Physica A 165 (1990) 72-91
North-Holland
CHAOTIC MOTION OF A HARMONICALLY BOUND CHARGED
PARTICLE IN A MAGNETIC FIELD, IN THE PRESENCE
OF A HALF-PLANE BARRIER
B e r n a r d J. G E U R T S 1 a n d F r e d e r i k W. W I E G E L
Center for Theoretical Physics, Twente University, P.O. Box 217, 7500 AE Enschede,
The Netherlands
R i c h a r d J. C R E S W I C K
Department of Physics and Astronomy, University of South Carolina, Columbia,
SC 29208, USA
Received 15 August 1989
The motion in the plane of an harmonically bound charged particle interacting with a
magnetic field and a half-plane barrier along the positive x-axis is studied. The magnetic field
is perpendicular to the plane in which the particle moves. This motion is integrable in
between collisions of the particle with the barrier. However, the overall motion of the particle
is very complicated. Chaotic regions in phase space exist next to island structures associated
with linearly stable periodic orbits. We study in detail periodic orbits of low period and in
particular their bifurcation behavior. Independent sequences of period doubling bifurcations
and resonant bifurcations are observed associated with independent fixed points in the
Poincar6 section. Due to the perpendicular magnetic field an orientation is induced on the
plane and time-reversal symmetry is broken.
1. Introduction
I n this p a p e r we s t u d y t h e m o t i o n in t h e (x, y ) - p l a n e o f a classical c h a r g e d
p a r t i c l e which is h a r m o n i c a l l y b o u n d to t h e origin a n d i n t e r a c t s with a
h o m o g e n e o u s p e r p e n d i c u l a r m a g n e t i c field a n d a s t r a i g h t h a r d line a l o n g t h e
p o s i t i v e x-axis. T h o u g h this m o t i o n is fully i n t e g r a b l e in b e t w e e n collisions with
t h e b a r r i e r , t h e o v e r a l l m o t i o n o f t h e p a r t i c l e is v e r y rich a n d d e p e n d s
sensitively o n t h e initial c o n d i t i o n s a n d t h e s t r e n g t h o f t h e m a g n e t i c field. T h e
reflection o f t h e c h a r g e d p a r t i c l e at t h e b a r r i e r is t r e a t e d as s p e c u l a r . D u e to
t h e s e reflections t h e " a n g u l a r m o m e n t u m " is a d i s c o n t i n u o u s f u n c t i o n o f t i m e
which shows j u m p s at t h e collisions. T h e o n l y c o n s t a n t o f t h e m o t i o n is the
1New address: Philips Natuurkundig Laboratorium, Postbus 80000, 5600 JA Eindhoven, The
Netherlands.
0378-4371/90/$03.50 © Elsevier Science Publishers B.V.
(North-Holland)
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
73
total energy. Hence, the trajectory does not lie on a single "two-torus" in
phase space. Rather it can be depicted as jumping from one such torus to
another at every collision. It is shown graphically that phase space is filled with
chaotic regions next to islands of stability associated with periodic orbits. The
chaotic regions become more prominent as the strength of the magnetic field
increases. The bifurcation behavior of low periodic orbits is analyzed and it is
shown that if the eigenvalues of the linearized flow around a fixed point in the
Poincar6 section pass through a resonance the associated orbit bifurcates out of
the fixed point. Although this system cannot be described with an analytic
Hamiltonian we recover several typical results for systems which do have an
analytic Hamiltonian. We find independent fixed points and associated bifurcations in different regions of the magnetic field strength.
Dynamical systems with two degrees of freedom are often studied by relating
these systems to area-preserving mappings of the real plane onto itself [1-3]. In
the model under consideration such a reduction cannot be performed explicitly. Hence, the dynamical properties of the motion of the charged particle can
be monitored only by generating its complete motion and studying for instance
the intersections of the trajectories with the y = 0 plane. An orbit of the
particle consists of an infinite sequence of "arcs" connecting two points on the
barrier. We plotted an example of part of such an orbit in fig. 1. It clearly
demonstrates that the overall motion is very complicated in spite of the fact
1.0
,
,
'
,
'
'
l
'
'
'
'
[
'
'
'
'
l
'
'
'
//2
0.5
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0.0
-0.5
-1"0-1'.0. . . .
!
-0.5'
' ' 0.0 . . . .
01.5 . . . .
1.0
Yl
Fig. 1. A typical example of part of an orbit. The direction of the arrows indicates increasing time.
We used a = 0.5 and adopted coordinates as in eq. (2.3). The initial condition for this orbit is
Yl = 0.4, Yl = 0.3 and Y2 = 0. The barrier is shown as a straight horizontal line.
