MA354
Project 2: Investigating the Long Term Behavior of Dynamical Systems
Using Mathematica (Details / Supplemental)
Exercise 1: (TURN-IN) Investigating the Fixed Points and Stability of the
Linear Affine System an+1 = ran + b
Prepare a set of 4 graphs that illustrate the steady states, if they exist, for each case described in
the table above. On each graph, show several plots of the system verses time (n) for different
values of a0. Carefully choose values of a0 that (i) show the steady states, if any, and (ii)
illustrate the stability of those steady states. (Details below.) For each case, describe what is
going on with the system (typical behavior, existence of fixed points, their stability) in your own
words with a sentence or two.
(A) Graph illustrating the steady states for the dynamical system when r=1 and b=0.
The dynamical system for this case would be an1  an with initial condition a 0 .
Show several plots of the dynamical system a n verses n (that is, how the system evolves
over time) for different initial conditions a 0 .
Note: there are a couple ways you can model the dynamical system. You can define the
system recursively or explicitly:
Way 1 (Recursively define the system):
A[0]=5;
(* Here choose your own initial condition. *)
A[n_]:=A[n-1];
Table1=Table[A[n],{n,0,50}];
A[0]=6;
(* Here choose your own initial condition. *)
A[n_]:=A[n-1];
Table2=Table[A[n],{n,0,50}];
(* Add other initial conditions... *)
ListPlot[{Table1,Table2},PlotJoined->True,PlotRange{-1,15}]
Way 2 (Explicitly define the system):
The explicit solution to the system is Ak=a0*(1^k).
Plot[{5*1^k,6*1^k},{k,0,50},PlotRange->{-1,15}]
(*Use your own initial conditions.*)
(B) Graph illustrating the steady states for the dynamical system when r=1 and b≠0.
The dynamical system for this case would be an1  an  b with initial condition a 0 .
Choose an arbitrary value of b. Show several plots of the dynamical system a n verses n (that
is, how the system evolves over time) for different initial conditions a 0 .
Note: there are a couple ways you can model the dynamical system. You can define the
system recursively or explicitly:
Way 1 (Recursively define the system):
b=3;(*Choose any value for b.*)
A[0]=5;(*Here choose your own initial condition.*)
A[n_]:=A[n-1]+b;
Table1=Table[A[n],{n,0,50}];
A[0]=15;(*Here choose your own initial condition.*)
A[n_]:=A[n-1]+b;
Table2=Table[A[n],{n,0,50}];
(* Add other initial conditions… *)
ListPlot[{Table1,Table2},PlotJoined->True,PlotRange->{-1,150}]
Way 2 (Explicitly define the system):
The explicit solution to the system is Ak=a0*(1^k) + k*b;
Plot[{5*1^k+k*b,6*1^k+k*b},{k,0,50},PlotRange->{-1,150}]
(*Use your own b, initial conditions.*)
(C) Graph illustrating the steady states for the dynamical system when r≠1 and b=0.
The dynamical system for this case would be a n 1  ra n with initial condition a 0 .
Choose an arbitrary value of r. Show several plots of the dynamical system a n verses n
(that is, how the system evolves over time) for different initial conditions a 0 . Carefully
choose values of a0 that (i) show the steady states, if any, and (ii) illustrate the
stability of those steady states.
Note: there are a couple ways you can model the dynamical system. You can define the
system recursively or explicitly:
Way 1 (Recursively define the system):
r=2;(*Choose your own value for r.*)
A[0]=3;(*Choose a variety of values for initial conditions,but
also choose a few to show the stable points and their
stability.*)
A[n_]:=r*A[n-1];
Table1=Table[A[n],{n,0,10}];
A[0]=6;(*Choose a variety of values for initial conditions,but
also choose a few to show the stable points and their
stability.*)
A[n_]:=r*A[n-1];
Table2=Table[A[n],{n,0,10}];
(*Add other initial conditions…*)
ListPlot[{Table1,Table2},PlotJoined->True,PlotRange->{-1,150}]
Way 2 (Explicitly define the system):
The explicit solution to the system is Ak=a0*(1^k) + k*b;
Plot[{3*r^k,6*r^k},{k,0,10},PlotRange->{-1,150}]
(*Use your own r, initial conditions.*)
(D) Graph illustrating the steady states for the dynamical system when r≠1 and b≠0
The dynamical system for this case would be an1  ra n  b with initial condition a 0 .
Choose an arbitrary value of r and choose an arbitrary value of b. Show several plots of the
dynamical system a n verses n (that is, how the system evolves over time) for different initial
conditions a 0 .
Note: there are a couple ways you can model the dynamical system. You can define the system
recursively or explicitly:
Way 1 (Recursively define the system):
r=2;b=3;(*Choose any values for r and b.*)
SS=b/(1-r);
A[0]=SS;(*Choose a variety of values for initial conditions,but
also choose a few to show the stable points and their stability.*)
A[n_]:=r*A[n-1]+b;
Table1=Table[A[n],{n,0,10}];
A[0]=6
A[n_]:=r*A[n-1]+b;
Table2=Table[A[n],{n,0,10}];
ListPlot[{Table1,Table2},PlotJoined->True,PlotRange->{-15,150}]
Way 2 (Explicitly define the system):
The explicit solution to the system is a function of a0 (contrary to what I said in class) and is
somewhat complex…can you derive it or find it? (It’s written in at least one place in your notes.)
Plot[{ ?? , ??},{k,0,10}]
Use your own initial conditions, b and r.
Exercise 2: (TURN-IN)
Further Investigation of the Linear Affine System an+1 = ran + b
The fixed points of a dynamical system do not tell us everything about the system.
Consider the dynamical system when r1 and b=0 (this is case C in the previous exercise).
According to the table, only a=0 is a fixed point. (In Exercise 1, Part C, you determined whether
or not this fixed point is stable for the value of r that you chose.) Nevertheless, there is a variety
of other interesting behavior as the parameter r is varied. For each case below, prepare a graph
that describes the dynamical system for each of the following cases. (You’ve already done one of
the cases in Exercise 1 Part C -- now repeat for the other cases of r.) For each case, describe
what is going on with the system (typical behavior, existence of fixed points, their stability) in
your own words with a sentence or two.
r  1
i)
r  1
ii)
iii)  1  r  0
iv) 0  r  1
v) 1  r
Note: when r is negative, you’ll probably find it easiest to plot the system if it is recursively
defined .