Mudassir Lodi
Homework for Lecture 13 of Dr. Z.’s Dynamical Models in Biology class
Email the answers (either as .pdf file or .txt file) to ShaloshBEkhad@gmail.com by 8:00pm Monday, Oct. 18,, 2021. Subject: hw13 with an attachment hw13FirstLast.pdf and/or hw13FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
1.	Read and understand the Maple Code https://sites.math.rutgers.edu/~zeilberg/Bio21/M13.txt .
2.	If ai is the i-th digit of your RUID, find the fixed points, and the stable fixed points of the following first-order system of two quantities x(n) and y(n)
X(n) = (2x(n-1)^2 + x(n-1) + 1)/(2x(n-1)^2 + x(n-1) + 5)
Y(n) = 3x(n-1)^2 + x(n-1) + 1/x(n-1)^2 + x(n-1) + 8

If it has a stable fixed point, confirm it using Orb2 with intial conditions x(0) = a2 + 0.5, y(0) = a4 + 0.5.

3.	By running RT2(x,y,d,K) with d=1 and K=100, generate 20 random transformations, and find, for each the list of stable equilibria (if they exist). Then confirm it with Orb2 with random initial conditions.
Warning: The Maple code sometimes misses some solutions (the solve command does not always give all solutions) SFP2 may give (erroneusly) the empty list.
[(93. + 45.*y + 96.*x)/(6. + 98.*y + 59.*x), (44. + 100.*y + 38.*x)/(69. + 27.*y + 96.*x)]
[(17. + 90.*y + 34.*x)/(18. + 52.*y + 56.*x), (43. + 83.*y + 25.*x)/(90. + 93.*y + 60.*x)]
[(93. + 14.*y + 50.*x)/(47. + 8.*y + 46.*x), (44. + 9.*y + 77.*x)/(59. + 16.*y + x)]
[(70. + 77.*y + 39.*x)/(92. + 71.*y + 67.*x), (78. + 51.*y + 53.*x)/(12. + 19.*y + 63.*x)]
[(40. + 90.*y + 3.*x)/(49. + 49.*y + 67.*x), (74. + 90.*y + 74.*x)/(27. + 98.*y + 72.*x)]
[(2. + 73.*y + 85.*x)/(41. + 4.*y + 44.*x), (13. + 19.*y + 10.*x)/(15. + 64.*y + 9.*x)]
[(12. + 52.*y + 25.*x)/(72. + 90.*y + 18.*x), (43. + 55.*y + 40.*x)/(17. + 70.*y + 52.*x)]
[(81. + 87.*y + 34.*x)/(85. + 9.*y + 68.*x), (83. + 63.*y + 100.*x)/(70. + 36.*y + 36.*x)]
[(10. + 40.*y + 66.*x)/(87. + 16.*y + 98.*x), (43. + 53.*y + 61.*x)/(47. + 28.*y + 75.*x)]
[(3. + 5.*y + 11.*x)/(37. + 75.*y + 4.*x), (91. + 22.*y + 40.*x)/(58. + 93.*y + 98.*x)]
[(11. + 30.*y + 6.*x)/(32. + 40.*y + 24.*x), (80. + 96.*y + 11.*x)/(23. + 41.*y + 52.*x)]
[(58. + 67.*y + 81.*x)/(65. + 69.*y + 2.*x), (36. + 61.*y + 84.*x)/(96. + 94.*y + 31.*x)]
[(81. + 31.*y + 54.*x)/(67. + 59.*y + 66.*x), (12. + 49.*y + 90.*x)/(35. + 15.*y + 26.*x)]
[(100. + 24.*y + 8.*x)/(63. + 78.*y + 23.*x), (73. + 22.*y + 32.*x)/(98. + 9.*y + 53.*x)]
[(3. + 98.*y + 69.*x)/(3. + 73.*y + 88.*x), (37. + 60.*y + 94.*x)/(52. + 16.*y + 29.*x)]
[(51. + 3.*y + 45.*x)/(67. + 40.*y + 71.*x), (74. + 49.*y + 60.*x)/(69. + 33.*y + 30.*x)]
[(1. + 83.*y + 9.*x)/(64. + 43.*y + 57.*x), (52. + 62.*y + 46.*x)/(76. + 9.*y + 53.*x)]
[(37. + 88.*y + 50.*x)/(37. + 76.*y + 95.*x), (8. + 92.*y + 92.*x)/(2. + 97.*y + 44.*x)]
[(9. + 30.*y + 14.*x)/(79. + 73.*y + 21.*x), (78. + 49.*y + 93.*x)/(15. + 56.*y + 69.*x)]
[(17. + 21.*y + 42.*x)/(21. + 5.*y + 58.*x), (3. + 86.*y + 55.*x)/(97. + 4.*y + 92.*x)]
Added Oct. 17, 2021: I wrote a procedure FP2drz that finds all the fixed points (the Maple solve command sometimes does not find all of them), and then SFP2drz that does the same thing as SFP2 but is more reliable. You are welcome to use the latter.