74
B.J. Geurts et al. / Chaotic motion o f harmonically bound charged particle
that the constituent arcs are known explicitly. The present model may be
looked upon as a simple model for an electron in a solid which is harmonically
bound to one of the lattice sites. A magnetic field is applied and due to some
topological constraint the electron cannot encircle the lattice site. We demonstrate that the properties of this system depend sensitively on the magnitude of
the magnetic field and the initial conditions characterizing the motion. Essential for this system is that the perpendicular magnetic field induces an orientation on the plane which causes the time-reversal symmetry, usually observed in
Hamiltonian systems, to be broken [4, 5]. In fact, a reversal of time must be
accompanied by a reversal of the magnetic field in order to preserve a given
trajectory as a solution of the equations repetitive. Hence the induced mapping
on the surface of section y = 0 is not reversible and no symmetry lines exist [6, 7].
The classical and quantum mechanical aspects of a variety of "billiard" and
"stadium" systems have been subject of a considerable amount of theoretical
studies (see e.g. refs. [ 4 , 5 , 8 , 9] and references therein). The connection
between classical systems, whose dynamics is chaotic in certain regions of
phase space, and the corresponding quantized systems is, however, still a
matter of much debate. The model studied in this paper furnishes an example
of such a classically chaotic system whose quantum mechanical analogue seems
tractable. The latter is being studied presently and will be published elsewhere.
It was recently pointed out by one of us [10] that the present system might
show a classical dynamics which is irregular in some sense. This claim is
substantiated in the work presented here. Thus our main object here is to
establish in what way the classical motion of the charged particle can be
regarded upon as "chaotic". In a subsequent paper we will address the
quantum mechanical analogue of this system and investigate what " r e m n a n t s "
of the classical chaos survive in the quantum domain. We concentrate on
periodic orbits which furnish a " b a c k b o n e " for characterizing the detailed
structures observed on the surface of section. First we derive the equations of
motion and integrate the motion between collisions (section 2). We also
present some typical examples of the Poincar6 sections generated with this
solution. In section 3 we describe the algorithm used to find periodic orbits as a
function of the magnetic field strength. Then, in section 4 we present the
results obtained for the resonant bifurcations, including period doubling. We
summarize our findings in section 5.
2. The motion and Poincar~ sections
In this section we first derive and solve the equations of motion for the
particle's trajectory between collisions with the barrier. Then some Poincar6
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
75
sections characterizing the overall motion, with reflections of the charged
particle at the barrier included, are described.
Consider a charged particle moving in the (x, y)-plane interacting with an
homogeneous magnetic field in the positive z-direction. The charged particle is
bound harmonically to the origin and reflects specularly when colliding with the
barrier situated along the positive x-axis. The Lagrangian describing the
motion of a charged particle in a homogeneous magnetic field and a harmonic
potential can be written as
~=2{
[(dx]2
( dy]2]
L\ dt /
-d-t,/ J
+
to2(X2 + Y2) --
-
g2(x dy
-tit
dx
- Y -d-7)} '
(2.1)
where g2 = - e B / m c with e the particle's charge, B the magnetic induction, m
the particle's mass, c the velocity of light and to the harmonic frequency. In
(2.1) we use the gauge for the vector potential A such that A = ( B / 2 ) ( - y , x , 0).
