TFY4305 solutions exercise set 5 2014
Problem 3.5.4
a) Looking at Fig. 1 in the textbook, we see that the restoring force is given
F = ( h2 + x2 L0 )k .
(1)
The force is in the direction of the spring. We need the component of the force along the
wire, wh
TFY4305 solutions exercise set 13
2014
Problem 7.3.1
The dynamics is given by the equations
x = x y x(x2 + 5y 2 ) ,
y = x + y y(x2 + y 2 ) .
(1)
(2)
a) The Jacobian matrix is given by
1 3x2 5y 2 1 10xy
1 2xy
1 x2 3y 2
A(x, y) =
!
.
(3)
Evaluated at the or
TFY4305 solutions exercise set 11
2014
Problem 6.6.5
a) The equation that governs the dynamics is
x + f (x)
+ g(x) = 0 .
(1)
If we define y = x,
we can write Eq. (1) as
x = y ,
y = f (y) g(x) .
(2)
(3)
Under the transformation t t and y y, both sides of
TFY4305 solutions exercise set 14
2014
Problem 7.2.16
We can use the counterexample
1
r log r ,
2
= 1 .
r =
(1)
(2)
Choosing g = 1 in Dulacs theorem, we obtain
1 2
1
r log r +
1
2r r
r
= (log r + 1) .
x =
(3)
This expression is positive in the region
TFY4305 solutions exercise set 18
2014
Exam 2012 problem 3
a) Clearly x = 0 is a fixed point of the tent map t(x). The stability is given by |t0 (0)|. Since
t0 (0) = r, we find that x = 0 is stable for r < 1. For r = 1, x = 0 is marginally stable.
Moreove
TFY4305 solutions exercise set 17
2014
Problem 10.3.2
The logistic map is given by
xn+1 = rxn (1 xn ) .
(1)
f [f (x)]0 = f 0 (p)f 0 (q)
= r(1 2p)r(1 2q)
(2)
a) The superstability is given by
!
= 0.
Thus either p =
1
2
or q = 12 .
b) The points are given b
TFY4305 solutions exercise set 15
2014
Problem 8.2.1
The biased van der Pol oscillator reads
x + (x2 1)x + x = a .
(1)
This equation can be rewritten as
x = y ,
y = (x2 1)y x + a .
(2)
(3)
The fixed point is (a, 0) and the Jacobian matrix is
A(x, y) =
0
1
TFY4305 solutions exercise set 20
2014
Problem 11.2.4
a) The set of rational numbers has zero measure (see 12.2.2). We first make a list of the
rationals, Q = c1 , c2 , . and cover the first element by an interval of length , the second by
an interval of
TFY4305 solutions exercise set 21
2014
Problem 11.3.7
a) and b) See Fig. 1.
Figure 1: The von Koch snowflake at various stages.
c) At each stage, the length is increased by 34 , i. e. Ln = 43 Ln1 and therefore Ln = ( 43 )n L,
where L is the original lengt
TFY4305 solutions exercise set 22
2014
Problem 11.3.2
Figure 1: A generalized Cantor set with = a = 0.2.
If we scale the original segment by a factor of (1/2 /2), we need two segments to cover
the next iterate. Thus the fractal dimension is
d =
ln 2
ln [2
TFY4305 solutions exercise set 1 2014
Problem 2.2.3
The equation reads
x = x(1 x2 ) ,
(1)
The fixed points are the solutions to x(1 x2 ) = 0. This yields
x = 1 .
x=0,
(2)
Furthermore, f 0 (x) = 1 3x2 Since f 0 (x = 0) = 1, x = 0 is an unstable fixed point
TFY4305 solutions exercise set 12
2014
Problem 7.1.8
The dynamics is governed by the second-order equation
x + ax(x
2 x 2 1) + x = 0 ,
(1)
where a > 0 is a constant. The equation can be written as
x = y ,
y = x ay(x2 + y 2 1) .
(2)
(3)
a) The only fixed
TFY4305 solutions exercise set 3 2014
Problem 3.6.2
a) The dynamics is governed by the equation
x = h + rx x2 ,
(1)
where h and r are parameters.
The fixed points are found by solving h + rx x2 = 0 which yields
r r2 + 4h
x =
.
2
(2)
i) h = 0:
In this case
TFY4305 solutions exercise set 7 2014
Problem 5.2.12
a) The equation that governs the dynamics is
I
LI + RI +
C
= 0.
(1)
We introduce the variables x = I and y = I = x.
