In applied mathematics, in particular the
context of nonlinear system analysis, a phase
plane is a visual display of certain characteristics
of certain kinds of differential equations;
a coordinate plane with axes being the values
of the two state variables, say (x, y), or
(q, p) etc. (any pair of variables). It is
a two-dimensional case of the general n-dimensional
phase space.
The phase plane method refers to graphically
determining the existence of limit cycles
in the solutions of the differential equation.
The solutions to the differential equation
are a family of functions. Graphically, this
can be plotted in the phase plane like a two-dimensional
vector field. Vectors representing the derivatives
of the points with respect to a parameter
(say time t), that is (dx/dt, dy/dt), at representative
points are drawn. With enough of these arrows
in place the system behaviour over the regions
of plane in analysis can be visualized and
limit cycles can be easily identified.
The entire field is the phase portrait, a
particular path taken along a flow line (i.e.
a path always tangent to the vectors) is a
phase path. The flows in the vector field
indicate the time-evolution of the system
the differential equation describes.
In this way, phase planes are useful in visualizing
the behaviour of physical systems; in particular,
of oscillatory systems such as predator-prey
models (see Lotka–Volterra equations). In
these models the phase paths can "spiral in"
towards zero, "spiral out" towards infinity,
or reach neutrally stable situations called
centres where the path traced out can be either
circular, elliptical, or ovoid, or some variant
thereof. This is useful in determining if
the dynamics are stable or not.Other examples
of oscillatory systems are certain chemical
reactions with multiple steps, some of which
involve dynamic equilibria rather than reactions
that go to completion. In such cases one can
model the rise and fall of reactant and product
concentration (or mass, or amount of substance)
with the correct differential equations and
a good understanding of chemical kinetics.
== Example of a linear system ==
A two-dimensional system of linear differential
equations can be written in the form:
d
x
d
t
=
A
x
+
B
y
d
y
d
t
=
C
x
+
D
y
{\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=Ax+By\\{\frac
{dy}{dt}}&=Cx+Dy\end{aligned}}}
which can be organized into a matrix equation:
d
d
t
(
x
y
)
=
(
A
B
C
D
)
(
x
y
)
d
x
d
t
=
A
x
.
{\displaystyle {\begin{aligned}&{\frac {d}{dt}}{\begin{pmatrix}x\\y\\\end{pmatrix}}={\begin{pmatrix}A&B\\C&D\\\end{pmatrix}}{\begin{pmatrix}x\\y\\\end{pmatrix}}\\&{\frac
{d\mathbf {x} }{dt}}=\mathbf {A} \mathbf {x}
.\end{aligned}}}
where A is the 2 × 2 coefficient matrix above,
and x = (x, y) is a coordinate vector of two
independent variables.
Such systems may be solved analytically, for
this case by integrating:
d
y
d
x
=
C
x
+
D
y
A
x
+
B
y
{\displaystyle {\frac {dy}{dx}}={\frac {Cx+Dy}{Ax+By}}}
although the solutions are implicit functions
in x and y, and are difficult to interpret.
=== Solving using eigenvalues ===
More commonly they are solved with the coefficients
of the right hand side written in matrix form
using eigenvalues λ, given by the determinant:
det
(
A
−
λ
I
)
=
0
{\displaystyle \det(\mathbf {A} -\lambda \mathbf
{I} )=0}
and eigenvectors:
A
x
=
λ
x
{\displaystyle \mathbf {A} \mathbf {x} =\lambda
\mathbf {x} }
The eigenvalues represent the powers of the
exponential components and the eigenvectors
are coefficients. If the solutions are written
in algebraic form, they express the fundamental
multiplicative factor of the exponential term.
Due to the nonuniqueness of eigenvectors,
every solution arrived at in this way has
undetermined constants c1, c2, ... cn.
The general solution is:
x
=
[
k
1
k
2
]
c
1
e
λ
1
t
+
[
k
3
k
4
]
c
2
e
λ
2
t
.
{\displaystyle x={\begin{bmatrix}k_{1}\\k_{2}\end{bmatrix}}c_{1}e^{\lambda
_{1}t}+{\begin{bmatrix}k_{3}\\k_{4}\end{bmatrix}}c_{2}e^{\lambda
_{2}t}.}
where λ1 and λ2 are the eigenvalues, and
(k1, k2), (k3, k4) are the basic eigenvectors.
The constants c1 and c2 account for the nonuniqueness
of eigenvectors and are not solvable unless
an initial condition is given for the system.
The above determinant leads to the characteristic
polynomial:
λ
2
−
(
A
+
D
)
λ
+
(
A
D
−
B
C
)
=
0
{\displaystyle \lambda ^{2}-(A+D)\lambda +(AD-BC)=0}
which is just a quadratic equation of the
form:
λ
2
−
p
λ
+
q
=
0
{\displaystyle \lambda ^{2}-p\lambda +q=0}
where;
p
=
A
+
D
=
t
r
(
A
)
,
{\displaystyle p=A+D=\mathrm {tr} (\mathbf
{A} )\,,}
("tr" denotes trace) and
q
=
A
D
−
B
C
=
det
(
A
)
.
{\displaystyle q=AD-BC=\det(\mathbf {A} )\,.}
The explicit solution of the eigenvalues are
then given by the quadratic formula:
λ
=
1
2
(
p
±
Δ
)
{\displaystyle \lambda ={\frac {1}{2}}(p\pm
{\sqrt {\Delta }})\,}
where
Δ
=
p
2
−
4
q
.
{\displaystyle \Delta =p^{2}-4q\,.}
=== Eigenvectors and nodes ===
The eigenvectors and nodes determine the profile
of the phase paths, providing a pictorial
interpretation of the solution to the dynamical
system, as shown next.
The phase plane is then first set-up by drawing
straight lines representing the two eigenvectors
(which represent stable situations where the
system either converges towards those lines
or diverges away from them). Then the phase
plane is plotted by using full lines instead
of direction field dashes. The signs of the
eigenvalues will tell how the system's phase
plane behaves:
If the signs are opposite, the intersection
of the eigenvectors is a saddle point.
If the signs are both positive, the eigenvectors
represent stable situations that the system
diverges away from, and the intersection is
an unstable node.
If the signs are both negative, the eigenvectors
represent stable situations that the system
converges towards, and the intersection is
a stable node.The above can be visualized
by recalling the behaviour of exponential
terms in differential equation solutions.
=== Repeated eigenvalues ===
This example covers only the case for real,
separate eigenvalues. Real, repeated eigenvalues
require solving the coefficient matrix with
an unknown vector and the first eigenvector
to generate the second solution of a two-by-two
system. However, if the matrix is symmetric,
it is possible to use the orthogonal eigenvector
to generate the second solution.
=== Complex eigenvalues ===
Complex eigenvalues and eigenvectors generate
solutions in the form of sines and cosines
as well as exponentials. One of the simplicities
in this situation is that only one of the
eigenvalues and one of the eigen vectors is
needed to generate the full solution set for
the system.
== See also ==
Phase line, 1-dimensional case
Phase space, n-dimensional case
Phase portrait