The equations of motion are readily derived and can be written as
)Jl + Yl = --0'3)2
Y2 + Y2
,
=
0"3)1
(2.2)
,
where we put
g2
x
0" = - - ,
Yl --
to
v~'
Y2 --
y
V-g'
K--
2E
mto
2
,
(2.3)
in which E is the conserved energy associated with the motion. Throughout we
restrict ourselves to a t> 0. The dot represents a derivative with respect to
r---tot. The motion is such that the total energy is conserved, which implies
2
2
.2
Yl + Y 2 + Y l
.2
+Y2 = 1 ,
(2.4)
showing that the trajectory in these scaled variables lies on the unit sphere in
the four-dimensional phase space. The interaction with the barrier Yl ~>0,
Y2 = 0 is described by a specular reflection, i.e. 3)2--+-3)2 at collision. It is
standard procedure to solve the system of equations (2.2) for the motion of the
particle between collisions and one obtains
Yl (r) = A cos(#+ r) + B sin(/,+ r) + C cos(/x_ r) + D sin(/x_ r ) ,
(2.5)
y2(r) = - B cos(/x+ r) + A sin(/x+ r) + D cos(/x_ r) - C sin(/z_ r ) ,
(2.6)
where the coefficients A, B, C, D are related to the initial conditions through
76
B.J. Geurts et al. / Chaotic motion o f harmonically bound charged particle
A~
92(0) + ~_yl(0)
c---
~_ + ~+
-9do)
/x_+/x+
,
B~
D
92(0) - ~ _ y 2 ( 0 )
~ - + ~+
,
(2.7)
91(o) + +y2(O)
(2.8)
/ z +p.+
The time constants /.~_ and tz+ are given by
p.+ = ½V2[(2 + o¢ 2) -t- (0~4 q-- 4 a 2 ) 1 / 2 ]
1/2 "
(2.9)
2
2
The conservation of energy (2.4) is related to the property /~+/~ = 1. Since
every orbit does collide with the barrier in the course of time we may assume
without loss of generality
= 0, implying B = D. The solution (2.5), (2.6)
describes the motion for 0 ~< r ~< ~ where ~ is the recurrence time defined as the
minimal positive value such that y2(~) = 0 and Yl (~) ~> 0. Hence, the recurrence
time is a function of the initial conditions. Since ~ cannot be determined
analytically we cannot express the end point ( y l ( ~ ) , 91(?)) of an arc in closed
form in terms of (y~(0), 91(0)). Hence, we cannot derive an explicit expression
for the mapping induced on the surface of section. If a = 0, i.e. in the absence
of a magnetic field one may show that ~ = 2w, independent of the initial
conditions. For small a one finds to leading order in a
y2(O)
,~ ~ 2-rr(1
aye(O))
(2.10)
292(0)
indicating that orbits starting at the barrier and initially moving " u p w a r d " have
a recurrence time ~ < 2,n whereas those initially moving " d o w n w a r d " are such
that ~ > 2 ~ . Using (2.10) one may also show that ] y ~ ( ' ~ ) - y l ( 0 ) [ as well as
19l(-~) - 91(0)] are G(a). Thus, in the case a = 0 every orbit is periodic and the
mapping induced on the barrier sending begin- to end points of the arcs is the
identity mapping. In general ( a > 0) the induced mapping is not the identity
mapping. H o w e v e r , by numerically determining the recurrence time -~ for
subsequent arcs constituting a given orbit we can generate Poincar6 sections of
the phase space. In order to generate a unique mapping we k e e p track of
intersections of the trajectories with the plane
= 0 provided Y2 > 0 if Yl < 0
and 92 < 0 if Y1/>0 before collisions. This implies that we only focus on
collisions with the barrier if the charged particle moves " d o w n w a r d " directly
before collisions and if it moves " u p w a r d " when passing the region Y2 = 0,
Yl < 0. The other intersections with the plane Y2 = 0 are disregarded. Before
presenting some examples we notice that the integrable part of the motion, i.e.
Y2
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
77
the connecting arcs, are further characterized by a second independent constant of motion given by
2(Yl)~2 - YzYl) - a ( y ~ + y~) =/3 = constant.
(2.11)
If a = 0 (2.11) is readily associated with the angular m o m e n t u m of the motion.
In the following we refer to (2.11) as the angular m o m e n t u m also if a > 0. The
property (2.11) will be used in section 4 to describe the fixed points of the
induced mapping in more detail.
The overall trajectory of the charged particle, i.e. with collisions included,
can be looked upon as lying on a two-torus given by (2.4), (2.11) with a
specific value assigned to fl for 0 ~< r <~ ~. At collision the value of fl is changed
instantaneously to some other value associated with the reflection ))2-->-Y2
and the trajectory proceeds on the new torus defined by that new value until
the next collision. This process is repeated ad infinitum.
We next present some examples of the Poincar6 sections at various values of
a. Fig. 2 a - d shows an example of the Poincar6 section together with some
enlargements for the case a = 0.3. One clearly recognizes two large islands of
stability associated with the fixed point, i.e. the period 1 orbit. Notice that a
single period 1 orbit gives rise to two island strucures (as do all other stable
periodic orbits). One is related to collisions with the barrier (Yl > 0) and the
second with intersections of the trajectory with the Y2 = 0 plane as Yl < 0. Both
can in this sense be identified. All relevant information about the system is
contained in either half of fig. 2a, i.e. Yl < 0 or Yl ~> 0, but we have plotted the
full section to emphasize the symmetry in the half space yl < 0. Also shown are
period 6 and 7 orbits. It is found that stable periodic orbits with an odd period
have one point on the negative yl-axis at sufficiently low a. The subsequent
enlargements shown in fig. 2 b - d exhibit the recurrence of the island structures
and chaotic orbits occuring in fig. 2a on smaller and smaller scales. It is of
interest to note that even at this low value of a large regions of phase space are
filled with chaotic/stochastic orbits. We find that such orbits are found predominantly near the "tip" of the barrier.
