Section 11.3: Nonlinear Systems (Theory)
In this section, we consider systems of differential equations of the form
dx1
dt
dx2
dt
..
.
dxn
dt
= f1 (x1 , x2 , . . . , xn )
= f2 (x1 , x2 , . . . , xn )
.
= ..
= fn (x1 , x2 , . . . , xn )
where fj is a real-valued function for j = 1, 2, . . . , n.
This system can be expressed in matrix form as
d~x
= F~ (~x),
dt
where


x1 (t)
 x2 (t) 


~x(t) =  .. 
 . 

and

f1 (x1 , x2 , . . . , xn )
 f2 (x1 , x2 , . . . , xn ) 


~
F (~x) = 
.
..


.
xn (t)
fn (x1 , x2 , . . . , xn )
This system of equations is called a first-order nonlinear system. A solution of this
system is an ordered n-tuple of functions (x1 (t), x2 (t), . . . , xn (t)) that satisfy each of the
differential equations.
In general, it is not possible to find explicit solutions of a nonlinear system of differential
equations. However, the long-term behavior of nonlinear systems can be determined by
analyzing the stability of the equilibrium points.
Definition: An equilibrium of a first-order nonlinear system
d~x
= F~ (~x)
dt
is a solution x̄ = hx¯1 , x¯2 , . . . , x¯n i from which there is no change. That is,
dx̄
= F~ (x̄) = ~0.
dt
An equilibrium is called locally stable if solutions that begin close to that equilibrium
approach the equilibrium and unstable if solutions that begin close to that equilibrium
move away from it.
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Stability (Analytical Approach)
Suppose that x∗ is an equilibrium of the nonlinear system
d~x
= F~ (~x).
dt
To determine the stability of this equilibrium, we linearize the sytem of differential equations
about x∗ by finding the linearization of the vector function F~ (~x) about x∗ . That is,
d~x
= F~ (~x)
dt
d~x
≈ F~ (x∗ ) + J(x∗ )(~x − x∗ )
dt
d~x
≈ J(x∗ )(~x − x∗ ).
dt
Here, J(x∗ ) denotes the Jacobian matrix of F~ (~x) evaluated at x∗ . Provided that ~x is close
to x∗ , this is a good approximation.
Setting ~z = ~x − x∗ , it follows that
d
d~z
=
(~x − x∗ )
dt
dt
d~x
=
dt
≈ J(x∗ )(~x − x∗ )
= J(x∗ )~z.
Therefore, we have the approximation
d~z
≈ J(x∗ )~z.
dt
Provided that ~x is close to x∗ (~z ≈ ~0), this is a good approximation.
If the nonlinear system is in equilibrium (~x = x∗ ), then ~z = 0 and the stability of the
equilibrium x∗ can be determined by analyzing the stability of ~z = 0 for the linear system
d~z
= J(x∗ )~z.
dt
That is, the stability of x∗ is determined by computing the eigenvalues of J(x∗ ).
More formally, the Hartman-Grobman Theorem states that the linear system has
the same behavior as the nonlinear system close to an equilibrium, provided that the real
parts of the eigenvalues of J(x∗ ) are non-zero.
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Example: Consider the nonlinear system
dx1
= −2 sin x1
dt
dx2
= −x2 ex1 .
dt
Determine the stability of (0, 0) and classify the equilibrium.
Example: Consider the nonlinear system
dx1
= 3x1 x2 − x1 + x2
dt
dx2
= x22 − x1 .
dt
Determine the stability of (0, 0) and classify the equilibrium.
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Example: Find all equilibria of the nonlinear system
dx1
= x1 x2 − x2
dt
dx2
= x1 + x2
dt
and determine the stability of each equilibrium.
4
Example: Find all equilibria of the nonlinear system
dx1
= 2x1 (5 − x1 − x2 )
dt
dx2
= 3x2 (7 − 3x1 − x2 )
dt
and determine the stability of each equilibrium.
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Example: Assume that a > 0. Find all equilibria of the nonlinear system
dx1
= 1 − ax1 x2
dt
dx2
= ax1 x2 − x2
dt
and determine the stability of each equilibrium.
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Stability (Graphical Approach)
Consider a general system of two nonlinear differential equations
dx1
= f1 (x1 , x2 )
dt
dx2
= f2 (x1 , x2 ).
dt
The curves in the x1 x2 -plane defined by f1 (x1 , x2 ) = 0 and f2 (x1 , x2 ) = 0 are called the
nullclines or zero isoclines of the system. These curves represent points in the x1 x2 plane where the respective differential equations are equal to zero. Therefore, the points of
intersection of the nullclines are equilibrium points.
Example: Consider the nonlinear system
dx1
= x1 (9 − x1 − 2x2 )
dt
dx2
= x2 (9 − 2x1 − x2 ).
dt
(a) Graph the nullclines.
(b) Show that (3, 3) is an equilibrium, and use the analytical approach to determine its
stability.
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Example: The nullclines of a general nonlinear system
dx1
= f1 (x1 , x2 )
dt
dx2
= f2 (x1 , x2 )
dt
are illustrated below. Assume that both x1 and x2 increase close to the origin and that f1
and f2 change sign when crossing their respective nullclines. Find the sign structure of the
Jacobian matrix at the equilibrium and determine its stability, if possible.
(a)
x2
dx2
0
dt
dx1
0
dt
x1
(b)
x2
dx1
0
dt
dx2
0
dt
x1
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Example: Consider the nonlinear system
dx1
= x1 (2 − x21 ) − x1 x2
dt
dx2
= x1 x2 − x2
dt
(a) Graph the nullclines.
(b) Show that (1, 1) is an equilibrium, and use the graphical approach to determine its
stability.
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