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)
Theorem: (Hartman-Grobman Theorem)
Suppose that x∗ is a hyperbolic equilibrium of the nonlinear system
d~x
= F~ (~x).
dt
Then the stability of x∗ can be determined from the linearized system
d~x
= J(x∗ )~x,
dt
where J(x∗ ) is the Jacobian matrix of F~ at x∗ .
Note: 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.
Example: Consider the nonlinear system
dx1
= −2 sin x1
dt
dx2
= −x2 ex1 .
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.
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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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