Figs. 3 a - d show a sequence of Poincar6 sections for increasing values of a,
and show that the fraction of phase space filled by chaotic orbits increases with
a. Also, as shown in fig. 3c, independent islands related to different period 1
orbits exist at one and the same a-value. For small a only one such period 1
orbit exists but as a is increased a second independent period 1 orbit appears.
At higher a the first turns unstable leaving again only one stable period 1 orbit
which, after again increasing a, in its turn gives rise to a period-doubling
sequence. The recurrence of stable period 1 orbits is quite remarkable. It
78
B . J . G e u r t s et al. I C h a o t i c m o t i o n o f h a r m o n i c a l l y
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o r b i t . A l s o s h o w n a r e p e r i o d 6 a n d 7 o r b i t s w h i c h b i f u r c a t e d o u t o f t h e f i x e d p o i n t at s o m e l o w e r
a-values. In (b)-(d) enlargements of some parts of the section are shown corresponding to the
b o x e s d r a w n in t h e p r e v i o u s f i g u r e in t h e s e q u e n c e .
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
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i m p l i e s a r e c u r r e n c e o f s t a b l e structures in p h a s e s p a c e after a p r e v i o u s l y
e x i s t i n g p e r i o d 1 orbit t u r n e d u n s t a b l e a n d g a v e rise to a p e r i o d - d o u b l i n g
s e q u e n c e . T h i s is d i s c u s s e d in m o r e detail in s e c t i o n 4 w h e r e w e d e t e r m i n e t h e
l o c u s o f stable p e r i o d i c orbits as a f u n c t i o n o f a. F i n a l l y , w e n o t i c e that
80
B.J. Geurts et al.
Chaotic motion of harmonically bound charged particle
1.0
....::~:::!:i: !i:f/i~ i : f..
i'1
0.5
?
iiii . . .
J
0.0
-0.5
-
'
i
!
:k+-: '.~'::~'',
_1.0_1. 0 ,
,
,
,
I
-0.5
,
,","
.';~:..~.
0.0
,,
a
0.5
1.0
Yl
1.0
,,
/ /
i":. ;.:." , 0 : ~ :~.. ,:-:. ~ ': :/.:.::." ~ ' ~ \
~ "!.
(~I 1'"
:::::::::::.:.
,:.,.<.:,i.::"".
..7~:;::".. . '
"""" ........
~:"~::
/~
,' ('O~)~w.~:~
>..:~ ....~,f-~
oo•
~:
-0,5-'.//
",d ~ ~
':+ .~;." , :{ - ::" :- :,:
~ i"
./;",
(.....
~..~<:..':.'.}: ".?;::..):.... ~ . : . -- ...... :,;.'"" :::
"..
^': ". "...
~ = : : : : : ~ - ' 7 .¢ :
".....\ k%\ ':":: :,I~ : :::,-:: :- \- ~ = ~ ..<"~.
"":~
"
-1"0-1.0 . . . .
':.'.:.
"
-0.5'
~
,".. 'i :::."., ,.
" .. "
: : " ::,;::
:5 : : ,'~ "
" "": ;" :0.0": " '
b
' 0'.5 . . . .
1.0
Y]
Fig. 3. Poincar6 sections at a = 0.15 (a); a = 0.5 (b); a = 0 . 9 (c); and a = 1.4 (d). Notice the
increase in the fraction of phase space filled with chaotic orbits as a increases. In (b) we r e m a r k
that t w o i n d e p e n d e n t stable period 3 orbits exist. T h e one with the smaller islands does not
bifurcate out of the fixed point, c o n t r a r y to the o t h e r one. M o r e o v e r , the latter bifurcates at a
lower a - v a l u e .
B . J . Geurts et al. /
1.0
,
,
Chaotic motion of harmonically bound charged particle
,
,
i
,
,.
,.
,--.I-.,,..~..-~,..
,
,
i
81
. . . .
,.". ')"-!';~?." . : ,. ~i;..:'.' ..'~:, ::...:: ~," ~ '~~ .~ ,'/!:~; .?~-: .?. : 7i...,
0.5
10
- ' -1.,
.... " ' ' : < : '
,
,
,
,
?' :~ : -!{~7:". ' : , . S ::: :: " "
I
- 0. 5
0. 0
. . . . . : ' L: i : 7 ',:.
I ,
0.5
Yl
1.0
1.0
-1 0
~
0
1¢1
Fig.
3.
(cont.)
elements of a stable periodic orbit approach the tip of the barrier as a is
increased and typically an additional osculating encounter with the barrier is
generated at sufficiently high a causing the stable orbit to disappear. This also
is considered in greater detail in section 4.
