CEE262C Lecture 2: Nonlinear ODEs
and Phase Diagrams
Overview
• Nonlinear chaotic ODEs: the damped nonlinear
forced pendulum
• 2nd Order damped harmonic oscillator
• Systems of ODEs
• Phase diagrams
– Fixed points
– Isoclines/Nullclines
References: Dym, Ch 7; Mooney & Swift, Ch 5.2-5.3; Kreyszig, Ch 4
CEE262C Lecture 2: Nonlinear ODEs and phase diagrams
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Forced pendulum
Frictional effect
m
m
g
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Free-body diagram
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Derivation of the governing ODE
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m
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Reduce and nondimensionalize!
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Governing nondimensional ODE
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Linearize
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The damped harmonic oscillator
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The particular solution
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Simulating the nonlinear system
pendulum.zip
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Phase plane analysis
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Direction field for a1=0.5
phasedirection.m
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24
Computing phase lines analytically
Solution in phase space
Elliptic Integral!
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Analytical Phase Lines for










CEE262C Lecture 2: Nonlinear ODEs and phase diagrams


21
Nullclines and fixed points
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Plotting nullclines and fixed points

q=0 (no acceleration)

increasing
friction



p=0 (no velocity)









Fixed points
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Behavior in the vicinity of fixed points
Suppose we have a nonlinear coupled set of ODEs in the form
du
 p (u , v)
dt
dv
 q (u , v)
dt
We can determine the behavior of this ODE in the vicinity of the
fixed points by analyzing the behavior of disturbances applied to
the fixed points such that
u  u0  u '
v  v0  v'
where the point u0 , v0 is a fixed point corresponding to
p(u0 , v0 )  q(u0 , v0 )  0
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Using the Taylor series expansion about the fixed point, we have
du
p
p
 p (u , v)  p (u0  u ' , v0  v' )  p (u0 , v0 )  u '
 v'
dt
u u0
v v0
dv
q
q
 q (u , v)  q (u0  u ' , v0  v' )  q (u0 , v0 )  u '
 v'
dt
u u0
v v0
Substitution into the ODEs gives
d
p
p
(u0  u ' )  p (u0 , v0 )  u '
 v'
dt
u u0
v v0
d
q
q
(v0  v' )  q (u0 , v0 )  u '
 v'
dt
u u0
v v0
Since the fixed points satisfy
p(u0 , v0 )  0
q(u0 , v0 )  0
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and
du0 dv0

 0 , then the perturbations satisfy
dt
dt
du '
p
p
 u'
 v'
dt
u u0
v v0
dv'
q
q
 u'
 v'
dt
u u0
v v0
In vector form, this is given by
 p

d  u '   u u0
   
q
dt  v' 

 u u0
The Jacobian matrix is given by
 p

J u0 , v0    u
 q

 u
p 

v v0  u ' 
 
q  v' 

v v0 
p 

v 
q 

v u u0 ,v v0
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The behavior of the solution in the phase plane in the vicinity of
the fixed points is determined by the behavior of the
eigenvalues of the Jacobian.
If
a b 

J u0 , v0   
c d 
then the eigenvalues of J are given by

   2  4
2
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two real negative
roots.
complex pair,
negative real part.
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two real positive
roots.
complex pair,
positive real part.
pure imaginary.
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Phase plane analysis for the
pendulum
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Underdamped
Critical or overdamped
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Spiral direction CW or CCW?
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Clockwise
c<0
Counterclockwise
c>0
33
Behavior around saddle point
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