| Home / lib / P_Physics / PNc_Nonlinear chaos / | ||
|
|
Size 3.8Mb Date Oct 3, 2003 |
this last, most interesting case when we study the Lorenz equations (Chapter 9)...
Is the bifurcation subcritical, supercritical,
or degenerate?
,M -1
Solution: The Jacobian at the origin is A =
,1 M
which has T = 2/i,
A...
When A > 0, we have a stable node if
266
BIFURCATIONS REVISITED
T^ - 4A = a^ - 4^ 1 - /^ > 0, i.e., if the damping is strong enough or if / is close
to 1; otherwise the sink is a stable spiral...
Specifically, the frequency tends to
zero like [ln(/ - 4)] , just as expected from the scaling law discussed in Section
8.4...
Starting on the outer equator, the trajectory moves onto the top surface, dives into the
hole, travels along the bottom surface, and then reappears on the outer equator,
two-thirds of the way around the torus...
For these parameter values, the sys-
teiT; has tv/o periodic aftractors, corresponding to forced osciilations confined to
the let; or riaht well...
(Don't try
to find the fixed point explicitly; use a graphical argument instead.)
c) Show that a Hopf bifurcation occurs at the positive fixed point if
4(b-2)
b\b + 2)
and b>2...
(Homoclinic bifurcation) Using numerical integration, find the value of
fi at which the system x = fix + y-x''', y = -x + fiy + 2x^ undergoes a
homoclinic bifurcation...
(Plotting Lissajous figures) Using a computer, plot the curve whose
parametric equations are x{t) = s.int, y{t) = s.incot, for the following rational and
irrational values of the parameter (o:
(a) ffl = 3 (b) (w = I (c) ffl = I
(d)ffl = V2 (e) co = n (f) ffl = i(l-hV5)...
BIFURCATIONS REVISITED
"T 8.7.9 Consider the vector field given in polar coordinates by r = r-r^, d=\...
The chambers are transparent, and the water has food coloring in it,
so the distribution of water around the rim is easy to see...
The real justification for the rule above is that it agrees with direct
measurements on the waterwheel itself, to a good approximation...
It turns out that (9) is equivalent to the Lorenz equations! (See Exercise 9.1.3.)
Before we turn to that more famous system, let's try to understand a little about (9)...
Simple Properties of the Lorenz Equations
In this section we'll follow in Lorenz's footsteps...
What happens immediately after the bifurcation, for r slightly greater than r^?
You might suppose that C* and C^ would each be surrounded by a small stable
limit cycle...
Of course, strange things could occur
for another reason—the electromechanical computers of those days were
unreliable and difficult to use, so Lorenz had to interpret his numerical results with
caution...
But how can this be, when the uniqueness theorem (Section 6.2) tells us that
trajectories can't cross or merge? Lorenz A963) gives a lovely explanation—the two
surfaces only appear to merge...
Of course, no measurement is perfect—
there is always some error \\5J\ between our estimate and the true initial state...
Then they rattle around chaotically for a while, but eventually
escape and settle down to C* or C~...
Before we leave the regime of small r, we note one other interesting
implication of Figure 9.5.1: for 24.06 <r < 24.74 , there are two types of attractors: fixed
points and a strange attractor...
Then, by hooking up the circuit to a
loudspeaker, Cuomo enabled us to hear the chaos—it sounds like static on the radio...
z- You should investigate the
consequences of choosing different initial conditions and lengths of integration...
By taking the limit in a certain way, all the dissipative terms in the
equations can be removed (Robbins 1979, Sparrow 1982)...
| © 2007 eKnigu | ||