Chapter 4. Systems of ODEs. Phase Plane.
Qualitative Methods
4.5 Qualitative Methods for Nonlinear Systems
• Qualitative methods are methods of obtaining qualitative information on solutions
without actually solving a system. These methods are particularly valuable for systems
whose solution by analytic methods is di!cult or impossible.
• Consider a nonlinear system
!
→ y1→ = f1 (y1 , y2 )
y = f (y), =↑ (1)
y2→ = f2 (y1 , y2 )
We assume that (1) is autonomous, that is, the independent variable t does not occur
explicitly.
• (1) may have several critical points. If P0 : (a, b) is a critical point with (a, b) not at
the origin (0, 0), then we apply the translation
y"1 = y1 ↓ a, y"2 = y2 ↓ b,
which moves P0 to (0, 0) as desired.
• We also assume that P0 is isolated, that is, it is the only critical point of (1) within
a (su!ciently small) disk with center at the origin. If (1) has only finitely many critical
points, that is automatically true.
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Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
Linearization of nonlinear systems
How can we determine the kind and stability property of a critical point P0 (0, 0) of
(1)? In most cases this can be done by linearization of (1) near P0 , writing (1) as
y→ = f (y) = Ay + h(y) and dropping h(y), as follows.
!
→ y1→ = a11 y1 + a12 y2 + h1 (y1 , y2 )
y = A(y) + h(y), (2)
y2→ = a21 y1 + a22 y2 + h2 (y1 , y2 )
Theorem 1. Linearization
If f1 and f2 in (1) are continuous and have continuous partial derivatives in a
neighborhood of the critical point P0 (0, 0), and if det A ↔= 0 in (2), then the kind
and stability of the critical point of (1) are the same as those of the linearized
system !
→ y1→ = a11 y1 + a12 y2
y = A(y), (3)
y2→ = a21 y1 + a22 y2
Exceptions occur if A has equal or pure imaginary eigenvalues; then (1) may have
the same kind of critical point as (3) or a spiral point.
Note that Taylor series of f (x, y) at (a, b)
f (x, y) = f (a, b) + [fx (a, b)(x ↓ a) + fy (a, b)(y ↓ b)]
1# $
+ fxx (a, b)(x ↓ a)2 + 2fxy (x ↓ a)(y ↓ b) + fyy (a, b)(y ↓ b)2
2!
+ ···
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Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
Example 1. Free Undamped Pendulum, Linearization
Figure shows a pendulum consisting of a body of mass m and a rod of length L.
Determine the locations and types of the critical points. Assume that the mass of
the rod and air resistance are negligible.
1. Set up the mathematical model
when ω is very small, we can approximate sin ω rather accurately by ω and obtain
↗ ↗
as an approximate solution A cos kt + B sin kt, but the exact solution for any
ω is not an elementary function.
2. Critical points
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Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
3. Linearization at critical points (0, 0)
4. Linearization at critical points (nε, 0), n = ±2, ±4, · · ·
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Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
5. Linearization at critical points (nε, 0), n = ±1, ±3, · · ·
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Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
Optional : Lotka-Volterra Population Model
Example 3. Predator-Prey Model
This model concerns two species, say, rabbits and foxes.
1. Rabbits have unlimited food supply. Hence, if there were no foxes, their number
y1 (t) would grow exponentially, y1→ = ay1
2. Actually, y1 is decreased because of the kill by foxes, say, at a rate proportional
to y1 y2 , where y2 (t) is the number of foxes. Hence y1→ = ay1 ↓ by1 y2 , where
a > 0 and b > 0.
3. If there were no rabbits, then y2 (t) would exponentially decrease to zero,
y2→ = ↓ly2 . However, y2 is increased by a rate proportional to the number of
encounters between predator and prey; together we have y2→ = ↓ly2 + ky1 y2 ,
where k > 0 and l > 0.
This gives the Lotka-Volterra System
y1→ = ay1 ↓ by1 y2
y2→ = ky1 y2 ↓ ly2
1. Linearization at (0, 0)
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Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
%l a
&
2. Linearization at k, b
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Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
Result of solution by solving nonlinear system. (Set a = 1, b = 0.01, k = 0.02, l = 1)
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