Governors School for the Sciences

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Governors School for the Sciences

• Mathematics

Day 3
2
MOTD Leonardo Fibonacci

• 1170 to 1250 (Italy)
• Popularized ancient mathematics
• Solved problems in algebra, geometry and number
  theory
• Best know for the Fibonacci sequence x(n1)
  x(n) x(n-1)

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Geometric Patterns

• Sequence 1, 2, 4, 8, 16, generated by the
  obvious rule A(n1) 2A(n)
• Any geometric sequence is expressed
  A(n1) r A(n)
• Identify r A(n1)/A(n) constant
• PGF is exponential A(n) rn A(0)

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1st Generalization

• A(n1)/A(n) r(n) nonconstant
• A(n1) r(n) A(n), A(0) A0
• Whats the PGF? A(1) r(0) A0 A(2)
  r(0) r(1) A0 A(3) r(0) r(1) r(2) A0
  A(n) r(0) r(1) r(n-1) A0

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Old vs. New
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Big Generalization Difference Equation

• Pattern generated by the rule x(n1)
  f(x(n)) with x(0) x0
• Called a difference equation or a dynamical
  system
• Iterates x0, f(x0), f(f(x0)),
• Write fk(x0) f(f(f(x0))) (k-times)
• Orbit O(x0)x0, f(x0), f2(x0), f3(x0),

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Big Question

• Given x0 and f, can you predict the behavior of
  the orbit O(x0)?
• Does it tend to one value? go off to
  infinity? oscillate between values? do none
  of the above?

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Linear 1st Order DE

• x(n1) a(n)x(n) c(n)
• c(n) 0 homogeneous else non-homogeneous
• Know if a(n) lt 1 and c(n) 0 then every orbit
  tends toward 0
• If a(n) a, alt1 and c(n) c then every orbit
  tends toward c/(1-a)

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General Answer

• Except for simple cases it is hard or impossible
  to find a solution of a DE and analyze orbits
  that way
• Instead look at Equilibrium Points
  Stability Theory

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Equlibrium Points

• Equilibrium Point Point x such that f(x) x
  (fixed point)
• If x(0) x, then x(k) x for all k
• Solve via algebra or by graphical technique
• Eg f(x) x2, solve x2 x, get two equillbrium
  points x1, x0

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Example of graph technique
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Stability Theory

• What happens if x(0) is near an equilibrium point
  x?
• If x(n) stays near x x is stable or
  attracting
• If x(n) moves away from x x is unstable
  or repelling
• Determine experimentally or by a Cobweb
  Diagram

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Experiments for f(x)x2
X(0) 0.9
X(0) 1.1
X(0) -0.1
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Cobweb Plot

• Plot y f(x) and y x on same axis
• Plot (x0,f(x0))
• Move horizontally to y x
• Move vertically to y f(x)

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Theory

• Worksheet Draw Cobwebs around Equilibrium
  Points
• How does angle of crossing between yx and yf(x)
  affect answer?

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Teams

• Team 3
• Austin Chu
• Michelle Sarwar
• Jennifer Soun
• Matt Zimmerman

• team one p
• Sam Barrett
• Clay Francis
• Michael Hammond
• Angela Wilcox

• Dr. Collins
• Charlie Fu
• Scott McKinney
• Steve White
• Lena Zurkiya

• Denominators of Doom
• Stuart Elston
• Chris Goodson
• Meara Knowles
• Charlie Wright

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Math Bowl Competition

• About 1 minute per question
• 5 questions
• 10 points right, 0 points wrong, 4 points for no
  answer
• Winning team gets additional 50 pts
• Today Team 1 vs. Team 2 Team 3 vs.
  Team 4

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Lab Today

• Study various types of DE to find
• Equilibrium points
• When stable/unstable
• Other patterns

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Done