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# SpTp MATH 3903

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This 4 page Class Notes was uploaded by Ms. Skylar Feil on Tuesday October 20, 2015. The Class Notes belongs to MATH 3903 at Southern Polytechnic State University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/225442/math-3903-southern-polytechnic-state-university in Mathematics (M) at Southern Polytechnic State University.

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Date Created: 10/20/15

Review Problems Part I 1 LetFx x2 1 Find the rst ve points on the orbit of 0 2 LetFx x2 3 Compute F2x and F3x 3 LetFx x2 Compute F2x F3x and F4 x What is the formula forFquot x 4 Find all real xed points for each of the following functions and classify them as attracting repelling or neutral 1 Fx x2 2 2 Fx3x2 3 Fx x3 4 Fxx3 3x 2x 0 S x lt 05 5 Let Dx be the doubling function 2x 1 05 S x lt 1 1 Discuss the behaVior of the resulting orbit of 02 under D 2 Discuss the behaVior of the resulting orbit of 07 under D 3 Find a 3cyc1e for the doubling function Dx in problem 4 6 Use graphical analysis to describe the fate of all orbits forF x x3 7 LetF x 1 x2 Verify that 0 lies in a 2cyc1e and classify the 2cy1ce as attracting repelling or neutral 8 Suppose that Fx has a neutral xed point at x0 withF39x0 1 Suppose also that F quotx0 gt 0 What can you say about x0 is x0 weakly attracting weakly repelling or neither Use graphical analysis and the concaVity of the graph of F x near x0 to support your answer 9 LetF x x2 x 2 Describe the bifurcation that occurs at A 0 10 ConsiderF x x3 xix Find a 2cyc1e and discuss the bifurcation at t 1 11 Prove that if F a b gt a b is continuous thenF has a xed point ina b 12 Each of the following functions has a neutral xed point Find this xed point and using graphical analysis with an accurate graph determine if it is weakly attracting weakly repelling or neither 1 F x x x2 2 Fxxx3 Review Problems Part 11 14 Consider the quadratic family Q5 x x2 c 1 Ifc gt i what will happen to any orbit 2 Ifc how many xed points does Qx have Find the xed points describe the nature of the xed points attracting repelling or neutral 3 Ifc lt how many xed points does Qx have Find the xed points describe the nature of the xed points attracting repelling or neutral 4 Ifc lt 3 Qx has a2cyc1e given by q 1 4c 3 1 5 3 q E 1 x 40 3 Prove that this cycle is attracting if Z lt c lt 2 neutral 5 5 ifc 7andre ellin ifclt i 4 p g 4 56 Consider the logistic familyFx xix1 x 5 For which values of A does F A have an attracting xed point at x 0 6 For which values of A does F A have a nonzero attracting xed point 1 3x if x S 7 79 Consider the function Tx 12 3 3x if x gt i 2 7 Find the xed points forT 8 Prove that lies on a 3cyc1e for T l 2 9 Show that if x e g 3 then Tquotx gt oo as n gt oo 10 Consider the shift mapO39 Z gt Z 1 Find the xed points fora 2 Find all 2cyc1es fora 3 List all 3cycles fora 4 List all 4cycles fora 5 Find a point in 2 such that its orbit is dense in Z 11 Computer ds t where 1 FE r a 2 s r 12 Find all points in 2 whose distance from 0000 is exactly 13 Give an example ofa sequence midway between 0000 and 1111 Give a second such example Are there any other such points Why or why not 14LetM01sele00sl1 andMlmSEZSO1SI0SZ1 1 What is the minimum distance between a point in M01 and a point ian01 Give an example of two sequences that are this close to each other 2 What is the minimum distance between a point in M 01 and a point inM 101 Give an example of two sequences that are this far apart 15 Let S be the set of all real numbers in 0 1 except those ofthe form 1 for n1 2 3 n Is S dense in 0 1 16 What are the three properties of a chaotic system Give two examples of chaotic system 17 Show that the shift map 039 Z gt Z is chaotic 18 Let Vx2x2 Find the xed points of V Compute an explicit formula for V2 and a 2cycle for V Show that V is chaotic 19 LetQxx2cA gtAwhere clt 2 aZ gtZ and SA gtZ bethe itinerary map 1 Prove that S 0 Q5quot 039quot o S for any positive integer n 2Prove that if s is a periodic point of period n for039 then S391s is a periodic point of period n for Q5 20 Prove that a continuous function that has a ncycle where ngt1 must also have a fixed point 21 Can a continuous function on R have a periodic point of period 3 and not one of period 2 Why 22 Can a continuous function on R have a periodic point of period 12 and not one of period 6 Why 23 Can a continuous function on R have a periodic point of period 48 and not one of period 56 Why 24 Give an example ofa function F 0 1 gt 0 1 that has a periodic point ofperiod 3 and no other periodic points Can this happen 25 Let La 2 12 where a p elm 1 What will happen to any orbit when p lt 1 2 What will happen to any orbit when p gt 1 3 What will happen to any orbit when p 1 and r is rational 4 What will happen to any orbit when p 1 and r is irrational 26 ForQz z2 c where z is a complex variable and c is a complex constant 1 What is the filled Julia set 2 What is the Julia set 3 What is the Mandelbrot set 4 Describe the lled Julia set and the Julia set whenc 0 c 2 and c gt 2

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