AMath 584, Autumn 2015
Sample Solutions for Assignment 4.
Reading: Lectures 9-11 in the text.
1. p. 68, Exercise 9.2.
(a) Since A is upper triangular, its eigenvalues are its diagonal entries, which
are all 1s. The determinant of A is the product of the e
16
Period doubling route to chaos
We now study the "routes" or "scenarios" towards chaos.
We ask: How does the transition from periodic to strange attractor occur?
The question is analogous to the study of phase transitions: How does a solid
become a m
17
Intermittency (and quasiperiodicity)
In this lecture we discuss the other two generic routes to chaos, intermittency and quasiperiodicity. Almost all our remarks will be on intermittency; we close with a brief de scription of quasiperiodicity. Definiti
AMath 584, Autumn 2015
Sample Solutions for Assignment 1.
Reading: Lectures 1-3 in the text.
1. p. 9, Exercise 1.1.
1. Double column 1:
2
1
B B
1
1
2. Halve row 3:
1
1
B
B
1/2
1
3. Add row 3 to row 1:
1
1
1
B
1
B
1
4. Interchange columns 1 and 4:
1
1
B B
AMath 584, Autumn 2015
Sample Solutions for Assignment 3.
Reading: Lectures 6-8 in the text.
1. p. 47, Exercise 6.1.
I 2P is unitary since (I 2P ) (I 2P ) = I 2P 2P + 4P P = I,
where the last equality follows because P = P and P = P 2 . Geometrically,
(I
AMath 584, Autumn 2015
Sample Solutions for Assignment 2.
Reading: Lectures 3-5 in the text.
1. p. 24, Exercise 3.5.
The same is true for the Frobenius norm since
m
E
2
F
n
2
|Eij | =
=
i,j
m
|ui |2 |j |2 =
v
|ui vj | =
i=1 j=1
m
n
2
n
|ui |2
i=1
i=1 j=1
HW #4 - Solutions of Selected Problems
5.1.9 (Page 141)
b) Follow the hint:
xx y y = x(y) y(x) = 0.
Notice that (x2 ) = 2xx, (y 2 ) = 2y y, integrating the above equation we obtain that
x2 y 2 = C,
where C is a constant.
c) The system can be written as
x
HW #2 - Solutions of The Rest of The Problems
Note: Whenever I say only one eigenvector, I mean (of course) up to scaling.
5.2.2 (Page 141)
a) The system can be written as follows:
x
y
=
1 1
1 1
x
y
To nd the eigenvalues of A we solve Det(A I) = 2 2 + 2 =
HW 2 - Solutions of the Rest of the Problems
4.1.3 (Page 113)
The xed points are at sin2 = 0, i.e. at = 0, /2, , 3/2. The vector eld looks as shown
below:
4.1.4 (Page 113)
The xed points are at sin3 = 0, i.e. at = 0, . The vector eld looks as shown below:
Solutions 1
2.2.5 There are no equilibria:
between 1/2 and 3/2.
1
2
f (c) 3 , so x(t) always moves to the right with speed
2
2.2.7 For x > 0, ex > 1 cos x, so f (x) = ex cos x > 0 has no equilibria in this range.
Note f (0) = e0 cos 0 = 1 1 = 0, so 0 is
HW #3 - Solutions of Selected Problems
3.6.2 (Page 86)
When h = 0 this system is the same as the one in Section 3.2. As h varies, the curves in
the following pictures move vertically.
r<0
r=0
r>0
V
V
V
x
x
x
a) If h < 0, the above curves are shifted down.
HW #2 - Solutions of Selected Problems
3.1.2 (Page 79)
The graph of the function y = coshx is shown below on the right, with dotted lines
indicating the values of r. Its clear that x moves to the right or left according to whether
y is below or above r. T
15
Lyapunov exponents
Whereas fractals quantify the geometry of strange attractors, Lyaponov ex ponents quantify the sensitivity to initial conditions that is, in effect, their most salient feature. In this lecture we point broadly sketch some of the math
14
Fractals
We now proceed to quantify the "strangeness" of strange attractors. There are two processes of interest, each associated with a measurable quantity: sensitivity to initial conditions, quantified by Lyaponov exponents. repetitive folding of att
13
Experimental attractors
In this brief lecture we show examples of strange attractors found in experi ments.
13.1
Rayleigh-Bnard convection e
BPV, Figure VI.25 Two dynamical variables, (T ) and (T ), represent time-dependent thermal gradients measured m
1
1.1
Pendulum
Free oscillator
To introduce dynamical systems, we begin with one of the simplest: a free oscillator. Specifically, we consider an unforced, undamped pendulum.
l
mg sin mg
The arc length (displacement) between the pendulum's current positio
2
Stability of solutions to ODEs
How can we address the question of stability in general? We proceed from the example of the pendulum equation. We reduce this second order ODE, g + sin = 0, l to two first order ODE's. Write x1 = , x2 = . Then x1 = x 2 g x
3
Conservation of volume in phase space
We show (via the example of the pendulum) that frictionless systems conserve volumes (or areas) in phase space. Conversely, we shall see, dissipative systems contract volumes. Suppose we have a 3-D phase space, such
Calculate
Pictorially
x1 x2 f = + = 0+0 x1 x2
x2
x1
Note that the area is conserved. Conservation of areas holds for all conserved systems. This is conventionally derived from Hamiltonian mechanics and the canonical form of equations of motion. In conser
5
5.1
Forced oscillators and limit cycles
General remarks
How may we describe a forced oscillator? The linear equation + + 2 = 0 is in general inadequate. Why?
(3)
Linearity if (t) is a solution, then so is (t), real. This is incompatible with bounded osc
6
6.1
Parametric oscillator
Mathieu equation
We now study a different kind of forced pendulum. Specifically, imagine subjecting the pivot of a simple frictionless pendulum to an alternating vertical motion:
rigid rod
This is called a "parametric pendulum,
7
Fourier transforms
Except in special, idealized cases (such as the linear pendulum), the precise oscillatory nature of an observed time series x(t) may not be identified from x(t) alone. We may ask How well-defined is the the dominant frequency of oscil
8
Poincar sections e
The dynamical systems we study are of the form d (t) = F ( , t) x x dt Systems of such equations describe a flow in phase space. The solution is often studied by considering the trajectories of such flows. But the phase trajectory is
9
Fluid dynamics and Rayleigh-Bnard convection e
In these lectures we derive (mostly) the equations of viscous fluid dynamics. We then show how they may be generalized to the problem of RayleighBnard convection-the problem of a fluid heated from below. La
10
Introduction to Strange Attractors
Thus far, we have studied only classical attractors such as fixed points and limit cycles. In this lecture we begin our study of strange attractors. We emphasize their generic features.
10.1
Dissipation and attraction
11
Lorenz equations
In this lecture we derive the Lorenz equations, and study their behavior. The equations were first derived by writing a severe, low-order truncation of the equations of R-B convection. One motivation was to demonstrate SIC for weather
12
Hnon attractor e
The chaotic phenomena of the Lorenz equations may be exhibited by even simpler systems. We now consider a discrete-time, 2-D mapping of the plane into itself. The e points in R2 are considered to be the the Poincar section of a flow in