Ordinary Differential Equations (ODEs)
• Differential equations are the ubiquitous, the
lingua franca of the sciences; many different
fields are linked by having similar differential
equations
• ODEs have one independent variable; PDEs
have more
• Examples: electrical circuits
Newtonian mechanics
chemical reactions
population dynamics
economics… and so on, ad infinitum
Example: RLC circuit
di 1
Ri L   i dt  V
dt C
2
d q R dq 1
V


q
2
dt
L dt LC
L
To illustrate: Population dynamics
• 1798 Malthusian catastrophe
• 1838 Verhulst, logistic growth
• Predator-prey systems, Volterra-Lotka
Population dynamics
• Malthus:
dN
 rN
dt
• Verhulst:
Logistic
growth
dN
N
 r (1 ) N
dT
K
N
N  N 0e


1
N0
N 0 rt
 (1  )e
K
K
N0
rt
Population dynamics
Hudson Bay Company
Population dynamics
V .Volterra, commercial fishing in the Adriatic
x1  biomass of predators
x2  biomass of prey
x1
 b12 x2  a1
x1
x2
 a2  b21x1
x2
In the x1-x2 plane
dx2 x2 a2  b21x1
 (
)
dx1 x1 b12 x2  a1
State space
dx2 x2 a2  b21x1
 (
)
dx1 x1 b12 x2  a1
Integrate analytically!
b12 x2  a1 ln x2  const.  b21x1  a2 ln x1
Produces a family of concentric
closed curves as shown… How to
compute?
Population dynamics
x1
 b12 x2  a1
x1
self-limiting term
x2
 a2  b21x1  c22 x2  stable focus
x2
Delay  limit cycle
As functions of time
Do you believe this?
• Do hares eat lynx, Gilpin 1973
Do Hares Eat Lynx?
Michael E. Gilpin
The American Naturalist, Vol. 107, No. 957 (Sep. - Oct.,
1973), pp. 727-730
Published by: The University of Chicago Press for The
American Society of Naturalists
Stable URL: http://www.jstor.org/stable/2459670
Putting equations in state-space form
y   y   y  f (t )
 x  Ax  bf (t )
x1  y; x2  y ; so
x1  x2
x2  y  f (t )  y  y
 f (t )   x2   x1
Traditional state space:
x vs. x plane
Example: the (nonlinear) pendulum
  ( g / ) sin   0
McMaster
Linear pendulum: small θ
For simplicity, let g/l = 1
    0
  sin t
  cos t
 2   2  1
Circles!
Pendulum in the phase plane
Varieties of Behavior
• Stable focus
• Periodic
• Limit cycle
Varieties of Behavior
•
•
•
•
Stable focus
Periodic
Limit cycle
Chaos
…Assignment
Numerical integration of ODEs
• Euler’s Method  simple-minded, basis of
many others
• Predictor-corrector methods  can be
useful
• Runge-Kutta (usually 4th-order)
workhorse, good enough for our work,
but not state-of-the-art
Criteria for evaluating
• Accuracy  use Taylor series, big-Oh,
classical numerical analysis
• Efficiency  running time may be hard to
predict, sometimes step size is adaptive
• Stability  some methods diverge on
some problems
Euler
• Local error = O(h2)
• Global accumulated) error = O(h)
(Roughly: multiply by T/h )
Euler
• Local error = O(h2)
• Global (accumulated)
error = O(h)
x1  x0
x0  f ( x0 ) 
h
x1  x0  h f ( x0 )
Euler step
Euler
• Local error = O(h2)
• Global (accumulated)
error = O(h)
2
h
x1  x0  h f ( x0 ) 
f ( )
2
Euler step
Taylor’s series
with remainder
Second-order Runge-Kutta
(midpoint method)
• Local error = O(h3)
• Global (accumulated) error = O(h2)
Fourth-order Runge-Kutta
• Local error = O(h5)
• Global (accumulated) error = O(h4)
Additional topics
• Stability, stiff systems
• Implicit methods
• Two-point boundary-value problems
shooting methods
relaxation methods