C k1 + C (kte 2 + η e ).
The phase portrait shares characteristics with that of a node. With
only one eigenvector, it is a degenerated▯looking node that is a cross
between a node and a spiral point (see case 6 below). The trajectories
either all diverge away from the critical point to infinite▯distant away
(when r > 0), or all converge to the critical point (when r < 0). This
type of critical point is called an improper node. It is asymptotically
stable if r < 0, unstable if r > 0.
© 2008 Zachary S Tseng D▯2 ▯ 10 A repeated real eigenvalue, one linearly independent eigenvector
Type: Improper Node
Stability: It is unstable if the eigenvalue is positive; asymptotically
stable if the eigenvalue is negative.
© 2008 Zachary S Tseng D▯2 ▯ 11 Case III. Complex conjugate eigenvalues
The general solution is
λ t λ t
x= C e1 (acos(▯t) −bsin(▯t) + C e ) 2 ( asin(▯t) +bcos(▯t) )
5. When the real part λ is zero.
In this case the trajectories neither converge to the critical point nor
move to infinite▯distant away. Rather, they stay in constant, elliptical
(or, rarely, circular) orbits. This type of critical point is called a
center. It has a unique stability classification shared by no other:
stable (or neutrally stable). It is NOT asymptotically stable and one
should not confuse them.
6. When the real part λ is nonzero.
The trajectories still retain the elliptical traces as in the previous case.
However, with each revolution, their distances fromthe critical point
grow/decay exponentially according to the terme . Therefore, the
phase portrait shows trajectories that spiral awayfrom the critical
point to infinite▯distant away (when λ > 0). Or trajectories that spiral
toward, and converge to the critical point (when λ < 0). This type of
critical point is called a spiral point. It is asymptotically stable if λ <
0, it is unstable if λ > 0.
© 2008 Zachary S Tseng D▯2 ▯ 12 Complex eigenvalues, with real part zero (purely imaginary numbers)
Stability: Stable (but not asymptotically stable); sometimes it is
referred to as neutrally stable.
© 2008 Zachary S Tseng D▯2 ▯ 13 Complex eigenvalues, with nonzero real part
Type: Spiral Point
Stability: It is unstable if the eigenvalues have positive real part;
asymptotically stable if the eigenvalues have negative real part.
© 2008 Zachary S Tseng D▯2 ▯ 14 Summary of Stability Classification
Asymptotically stable – All trajectories of its solutions converge to the
critical point as t → ∞. A critical point is asymptotically stable if all of A’s
eigenvalues are negative, or have negative real part for complex eigenvalues.
Unstable – All trajectories (or all but a few, in the case of a saddle point)
start out at the critical point at t → − ∞, then move away to infinitely distant
out as t → ∞. A critical point is unstable if at least one of A’s eigenvalues is
positive, or has positive real part for complex eigenvalues.
Stable (or neutrally stable) – Each trajectory move about the critical point
within a finite range of distance. It never moves out to infinitely distant, nor
(unlike in the case of asymptotically stable) does it ever go to the critical
point. A critical point is stable if A’s eigenvalues are purely imaginary.
In short, as t increases, if all (or almost all) trajectories
1. converge to the critical point → asymptotically stable,
2. move away from the critical point to infinitelyfar away → unstable,
3. stay in a fixed orbit within a finite (i.e., bounded) range of distance away
from the critical point → stable (or neutrally stable).
© 2008 Zachary S Tseng D▯2 ▯ 15 Nonhomogeneous Linear Systems with Constant Coefficients
Now let us consider the nonh