Governor’s School for the Sciences

MathematicsDay 3

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(n+1) = x(n) + x(n-1)

Geometric Patterns

• Sequence 1, 2, 4, 8, 16, … generated by the obvious rule A(n+1) = 2A(n)

• Any geometric sequence is expressed A(n+1) = r A(n)

• Identify r = A(n+1)/A(n) [constant]• PGF is exponential: A(n) = rn A(0)

1st Generalization

• A(n+1)/A(n) = r(n) [nonconstant]• A(n+1) = r(n) A(n), A(0) = A0

• What’s 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

Old vs. New

Big Generalization: Difference Equation

• Pattern generated by the rule: x(n+1) = 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), …}

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?

Linear 1st Order DE

• x(n+1) = a(n)x(n) + c(n)• c(n) = 0: homogeneous; else: non-

homogeneous• Know if |a(n)| < 1 and c(n) = 0

then every orbit tends toward 0• If a(n) = a, |a|<1 and c(n) = c then

every orbit tends toward c/(1-a)

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

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: x*=1, x*=0

Example of graph technique

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

Experiments for f(x)=x2

X(0) = 0.9

X(0) = 1.1

X(0) = -0.1

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)

Xsqr1.jpg

Theory

• Worksheet: Draw Cobwebs around Equilibrium Points

• How does angle of crossing between y=x and y=f(x) affect answer?

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

• 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

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

Lab Today

Study various types of DE to find:1. Equilibrium points2. When stable/unstable3. Other patterns

Done