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Strogatz S.H. Nonlinear Dynamics And Chaos (Perseus, 1994)(KA)(T)(505s)_PNc_.djvu |
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are certainly no closed orbits for ii> 0...
OSCILLATING CHEMICAL REACTIONS
257
x = 0
Figure 8.3.2
We can't apply the Poincare-Bendixson theorem yet, because there's a fixed
point
a/5...
Saddle-node Bifurcation of Cycles
A bifurcation in which two limit cycles coalesce and annihilate is called a fold
or saddle-node bifurcation of cycles, by analogy with the related bifurcation of
fixed points...
The scahng of the period in that case is obtained by estimating
the time required for a trajectory to pass by a saddle point (see Exercise 8.4.12 and
Gaspard 1990)...
Here a > 0 on physical grounds, and we
may choose 7^0 without loss of generality (otherwise, redefine 0 —> -0)...
There
are two qualitatively different cases, depending on whether the slope is a rational
or an irrational number...
The red points show the evolution of
o small blob o? 10,000 nearby initio! conditions, of times i~2, 6, 9, arid 15...
Therefore, back on the torus, the trajectories of A) approach a stable
phase-locked solution in which the oscillators are separated by a constant phase
difference 0 *...
Plot typical phase
portraits above and below the Hopf bifurcation,
8.2.10 (Bacterial respiration) Fairen and Velarde A979) considered a model for
respiration in a bacterial culture...
a) Plot the nullclines r' = 0 and <p' = 0 in the phase plane, and study how their
intersections change as the detuning a is increased from negative values to large
positive values...
a) Show that the system has a solution r= r^, 6 = (Og, corresponding to uniform
circular motion at a radius r^ and frequency (Og...
Can you say anything about their stability? (Hint: Regard the system as a
vector field on a cylinder: i=l, 9 = sint-sin9...
The same equations also arise in models of lasers and dynamos, and as we'll see in
Section 9.1, they exactly describe the motion of a certain waterwheel (you might
like to build one yourself)...
and 62 is M(t) = m(9, t) dO
Je,
Q{9) = inflow (rate at which water is pumped in by the nozzles above
position G)
r = radius of the wheel
K = leakage rate
V = rotational damping rate
/ = moment of inertia of the wheel
The unknowns are m{d,t)a.nd @(t)...
All trajectories starting in the blob end up somewhere in this limiting set;
later we'll see it consists of fixed points, hmit cycles, or for some parameter
values, a strange attractor...
They represent left- or right-turning convection rolls (analogous to the steady
rotations of the waterwheel)...
It turns out that they are linearly stable for
1 < r < r„
(T((T + ^ + 3)
(assuming also that ct-'^-1>0)...
Lorenz discovered that a wonderful structure emerges if the solution is
visualized as a trajectory in phase space...
This means that two trajectories starting very close together will rapidly diverge
from each other, and thereafter have totally different futures...
Aperiodic long-term behavior" means that there are trajectories which
do not settle down to fixed points, periodic orbits, or quasiperiodic
orbits as ? —> °° ...
There are stable fixed
points at the endpoints (±1,0) of / and a saddle point at the origin...
Moreover, the extent to which this distance is
exceeded appears to determine the point at which the next spiral is entered; this
in turn seems to determine the number of circuits to be executed before
changing spirals again...
At r^ = 24.74 the fixed points lose stability by absorbing an unstable limit cycle in
a subcritical Hopf bifurcation...
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