Exercise 2.6
2.6.1
Impossibility of Oscillation
Explain this paradox: a simple harmonic oscillation
is a system that oscillates in one
dimension (along the xaxis). But the text says that onedimensional systems cannot oscillate.
Solution:
The equation
is
4.5.3 (Excitable systems) Suppose you stimulate a neuron by injecting it with a pulse of current. If the
stimulus is small, nothing dramatic happens: the neuron increases its membrane potential slightly, and
then relaxes back to its resting potential. How
Solving Equations on the Computer:
2.8.2 Sketch the slope field for the following differential equations. Then "integrate" the equation
manually by drawing trajectories that are everywhere parallel to the local slope.
a) =
b) = 1 2
c) = 1 4(1 )
d) = sin
Nonlinear Dynamics
Some exercises and solutions
S. Strogatz Nonlinear dynamics and chaos
Dominik Zobel
dominik.zobel@tuharburg.de
Please note: The following exercises should but mustnt be correct.
If you are convinced to have found an error, feel free to
Egye 332 probset 2
4.1.2 For each of following vector fields, find and classify all the fixed points, and sketch the
phase portrait on the circle
=1+2 cos
0=1+ 2cos
1
=cos
2
2 4
,
Fixed points: =
3 3
d
2
=
Stability check:
= 1.73, stable point
dt
3
d
4
2.2 Fixed Points and Stability
Analyze the following equations graphically. In each case, sketch the vector field on the real line, find all
the fixed points, classify their stability, and sketch the graph of x(t) for different initial conditions. Then
tr
2.3
Solve
Population Growth
(
) analytically for an arbitrary initial condition
.
a) Separate variables and integrate using partial fractions.
Solution:
(
)
(
(
)
)
Therefore,


(
)
At t = 0, N(0) = N0. Solving for C,
(
)
Solving for N,
( (
)
( (
)
(
)
Exercise 2.1
A Geometric Way of Thinking
In the next three exercises, interpret
2.1
as a flow on the line.
Find all the fixed points on the flow.
Solution:
Fixed points:
x = n where n
2.1.2
At which points x does the flow have greatest velocity to the r
Egye 332 probset 2
4.1.2 For each of following vector fields, find and classify all the fixed points, and sketch the
phase portrait on the circle
Fixed points:
Stability check:
= 1.73, stable point
= 1.73, unstable point
4.1.8
a)
, hence, for every poin
5.3.6 (Romeo the robot) Nothing could ever change the way Romeo feels about Juliet = 0 , = +
. Does Juliet end up loving him or hating him?
In matrix form:
=[
0
0
]
= 0 0 = [ ]
0
= = [ ]
1
There exists a line attractor along 0 . R stays constant.
b
a
b
4.3.9 (Alternative derivation of scaling law) For systems close to a saddlenode bifurcation, the scaling
1
law Tbottleneck ~ ( 2 ) can also be derived as follows.
a) Suppose that x has a characteristic scale ( ), where a is unknown for now. Then = ,
wher
4.3.6 Draw the phase portrait as function of the control parameter . Classify the bifurcations that occur
as varies, and find all the bifurcation values of .
= + sin + cos 2
= + sin + cos 2 = + sin + 1 2 sin2
For   < 2, fixed points satisfy:
sin =
1
Chapter 2
Basic Navigational Mathematics,
Reference Frames and the Earths
Geometry
Navigation algorithms involve various coordinate frames and the transformation of
coordinates between them. For example, inertial sensors measure motion with
respect to an