1. Governor’s 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(n+1) = x(n) + x(n-1)

3. 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) = r n A(0)

4. 1 st Generalization A(n+1)/A(n) = r(n) [nonconstant] A(n+1) = r(n) A(n), A(0) = A 0 What’s the PGF? A(1) = r(0) A 0 A(2) = r(0) r(1) A 0 A(3) = r(0) r(1) r(2) A 0 … A(n) = r(0) r(1) … r(n-1) A 0

5. Old vs. New

6. Big Generalization: Difference Equation Pattern generated by the rule: x(n+1) = f(x(n)) with x(0) = x 0 Called a difference equation or a dynamical system Iterates: x 0, f(x 0 ), f(f(x 0 )), … Write: f k (x 0 ) = f(f(…f(x 0 ))…) (k-times) Orbit O + (x 0 )={x 0, f(x 0 ), f 2 (x 0 ), f 3 (x 0 ), …}

7. Big Question: Given x 0 and f, can you predict the behavior of the orbit O + (x 0 )? Does it tend to one value? go off to infinity? oscillate between values? do none of the above?

8. Linear 1 st 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)

9. 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

10. 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) = x 2, solve x 2 = x, get two equillbrium points: x*=1, x*=0

11. Example of graph technique

12. 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

13. Experiments for f(x)=x 2 X(0) = 0.9 X(0) = 1.1 X(0) = -0.1

14. Cobweb Plot Plot y = f(x) and y = x on same axis Plot (x 0,f(x 0 )) Move horizontally to y = x Move vertically to y = f(x)

15. Theory Worksheet: Draw Cobwebs around Equilibrium Points How does angle of crossing between y=x and y=f(x) affect answer?

16. Teams Team 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

17. 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

18. Lab Today Study various types of DE to find: 1.Equilibrium points 2.When stable/unstable 3.Other patterns

19. Done