82
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
In the next section we describe the algorithm used to determine the locus of
stable periodic orbits as a function of a.
3. Determination of stable periodic orbits
As was shown in the previous section the motion of the charged particle
depends sensitively on the initial conditions and the magnitude of the magnetic
field. This motion was shown to be either quite regular, i.e. a periodic orbit or
a " r o t a t i o n " about a periodic orbit, or irregular having stochastic features. The
total phase space was found to consist of a complicated mixture of these two
components [11]. In order to characterize the regular components in more
detail we focus attention on the locus of stable periodic orbits and the
bifurcation of these periodic orbits. The algorithm with which we may track a
stable desired periodic orbit as a function of a is based on the observation that
in the neighborhood of a periodic orbit in the section the orbits are arranged in
an "island" structure [1,2]. Since we do not have an explicit expression for the
induced mapping we employ this "geometrical" structure in order to locate a
given periodic orbit.
The algorithm proceeds as follows. Consider a point x close to a periodic
point of period q in the section. If it is close enough and the desired periodic
orbit is stable, the orbit generated with x as initial point will be such that each
qth iterate lies close to the desired q-periodic point. By determining the
"q-orbit average" ~(x) defined as
N
£(x) = lim ~ x j ,
xj =- TqJ(x),
(3.1)
N---~¢¢ j = 0
we obtain an improved estimate of the desired periodic point. In (3.1) we
symbolically denote the induced mapping by T. Using J~(x) as our new starting
point we can iterate this construction and obtain increasingly accurate estimates of the desired periodic point. Typically in about 5-10 iterations the
periodic point is found with an error of 0(10-7). In practice of course we
obtain only an estimate of the q-orbit average. The determination of J?(x) is
continued unless two subsequent estimates differ less than some bound e.
Typically, the algorithm works best if e is about 100 times the accuracy desired
for the periodic point, i.e. (7(10-5). Finally, we notice that a certain minimal
number of iterations is required in order to ensure that a complete encirclement about the q-periodic point is performed so that the q-orbit average
determined actually takes the "island structure" into account.
The minimal number of iterations required is related to the winding number
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
of the periodic point. Let X q denote a q-periodic point, i.e~. Tq(xq)
for a point x close to Xq we have
Tq(x)
= Xq
~- OTq(xq) (x
-- Xq)
"~- ' '
" ,
= Xq,
83
then
(3.2)
DTq(xq) denotes the linearized mapping in xq. If A1, A2 denote the
eigenvalues o f OTq(xq) then the orbit of (x - Xq) under the linearized mapping
encircles the origin with a shift in the angle per iteration given by
where
l
q=
/im(A1) ]
arctanl~ Re(e(A1)/'
(Im(A1) )
( ' n + arctan Re(A1) / ,
Re(A1) > 0,
(3.3)
Re(A1)<0,
(3.4)
where we used /~2 = - - ~ 1 and the fact that T is area-preserving, i.e. /~i/~ 2 = 1. In
(3.3), (3.4) we assume that I m ( A x ) > 0. The winding number of the periodic
point, p, is defined as p---q~/2,n" and the minimal number of iterations (N)
required to ensure complete encirclement of the q-orbit of x around Xq is taken
6(1/p). Typically N--3/2p was found to be adequate. The linearized mapping
DTq(xq) was determined by numerically differentiating T q. The eigenvalues
could be determined accurately to 6(10-3), which was sufficient for our
purposes.
By varying a and taking as initial guess for the algorithm a previously
determined periodic point or an extrapolation based on a few such points the
"locus" of a period q orbit was determined, given by {Xq(a),
T(xq(a)) . . . . .
Tq-l(xq(a))). The construction was stopped if either A1 = / ~ 2 =
- 1 , which is generally associated with a period-doubling bifurcation, or if the
program failed to find a q-periodic point in spite of a sufficiently reduced step
in o~. As we will discuss in more detail in the next section the latter situation
appears when a new osculating encounter of the orbit with the barrier is
generated due to "deformations" of the orbit as a is increased. In that case the
period of the orbit as given by the number of different encounters with the
barrier per cycle is changed.
Finally, we notice that the algorithm requires an initial guess at some
a-value of a period q point. In order to obtain such a guess use was made of
the observation that typically a period q orbit with winding number p/q about
the fixed point (p, q relative prime positive integers) bifurcates out of the fixed
point at an a-value such that the winding number of the fixed point was equal
to p/q. This gives an accurate guess for an appropriate a-value and a region in
which to look for period q points.
In the next section we present results obtained for low periodic orbits based
on the above algorithm.
