| Home / lib / P_Physics / PNc_Nonlinear chaos / | ||
Strogatz S.H. Nonlinear Dynamics And Chaos (Perseus, 1994)(KA)(T)(505s)_PNc_.djvu |
|
Size 3.8Mb Date Oct 3, 2003 |
bifurcations encountered in practice, the limit cycle is elliptical, not circular, and
its shape becomes distorted as IJ...
■
EXAMPLE 8.3.2:
Using numerical integration, show that a Hopf bifurcation occurs at ^ = b^...
A few of their scientific apphcations are discussed in
Sections 8.5 and 8.6 and in the exercises...
HYSTERESIS IN THE DRIVEN PENDULUM 265
We will phrase our discussion in terms of the Josephson junction, but will
mention the pendulum analog whenever it seems helpful...
But we haven't excluded the possibility that P{y) = y on some in-
268
BIFURCATIONS REVISITED
terval, in which case there would be a band of infinitely many closed orbits...
So it's
plausible that an infinite-period bifurcation
should also occur if a is large but finite...
Now recall from Section 4.6 that the junction's dc-voltage is proportional to
its oscillation frequency...
The second possibility is that the slope is irrational (Figure 8.6.6).Then the
flow is said to be quasiperiodic...
EXAMPLE 8.7.1:
Consider the vector field given in polar coordinates by r = r(l - r"), 6 = 1...
Then there is a basis of eigenvectors \e.] and so we can write
n-\ I- -'J
Vq = ^ VjCj for some scalars Vj...
(This is a
common joke played by the chemical oscillator community; there is also the
"Oregonator," "Palo Altonator," etc.) In dimensionless form, its kinetics are
x = l-(b + l)x + cix^y
y = bx- ax^y
where a,b>0 are parameters and x,y>0 are dimensionless concentrations...
(No periodic solutions for averaged system) Regard (r,0) as polar
coordinates in the phase plane...
By numerical
computation of the phase portrait, verify that if a is fixed and sufficiently small, the
system's stable limit cycle is destroyed in a homoclinic bifurcation as / decreases...
A generalization of this system to A^» 1 phases has been proposed as a model
of switching in charge-density waves (Strogatz et al...
You
may have encountered it if you've studied fluids, electrostatics, or chemical
engineering...
Hence, after the transients decay, the equation of motion is
/ft) = damping torque + gravitational torque...
Suppose we encase a repeller with a
closed surface of initial conditions nearby in phase space...
He began integrating from the initial condition @, 1, 0), close to the saddle
point at the origin...
CHAOS ON A STRANGE ATTRACTOR 323
"Deterministic" means that the system has no random or noisy inputs or
parameters...
Eventually the
trajectories will settle down into a periodic behavior—it just happens that the period
is incredibly long, much longer than you've tried in your computer...
Figure 9.5.3
332
LORENZ EQUATIONS
Other names used for transient chaos are metastable chaos (Kaplan and Yorke
1979) orpre-turbulence (Yorke and Yorke 1979, Sparrow 1982)...
First he
showed us how to make the chaotic mask, using an electronic implementation of
the Lorenz equations (Figure 9.6.1)...
= 5u(t) V, - bw^
where we have written u(t) to emphasize that the receiver is driven by the chaotic
signal u(t) coming from the transmitter...
(Behavior of higher modes) In the text, we showed that three of the
waterwheel equations decoupled from all the rest...
Specifically, assume b = \ as in the waterwheel, and then plot the pitchfork and Hopf
bifurcation curves in the (CT, r) parameter plane...
Numerically integrate the equations,
and plot the solutions in whatever way seems most revealing...
If the receiver has synchronized with the drive, then
x^(t)''x(t) and so m(t) may be recovered as m(t) = s(t)-x^(t)...
| © 2007 eKnigu | ||