The equation can then be written as
x = y ,
R
1
y = y
x.
L
LC
(2)
(3)
b) The matrix
TFY4305 solutions exercise set 2 2014
Problem 2.4.8
Gompertz equation for tumor growth reads
N = aN ln(bN ) ,
(1)
where a, b > are parameters. The fixed points N are given by
f (N ) = aN ln(bN )
= 0.
(2)
This yields N = 0 and N = 1/b. The stability of the
TFY4305 solutions exercise set 9 2014
Problem 6.5.6
The dynamics of an epidemic is given by
x = kxy ,
y = kxy ly ,
(1)
(2)
where k, l > 0 are constants. x(t) is the size of the healthy population and y(t) is the size
of the sick population. The rate of ch
TFY4305 solutions exercise set 10
2014
Problem 6.5.12
The dynamics is governed by the set of equations
x = xy ,
y = x2 .
(1)
(2)
dE
d 2
=
x + y2
dt
dt
= 2(xx + y y)
= 0,
(3)
a) One finds
where we in the last line have used the expressions for x and y.
Th
TFY4305 solutions exercise set 4 2014
Problem 3.4.5
The dynamics is governed by the equation
x = r 3x2 ,
q
where r is a parameter. The fixed points are x =
(1)
r
,
3
and exist only for r 0. Moreover
q
f 0 (x) = 6x and f 0 (x ) = 6 3r . The positive fixed
TFY4305 solutions exercise set 6 2014
Problem 5.1.9
a) The dynamics of the system is governed by the equations
x = y ,
y = x .
(1)
(2)
The velocity field v = (y, x) is shown in Fig. 1.
1.0
0.5
0.0
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
Figure 1: Velocity field f
TFY4305 solutions exercise set 16
2014
Problem 8.2.11
a) The damped Duffing equation reads
x + x + x x3 = 0 .
(1)
This can be written as
x = y ,
y = y x + x3 .
(2)
(3)
The fixed points are given by (0, 0) and (1, 0). The Jacobian matrix reads
0
1
2
1 + 3x
Assignment 420-16-5
To be returned September 29, 2016
Main part
Solve
1. (5) x(3) 6
x + 12x 8x = 0
2. (5) x + 2x + x = et
Find a periodic solution with the frequency , when exists, and plot
the graph of its amplitude as a function of :
3. (10) x + x = sin
Assignment 420-16-3
To be returned September 15, 2016
Main part
In the Problems 1 5 find and plot the phase curves.
1. ( 4) x = 2 x
2. (4 ) x = k 2 x
3. (6 ) x = x3 x
4. (6 ) x = sin x
5. ( 7+6+6) x = sin x + a, a = 21 , 1, 2
6. ( 7) Is the following vect
Prelim 4200-13-2
To be returned November 14, 2013
Notes and books are allowed; any collaboration is not allowed.
1. (10) Find the image of the unit square [0, 1] [0, 1] under the action of
2 2
eA , A =
1 0
2. (10) Plot the orbits of a system x = Ax for
0
Assignment 420-13-9
To be returned November 21, 2013
Main part
1. (10) (HSD 8.7) Plot the orbits of a system written in polar coordinates:
r = r r3 , = sin2 + a. Describe the global bifurcation at a = 1.
2. (12) (HSD 8.9) Plot the orbits of a system writt
Assignment 4200-13-8
To be returned November 7, 2013
Main part
1. (12) Is the space of continuous functions on [0, 1] with the norm f =
1
|f (x)|dx complete?
0
2. (12) Prove that in assumption of the Banach Fixed Point Principle, the
distance between x an
Assignment 420-13-7
To be returned October 31, 2013
Main part
1. (7 each) Find the solutions and plot the orbits in the x-plane for the
following Cauchy problems: x = Ax, x(0) = a, x(0) = b
a)A =
53
, a=
35
0
0
= 0, b =
1
0
b)A =
53
, a = e1 , b = 0
35
c)
Assignment 420-13-6
To be returned October 10, 2013
Main part
1. (6) Prove that the image of a cube under the action of a linear operator
is a parallelepiped, probably of lower dimension.
2. (8) Prove that the image of a ball under the action of a non-deg
Assignment 420-13-5
To be returned October 3, 2013
Main part
Solve
1. (5) x + 2x + x = 0
2. (5) x(3) 6 + 12x 8x = 0
x
3. (5) x + 2x + x = et
Find a periodic solution with the frequency , when exists, and plot
the graph of its amplitude as a function of :