84
B.J. Geurts et al. / Chaotic motion o f harmonically bound charged particle
4. Resonant bifurcations
First we turn to period 1 orbits and demonstrate that a large n u m b e r of
independent period 1 orbits exists, each giving rise to their own perioddoubling sequence. Then we consider low periodic orbits with period up to 4,
which bifurcate out of these fixed points of T in a way which we refer to as
" r e s o n a n t " , i.e. at a - v a l u e s at which the winding n u m b e r of the fixed point is
equal to a desired value of p/q. Finally, we consider in m o r e detail the
" d i s a p p e a r e n c e " of period q orbits due to the formation of a new osculating
encounter with the barrier at some critical a - v a l u e . This disappearance of
periodic orbits is related to the steric hindrance associated with the barrier.
As r e m a r k e d in section 2 all orbits are period 1 orbits if a = 0. Using the
"constant of m o t i o n " (2.11) in polar coordinates, i.e.
2r20 - a r 2 = / 3 ,
(4.1)
one notices that O has a definite sign and hence all these orbits (a = 0) encircle
the origin (0 denotes the angle with the positive yl-axis, anti-clockwise). If
a > 0 this can no longer be shown as easily, but numerical evidence indicates
that all period 1 orbits consist of two different arcs connecting two points on
either side of the barrier• This class of orbits shows a r e m a r k a b l e symmetry•
Let the two arcs be characterized by "constants of m o t i o n " /31 and /32. In
addition let yj. . . .,. yj b denote the coordinates of the period 1 orbit on the u p p e r
•
•
~°a
~b
side (u), after (a) and before (b) the reflection and similarly y [ , y [ for those
points on the lower side ( Q , then obviously
u,a • u,a
Y2 -
a(Y~'a) 2 = ~zYle,b Y2
.f,b
a t,,Y le,b-~2
) ,
(4.2)
/32 = zYl Y2 - a(Y~'a) 2 = 2y~'bfi~ ' b - a(Y~'b) 2"
(4.3)
/31 = 2 y l
-
-
+,a - f,a
Since
yl,a =
y~,b,
yf,a = yf,b,
y~,a=
u,b
--Y2
,
Y2 = - - Y 2 ' ,
f,a
.eb
(4.4)
(4.5)
one readily shows by considering/31 +-/32 that
yl,a =y~l,a,
))2,a = _ _ ~ , a
and combination with (2.4) gives
(4.6)
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
I •
)~l,a = -+ Y.&a
85
(4.7)
In fact, analysis of the full solution (2.5), (2.6) using (4.6), (4.7) results in the
global property
y1(~-) = yi(~ - ~-),
y2(~') = - y 2 ( ~ - ~'),
(4.8)
where ~ is the recurrence time. Hence, we find that the period 1 orbits are
invariant under reflection about the Yl-axis- This implies that the period 1 orbit
points in the Poincar6 section also possess this symmetry in the half space
yl < 0 (see figs. 2a and 3 a - d ) . Thus, if there is a single such point in the region
y~ < 0, then it lies on the negative yl-axis, i.e. the orbit crosses the y~-axis at
right angles. If there is more than one such point, then these are found in pairs
(ya,)~1) and (ya, - y ~ ) provided y~ < 0 . Numerical evidence indicates that all
periodic orbits are invariant under reflection about the Yl-axis, more specifically a complete cycle in the motion has this symmetry.
We next turn to the numerical results obtained for the loci of period 1 orbits
as a function of a. In fig. 4 we plotted a number of different loci each
parametrized with a and in fig. 5 we have shown Y1 and Yl of these loci as
functions of a. Note that each locus shown in fig. 4 gives rise to a set of two
corresponding curves in fig. 5. Hence, the Latin numerals in fig. 5 refer to both
curves directly beneath them, the lower showing y~ and the upper yl as
functions of a of the corresponding locus shown in fig. 4 with the same
numeral. Each period 1 orbit has a period-doubling sequence associated with
it. The different period 1 loci as shown in fig. 5 show a remarkable selfsimilarity, except for the locus labeled (I). Small bands exist between different
a-value intervals at which a period 1 locus is found. In these bands periodic
orbits resulting from period-doubling bifurcations are found and the dynamics
of the charged particle shows almost global chaos; roughly speaking the orbit
of almost any initial point on the barrier comes in the vicinity of almost any
point in the course of time. For a-values at which a period 1 locus is found the
section shows a (larger) island structure embedded in an almost uniform
chaotic sea for the loci II-V. At a-values larger than 2 . 1 9 1 6 . . . approximately,
no period 1 locus exists; the lowest stable periodic orbits found are period 2
orbits.
