Section 1 Exercises: Introduction

Complete these as you are going through part 1. Try and do questions from this regularly.
1
.
Think of five to ten different dynamical systems and write a few of them in the forums. What is interacting with what, and what is changing with time?
​
2
.
Watch Steven Strogatz’s first lecture linked on the first page of the notes from 8.12 minutes to 36 minutes in. Write in the forum anything which seems very strange, confusing, surprising, or just doesn’t make sense.
​
3
.
Read through some of the links about the following and put a link in the forum about the most interesting thing you've found.
​
1
.
Kepler’s laws
2
.
The three body problem
3
.
Edward Lorenz and the beginnings of chaos
​
4
.
Although we talk about Tycho and Sophia Brahe as being responsible for some of the most accurate astronomical observations of the era, many cultures had made amazing progress in observations even centuries before. Have a look into ancient Chinese, Indian and Arabic astronomy and write in the forum any particularly interesting things that you discover.
​
5
.
Put the following systems of differential equations in the form of first order systems of autonomous differential equations.
​
1
.
x''(t)+x'(t)=3
2
.
x''(t)+3x'(t)-4x(t)=sin(x
(t))
3
.
x''(t)+y(t)+x'(t)=
2
x(t)
,
y'(t)+x(t)=
2
x(t)
4
.
y''(t)-y'(t)+x(t)=
2
y(t)
,x'(t)=sin(y(t))
5
.
x'''(t)+x'(t)=x''
2
(t)
6
.
x'''(t)+x'(t)=x''
2
(t)
-
2
t
​
​
6
.
How many dimensions is the phase space of a particle moving freely on a two-dimensional surface?
​
7
.
Imagine one pendulum attached to another pendulum (creatively called a double pendulum). This system happens to be chaotic, but that’s by-the-by. You definitely don’t need to understand this code!!!
​
How many dimensions is the phase space of this system (you can imagine that the system is in swinging in a plane (ie. not in three dimensions))? ie. how many values would you need to capture what is happening at any one moment?
In[]:=
​​deqns=
m
1
′′
x1
[t]
x1[t]λ1[t]
l
1
-
(-x1[t]+x2[t])λ2[t]
l
2
,
m
1
′′
y1
[t]-g
m
1
+
y1[t]λ1[t]
l
1
-
(-y1[t]+y2[t])λ2[t]
l
2
,
m
2
′′
x2
[t]
(-x1[t]+x2[t])λ2[t]
l
2
,
m
2
′′
y2
[t]-g
m
2
+
(-y1[t]+y2[t])λ2[t]
l
2
;​​aeqns={
2
x1[t]
+
2
y1[t]

2
l
1
,
2
(-x1[t]+x2[t])
+
2
(-y1[t]+y2[t])

2
l
2
};​​ics={x1[0]1,y1[0]0,y1'[0]0,x2[0]1,y2[0]-1,y2'[0]0};​​params={g9.81`,
m
1
1,
m
2
1,
l
1
1,
l
2
1};​​soldp=First[NDSolve[{deqns,aeqns,ics}/.params,{x1,y1,x2,y2,λ1,λ2},{t,0,15},Method{"IndexReduction"{"Pantelides","ConstraintMethod""Projection"}}]];​​frames=Table[Graphics[{{PointSize[.025],{Red,Point[{x1[t],y1[t]}]},{Blue,Point[{x2[t],y2[t]}]},Line[{{0,0},{x1[t],y1[t]},{x2[t],y2[t]}}]}/.soldp,​​{Gray,Line[Map[Function[Evaluate[{x2[#],y2[#]}/.soldp]],Range[0,t,0.025]]]}},PlotRange{{-2,2},{-2,0}},AxesTrue,TicksFalse,ImageSize500],{t,0,10,0.25}];​​ListAnimate[frames,SaveDefinitionsTrue]
Out[]=
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