4.
(i)	Write three-variable analog of RT2(x,y,d,K), call it RT3(x,y,z,d,K).
(ii)	Write three-variable analog of Orb2(F,x,y,pt0,K1,K2) call it Orb3(F,x,y,z,pt0,K1,K2) .
(iii)	Write three-variable analog of Orb2(F,x,y,pt0,K1,K2) call it Orb3(F,x,y,z,pt0,K1,K2) .
(iv)	Write three-variable analog of FP2(F,x,y) call it FP3(F,x,y,z) .
(v)	Write three-variable analog of SFP2(F,x,y) call it SFP3(F,x,y,z) .

5.	By running your RT3(x,y,z,d,K) with d=1 and K=100, generate 10 random transformations, and find, for each the list of stable equilibria (if they exist). Then confirm it with your Orb3 with random initial conditions.
[(88.*y*z + 34.*x + 46.)/(49.*y*z + 61.*x + 68.), (86.*y*z + 42.*x + 21.)/(33.*y*z + 77.*x + 5.)]
(58.*y*z + 98.*x + 98.)/(65.*y*z + 29.*x + 29.), (29.*y*z + 34.*x + 35.)/(44.*y*z + 60.*x + 66.)
[(32.*y*z + 85.*x + 83.)/(68.*y*z + 59.*x + 100.), (76.*y*z + 92.*x + 40.)/(17.*y*z + 50.*x + 39.)]
(20.*y*z + 18.*x + 78.)/(51.*y*z + 34.*x + 18.), (10.*y*z + 52.*x + 78.)/(13.*y*z + 87.*x + 100.)
[(37.*y*z + 92.*x + 13.)/(69.*y*z + 62.*x + 97.), (60.*y*z + 46.*x + 38.)/(61.*y*z + 80.*x + 78.)]
[(72.*y*z + 48.*x + 3.)/(41.*y*z + 46.*x + 9.), (35.*y*z + 88.*x + 78.)/(26.*y*z + 27.*x + 79.)]
[(83.*y*z + 55.*x + 76.)/(72.*y*z + 95.*x + 35.), (55.*y*z + 81.*x + 100.)/(88.*y*z + 20.*x + 38.)]
[(17.*y*z + 68.*x + 16.)/(48.*y*z + 67.*x + 79.), (86.*y*z + 92.*x + 98.)/(33.*y*z + 55.*x + 74.)]
[(82.*y*z + 25.*x + 17.)/(24.*y*z + 60.*x + 94.), (17.*y*z + 14.*x + 74.)/(87.*y*z + 79.*x + 12.)]
(7.*y*z + 69.*x + 64.)/(83.*y*z + 3.*x + 90.), (16.*y*z + 84.*x + 48.)/(41.*y*z + 53.*x + 63.)
Warning: The Maple code sometimes misses some solutions (the solve command does not always give all solutions) your SFF3 may give (erroneusly) the empty list.