The resonant bifurcations of periodic orbits out of the fixed points of T will
be considered next. In fig. 6 we have shown the winding number of the fixed
point labeled (I) in fig. 4 as a function of a. One clearly recognizes an initially
almost linear increase in p as a function of a. At a ~ 0.705 the winding number
of the fixed point is equal to ½, i.e. both eigenvalues are - 1 . No perioddoubling bifurcation appears since the linearized mapping can still be diagonalized at a ~ 0.705. Using this figure one may estimate a-values at which for
86
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
1.00
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
~)1 0.75
0.50
vA
0.25
0.00
-0.25
-0.513
-0.75
-1.00
( I I ) / f
/
~
(I)
(III)
I
.0 . . . .
i
,
i
0.2
i
I
i
I
r
i
I
0.4
i
i
0.6
i
,
I
,
i
i
0.8
i
1.0
91
Fig. 4. The locus of several period 1 orbits. Each locus is parametrized with a. The direction of the
arrows indicate increasing a. Notice that different loci exist at different a-values, as shown in fig.
5. The different loci are labeled according to the order in which they appear as a is increased.
1.00
. . . .
I
. . . .
Yl°'75
1
'
(II)
'
'
'
I
. . . .
(III)
I
(IV)
. . . .
(V)
0.25
~)10.0~
-0.2~
-0.5C
-0.7~
- | "0~I.0 . . . .
01.5 . . . .
li.0
1.5
2.0
2.5
Fig. 5. The orbit points Yl and 3~1 as a function of a for tile period 1 orbits shown in fig. 4. The end
of each set of curves is encountered at an a-value at which the associated period-doubling
sequence starts, i.e. the period 1 orbits shown turn unstable.
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
87
0.5
P
....
I ....
i ....
0.4
0.3
0.2
•
,
.
.
.
.
.
.
.
.
.
.
.
l
l
l
l
i
l
l
,
1.0
Fig. 6. T h e w i n d i n g n u m b e r p o f t h e fixed p o i n t l a b e l e d (I) in fig. 4, as a f u n c t i o n o f a .
example periodic orbits with winding number p/q about the fixed point
bifurcate. One merely has to invert fig. 6. Taking this together with the locus
plotted in figs. 4, 5 then gives "initial conditions" for the algorithm described in
section 3 for the locus of the desired periodic orbits. In figs. 7a,b we have
drawn two examples of these loci, for periods 3 and 4, respectively. We plotted
the originating period 1 locus as well for sake of reference. One clearly
recognizes that the locus of higher periodic orbits originates out of some point
on the period 1 locus and is represented by a set of q lines moving " o u t w a r d "
as o~ increases.
A well known example of resonant bifurcations is the period doubling
bifurcation which appears as A1 = A2 = - 1 and the linearized mapping cannot
be diagonalized. Thus, in these bifurcations the originating fixed point itself
turns unstable. In fig. 8 we show the period-doubling " t r e e " associated with a
specific fixed pont of T showing the locus of period 1, 2 and 4 orbits. The locus
of subsequent orbits in the sequence was also determined but was found to be
of a scale which was not suited for this figure. The scaling similarity usually
observed for these trees with analytic mappings was obtained in this model as
well.
We finally discuss the disappearance of periodic orbits other than through
period-doubling. Periodic orbits as shown in fig. 7a,b which bifurcate out of a
fixed point of T were found to "disintegrate" at sufficiently high a. The orbit in
the Ya-Y2 plane continuously " d e f o r m s " as a is increased and at some
88
B.J. Geurts et al. / Chaotic motion o f harmonically bound charged particle
1.00
bl
. . . .
)
. . . .
i
. . . .
i
. . . .
0.75
0.5(
0.2~'
0.00
-0.25
-0.50
-0.75
_1. i . . . . . . . . . . . . . . .
0.25
0.50
a
0.75
1.00
Yl
1.00
. . . .
I
. . . .
i
. . . .
I
. . . .
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.0~.
. . . .
0.25 . . . .
0.50 . . . .
0.75 . . . .
1.00
Y]
Fig. 7. (a) T h e l o c u s o f a p e r i o d 3 o r b i t w i t h w i n d i n g n u m b e r ½ a b o u t t h e f i x e d p o i n t , w h o s e l o c u s
is a l s o s h o w n . T h i s p e r i o d i c o r b i t b i f u r c a t e s at ~ = 0 . 4 7 6 . . . ( b ) T h e l o c u s o f a p e r i o d 4 o r b i t w i t h
w i n d i n g n u m b e r ~, b i f u r c a t i n g o u t o f t h e f i x e d p o i n t a t o~ = 0 . 3 6 8 . . . B o t h o r b i t s b i f u r c a t e d o u t o f
the fixed point labeled (I).
parameter-value t~ o n e of the arcs creates a n e w osculating encounter with the
barrier. This disappearence of period q orbits is related solely to the presence
of a barrier. The osculating encounter implies that our section is not transversal
to the flow for that specific orbit and specific position along the barrier. To
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
1"0
'
I
[
I
I
.
