> #OK to post #Tim Nasralla, hw24, 11/29/21
First Written: Nov. 2021
This is DMB.txt, A Maple package to explore Dynamical models in Biology (both
discrete and continuous)
accompanying the class Dynamical Models in Biology, Rutgers University.
Taught by Dr. Z. (Doron Zeilbeger)
The most current version is available on WWW at:
http://sites.math.rutgers.edu/~zeilberg/tokhniot/DMB.txt .
Please report all bugs to: DoronZeil at gmail dot com .
For general help, and a list of the MAIN functions,
type "Help();". For specific help type "Help(procedure_name);"
------------------------------
For a list of the supporting functions type: Help1();
For help with any of them type: Help(ProcedureName);
------------------------------
For a list of the functions that give examples of Discrete-time dynamical
systems (some famous), type: HelpDDM();
For help with any of them type: Help(ProcedureName);
------------------------------
For a list of the functions continuous-time dynamical systems (some famous)
type: HelpCDM();
For help with any of them type: Help(ProcedureName);
------------------------------
> #Question 3ii: Find when the differential equation (found in pdf) reaches the
> ground using dsolve
> dsolve({D(D(x))(t)-2*D(x)(t)+9.81=0,x(0)=100,D(x)(0)=0},x(t))
981 981 40981
x(t) = - --- exp(2 t) + --- t + -----
400 200 400
> evalf(solve(0 = -(981*exp(2*t))/400 + (981*t)/200 + 40981/400, t))
-20.88735984, 1.90989362
> #Since time cannot be negative, the solution for when the ball reaches the
> ground would approximately be 1.91 seconds
> #Question 5: Using the Orb function, find stable fixed points of the given
> discrete time dynamical systems. #i, f(x) = x+1 / x+2
> Orb([(x+1)/(x+2)], [x], [0.5], 1000, 1010)
[[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888],
[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888],
[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888]]
> Orb([(x+1)/(x+2)], [x], [0.7], 1000, 1010)
[[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888],
[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888],
[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888]]
> Orb([(x+1)/(x+2)], [x], [0], 1000, 1010)
[[4224696333392304878706725602341482782579852840250681098010280137314308584370\
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6571558458025510084617009128909], [51960329961200773146352548068399232049576\
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> #Seemingly, for function i, the only fixed point is 0.618.
> #ii, f(x) = 5/2*x - 5/2*x^2
> Orb([5/2*x-5/2*x^2], [x], [0], 1000, 1010)
[[0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0]]
> Orb([5/2*x-5/2*x^2], [x], [0.01], 1000, 1010)
[[0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995],
[0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995],
[0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995]]
> Orb([5/2*x-5/2*x^2], [x], [0.5], 1000, 1010)
[[0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005],
[0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005],
[0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005]]
> Orb([5/2*x-5/2*x^2], [x], [0.65], 1000, 1010)
[[0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990],
[0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990],
[0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990]]
> Orb([5/2*x-5/2*x^2], [x], [-0.5], 1000, 1010)
[[-Float(infinity)], [-Float(infinity)], [-Float(infinity)],
[-Float(infinity)], [-Float(infinity)], [-Float(infinity)],
[-Float(infinity)], [-Float(infinity)], [-Float(infinity)],
[-Float(infinity)], [-Float(infinity)], [-Float(infinity)]]
> #The only stable fixed point is seemingly x= 0.6, or 3/5s.
> #iii, f(x) = 7/2*x - 7/2*x^2
> Orb([7/2*x-7/2*x^2], [x], [0], 1000, 1010)
[[0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0]]
> Orb([7/2*x-7/2*x^2], [x], [0.1], 1000, 1010)
[[0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060],
[0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212],
[0.8749972637], [0.382819683]]
> Orb([7/2*x-7/2*x^2], [x], [0.5], 1000, 1010)
[[0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212],
[0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637],
[0.382819683], [0.8269407060]]
> Orb([7/2*x-7/2*x^2], [x], [0.8269407060], 1000, 1010)
[[0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060],
[0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212],
[0.8749972637], [0.382819683]]
> Orb([7/2*x-7/2*x^2], [x], [0.7], 1000, 1010)
[[0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683],
[0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060],
[0.500884212], [0.8749972637]]
> Orb([7/2*x-7/2*x^2], [x], [-0.5], 1000, 1010)
[[-Float(infinity)], [-Float(infinity)], [-Float(infinity)],
[-Float(infinity)], [-Float(infinity)], [-Float(infinity)],
[-Float(infinity)], [-Float(infinity)], [-Float(infinity)],
[-Float(infinity)], [-Float(infinity)], [-Float(infinity)]]
> #There are seemingly no stable fixed points for this function.