. «4W W'ULA-t
ét = "(‘2 "t)
é, - a, - 20u(t)¢,
e, = 52(1):, — be,
This is a linear system for e(I) . but it has a chaotic time-dependent coefﬁcient .0)
intwoternn'l'he ideaistoconstrucl aljlpunovfunction insucha waythattlte
chaos cancel: out. Here’s how:
f(.r.y) = ~3V/3x and ﬁx. y) = —3V/3y. These two equations may be “partially
integrated" to ﬁnd V. Use this procedure to ﬁnd V for the following gradient sys-
terns.
a) i: y’ syoosx. j 2xyrsinx
h) i=3:’-l— .'. y=-2xc“
7.1.7 Considerthe system i = y r 2
an opposing torque -It9. Then the equation of traction becomes bé+
mgLsino = F—Iro.
a) Does this equation give a well-deﬁned vector ﬁeld on the circle?
b) Nondirnensionalize the equation.
c) What does the pendulum do in the long run?
d) Show that many bif
rt,— l.l.l
prme transitions. and enticed a germtiorl ofphystcists to the study ofdynatttics. Fl»
ttllly. experimnlalisrs such as Gdtlub. Libclnber. Swimtzy. Linsay. Moon. and
Weﬂcrvelt new the new ideas about chaos in cxpetimmls on ﬂuids. chemical tuc-
ti
Note that this equation is invariant under the change of variables .r —t -x . That
is. if we replace it by —x and then cancel the resulting minus signs on both sides
of the equation. we get (1) back again. This invariance is the mathematical ex-
pression
Note that the definition 01 stable equilibrium is based on ma]! disturbances;
certain large disturbances may fail to decay. In Example 2.2.l. all small distur-
bances to r‘: —I will decay. but a large disturbance that sentk x to the right of
x I I will no
2.8 Solving Equations on the Computer
Throughout this chapter we have used graphical and analytical methods to analyne
ﬁrst—order systems. Every budding dynamicist should master a third tool: numeri-
cal methods. In the old days. numerical methods were im
rental. in which case there would he a hand of inﬁnitely many closed orbits
To nail downununiquenesspanofwrdairnwerecall frornSectionoﬂthl
there are two topologically different ltinds ofperiodic orbits on a cylinder: liki-
- The Poincare map is also calle
Iheﬁreflyuicdagainoolhenextbeatcycle.’l‘hispooestiscalledpha.r¢mlk-
rhmrrgftoephmﬁiﬂ.
Model
Ernrentrout and Rinml (I984) proposed a simple model of the ﬁreﬂy's flashing
rhythm and in response to stimuli. Suppose that 9(1) is the phase of the fueﬂy's
ﬂashi
from the mend two terms on the right-hand side. (Because of the assumption r < I,
the coefﬁcient of y’ is mom.) Thus the ﬁrst term reduces to —x’. which vanishes
only if x = 0.
The upshot is that V = 0 implies (x. y. z) = (0.0.0). ouieiwise v < 0. Hence t
molethanoneotttput z. foragivcninptll z_.0ntlleotherltand.thetllicltnessis
sostnall.andtlteleissomuehtobegainedhytreatingthegraphasaeurvenhalwe
will simply make this appoximation. keeping in ntilld that the Iubseqtleni analysis
is plausible but not rigoro
mmwm
The mechanisms ofdtetnical oscillaioos can be very complex. The 82 ractioo
is thouytt to involve more than twenty elementary reaction steps. but luckily may
of them equilibrate rapidly—this allows the kinetics to be reduced to as few as
three differe
through a small hole at the bottom of each chamber. and then collects underneath
the wheel. where it is pumped back up through the nozzles, This system provides a
steady input of Water.
The parameters can be changed in two ways. A brake on the wheel can b
6-6. -6.
=M—0’a—lK. +K,)sino. m
which isjust the munil’onn oscillator studied in Section 4.3. By drawing the new
dud pictuie (Figure 8.6.7). we see Ihlt there are No fixed poi-Is for (2) if
|aa.-m,|<x,+K, andnanei!|aa.-m,|>l(.+K,.Asaddle—nodebifuxcuionoc-
less population of the predator. and a 2 0 is a control parameter.
a) Sketch the nullclines in the ﬁrst quadrant x.y 20.
b) Show that the ﬁxed points are (0.0). (1.0) . and (am — a’) . and classify tltern.
c) Sketch the phase portrait for a > I . and show
a simpler analysis.)
a) Showthat xt-yvz: N.where N isconsunt.
b) Use the i Ind 1 equation to show that 1(1) =xuexp(-lz(r)/l). where
I. - 1(0).
c) Show that 2 satisﬁes the Tint-order equation 2' - [N - z - x. exp(- Irr/o].
d) Show that this equation can he
furcation. we mentioned theassumptiontlut a I 31/340,“ t 0. Tosee what can hap-
pen if 31734.,“ = 0. sketch the vector fields forllte following examples. and then plot
the fixed points a a function of r.
a) it I r1 - a:
b) x = r’ +x’
3.2 Tronuritieol Bifu