.
.
.
I
.
.
.
.
I
.
.
.
.
I
.
.
.
89
.
bl
0.5
0.0
_0.5d.0 . . . .
~ ....
0.2
i ....
0.4
01.6. . . .
0.8 . . . .
1.0
Yl
Fig. 8. Period-doubling " t r e e " associated with the fixed point labeled (I) s h o w i n g the locus of the
period 1, 2 and 4 orbits.
exemplify this with a specific oribt we plotted a period 3 orbit before and at
osculation with the barrier in fig. 9. Notice that the oribit is unstable at
osculation.
5. Discussion
We have demonstrated the existence of stochastic motion in a simple
mechanical model representing a charged particle harmonically bound to the
origin and interacting with a perpendicular magnetic field and a rigid half-plane
barrier. The trajectories of this charged particle in phase space were shown to
lie on a specific "two-torus" in between collisions with the barrier. These
collisions caused the trajectory to proceed on a (in general) different torus (the
only exception being the case ~x = 0). Poincar6 plots exhibit the intricate
mixture of regular and stochastic motion in phase space, characteristic of
non-integrable Hamiltonian systems. Several well known results [2, 3, 6, 8]
found for analytic nonlinear area-preserving mappings of the plane onto itself,
such as the resonant bifurcations and the period-doubling sequences, were
recovered for each of the independent fixed points of T. However, the
presence of the barrier gives rise to some unfamiliar properties such as the
distintegration of periodic orbits. This process is associated with the occurrence
of osculating encounters with the barrier as a is varied. For these orbits the
90
B.J. Geurts et al. / Chaotic motion o f harmonically bound charged particle
1.0
Y2
i
0.5
0.0
-0.5
Y -1"01
-1.9;.o
0
i
i
i
i
-d.5
J
i
i
i
I
o.o
i
i
i
~
I
0.5
i
i
i
i
1.o
Yl
Fig. 9. (a) P e r i o d 3 orbit at a = 0 . 5 8 . (b) P e r i o d 3 orbit at a = 0 . 6 3 9 4 7 2 6 . . . showing the osculating
encounter formed. This orbit bifurcated out of the fixed point labeled (I) in fig. 4 a t o~ = 0 . 4 7 6 . . .
section is, at specific points and parameter-values, not transversal to the flow.
However, since the surface of section coincides with the pysical barrier, even in
these singular points the section plot reflects the physical properties of the
orbits. The quantum mechanical aspects of this system are presently being
studied and will be published elsewhere.
B.J. Geurts et al. / Chaotic motion of harmonically bound charged particle
91
Acknowledgement
F r e d e r i k W i e g e l w o u l d like to t h a n k M i c h a e l B e r r y a n d J o h n H a n n a y for
discussions a n d hospitality d u r i n g this visit to the U n i v e r s i t y of Bristol in the
spring of 1988.
References
[1] M. Henon, Quart. Appl. Math. 27 (1969) 291.
[2] A.J. Lichtenberg and M.A. Liebermann, Regular and Stochastic Motion (Springer, New
York, 1983).
[3] R.S. MacKay, Thesis, Dept. of Astrophysics, Princeton, 1982.
[4] G.D. Logan and J.M. Knight, Preprint, Univ. of South Carolina, 1989.
[5] M. Robnik and M.V. Berry, J. Phys. A: Math. Gen. 19 (1986) 669.
[6] B.J. Geurts, Breaking of kam-circles in quadratic mappings; some phenomenology, L.J.I
R-84-297, 1984.
[7] J.M. Greene, R.S. MacKay, F. Vivaldi and M. Feigenbaum, Physica D 3 (1981) 468.
[8] M.V. Berry, in: Chaotic Behaviour of Deterministic Systems, G. Iooss, R.H.G. Helleman and
R. Stora, eds. (North-Holland, Amsterdam, 1983), p. 171.
[9] M.V. Berry and M. Robnik, J. Phys. A: Math. Gen. 19 (1986) 649.
[10] F.W. Wiegel, Feynman and Hibbs revisted, Paper presented at third Int. Conf. on Path
Integrals from meV to MeV, to appear, 1989.
[11] R.S. MacKay, J.D. Meiss and I.C. Percival, Physica D 13 (1984) 55.