VOLUME 74, NUMBER 8

PH YSICALREVIEW LETTERS

20 FEBRUARY 1995

Chaos from Orbit-Flip Homoclinic Orbits Generated in a Practical Circuit
Department

Hisa-Aki Tanakaof Electronics and Communication Engineering, Waseda University,(Received 15 March 1994)

Shinj uku ku, -Tokyo

169, JapanA new class of chaotic systems is generated in a practical, nonlinear, mutually coupled phase-lockedloop (PLL) circuit. Theoretical results make it possible to understand experimental results of such
PLL's on the onset of chaos using the geometry of the invariant manifolds, while the resultant simple
geometry and complex dynamics is expected to have applications in other areas, e. g. , power systems orinteracting bar magnets. The 1D map projected from a numerically obtained attractor indicates the existence of a piece-wise linear structure having a sensitive dependence on the bifurcation parameter fo2.
PACS numbers: 05.45.+b, 84. 30.Ng, 84.30.Qi, 84.30.Wp

Much research has been directed at investigating thedynamical behavior occurring when a homoclinic orbit
unfolds. Shil'nikov's theorem straightforwardly shows theexistence of the Smale horseshoe near a homoclinic orbit
under a specific eigenvalue condition of the linearized
matrix located at the saddle-focus fixed point [1]. On theother hand, for a homoclinic orbit arising from a saddle
fixed point located in a three-dimensional (3D) vector field,
studies aimed at elucidating this phenomena have only
been carried out primarily from a mathematical standpoint
in which the following generic conditions are considered
to hold: (1) The homoclinic orbit 1 (t) is tangent to theeigendirections e' and e' associated with the principal~~; (2) A' 4 ~A'~; andeigenvalues A' and A' as t(3) the stable manifold W'(0) at 0 transversely intersects
with the two-dimensional (2D) extended unstable manifold
W" (0) along the homoclinic orbit, where W'"(0) is
tangent at 0 to the linear space spanned by e' and e'.
Under conditions (1)—(3), nonexistence of homoclinic
doubling bifurcation is proven [2]. However, if one of theabove conditions is broken, complex dynamical behaviors
are known to occur [3 —6], namely, the homoclinic orbit iscalled orbit fiip or inclination fiip when conditions (1) or(3) are broken, respectively. Results have been obtained
regarding the Smale horseshoe in an unfolding of the orbithomoclinic
fiip homoclinic point [4] or inclination-fiip
point [5]. Kokubu and Oka [4] investigated the unfoldingof the orbit-flip homoclinic point and demonstrated theexistence of the Smale horseshoe and invariant foliation,
while Homburg, Kokubu, and Krupa [5] showed in an
unfolding of the inclination-Hip
homoclinic point that
N-homoclinic bifurcation occurs under some eigenvalueconditions of the linearized matrix located at the saddle
fixed point. lori's [6] detailed numerical study was thefirst to confirm such bifurcation for the orbit-Hip case using
a piecewise linear ordinary differential equation (ODE).
However, resultant chaos from the orbit-flip homoclinic
orbit has not yet been identified to occur in a concrete
system, i.e. , one possessing associated chaotic dynamics
unfolding from the orbit-Hip homoclinic orbit. This led tothe present Letter which describes a real, practical systemthat possesses chaotic behavior for modeling orbit-flip

0031-9007/95/74(8) /1339(4) $06.00

homoclinic bifurcation.
In addition, the system s Smale
horseshoe generation mechanism is explained.
Throughout this Letter, phase-locked loops (PLL's) areconsidered that incorporate a voltage controlled oscillator(VCO), a phase detector (PD) having triangular characteristics (Fig. 1), and a loop filter (LF) comprised of a simple RC filter with transfer function F(S) = 1/(1 + r5).
Such PLL's can be connected to form mutually coupled
PLL's, which are frequently used in practical communication systems to synchronize geographically separated timing clocks. The following fourth-order ODE describes thedynamics of the phases of such PLL's:+ 2/ixi + h(xi —x2) = 6,x2 + 2124rl /r2x2 + (rl /r2)h(x2xi(rl /r2) ~xl )(1)

where x~ and x2 are normalized variables from phases @~and @2 of PLL 1 and PLL 2, respectively.
g&, g2, r&t, r2,
and 6 are also normalized parameters of PLL 1 and PLL2, while h is a 2~-periodic triangular function (Fig. 1).
Equation (1) can be normalized to giveg[ =Plgz= p2~= Pth(qiotipi + yiqz)—P2h(q2 —qi) —~~p2 + y2,
p~ =picr Iy~2glcr2

212$rl/r2= 6, and y2 = —(rt/r2)6.

Pl

1

P2

rl/r2

Mutually coupled PLL's possess chaotic dynamics fora wide range of realistic parameters.
Although not synchronized, they are close to synchronization,i.e. , theyare marginally out of lock as described by Endo andChua [7]. These researchers carried out experiments using

FIG. 1. Triangular phase detector characteristics.

1995 The American Physical Society

1339

PH YS ICALVOLUME 74, NUMBER 8

REVIE% LETTERS

real PLL integrated circuit modules as well as performing corresponding numerical simulations.
Consequently,
they numerically showed the existence of the positive Liapunov exponent, and also described the resultant broadband power spectrum generated by the real circuits.
Physically, Eq. (2) describes a class of mutually coupled nonlinear oscillators with 2~-periodic odd functions.
It should be noted that symmetry reduces Eq. (2) to athird-order ODE, namely, by introducing Pi = pi + p2,P. = P& —P2, Q~ = q~ + q2, and Q& —q~ —q2, Eq. (2)becomes= Pi,
Q2 = Pz,(3a)

Qi

(3b)

Pi = —(niPi + n Pp)/2 —p h{Q2) + 7~,(3c)+ n+P2)/2 —P+h(Qz) + 7-, (3d)P2== n~ —n2, P+ = P~ + P2,
where n+n~ + n2, n—+72.
7&
72 and 7 — 71
P~P
P2 7+(n —Pl

Note the solut&on for P~, Pq, and Q2 can only be
determined by Eqs. (3b) —(3d). Next, we employ thetransformation
x = P&/P —P2/P+, y = Pq//3+, and=
which
reduces Eq. (3) to the following thirdz
Qq,order ODE:

x =AIx+A2y,1t

whereP-I/3A))= Bo + B~x + B2y —h(z),z=P+y,(4)= (n /P~ —n~/P )P /2, A2 = (P /Pp/2, B. =-7-/P- =~, -B =P /2/3- ,is considered here.
Since Eq. (4) is 2~ periodic with respect to z, let us
concentrate on the region D = ((x, y, z)I —~ —6 ~ z ~~ —6), in order to examine the geometric structure of
the defined vector field. Region D can be divided into
three segments, D+, Dp, andrespectively, defined by
D+ = t(x, y, z)I~/2
z —~ —~), Do = ((x, y, z)IIzI——~/2),
and
D = t(x, y, z)I —~ —6~/2),z
where the vector field in each segment is linear. Further,denote the planes f(x, y, z)Iz = +. 7r/2) as P+, andFrom Eq. (4) and the form of h(z), it folrespectively.
lows that a unique equilibrium point at 0+, Op, 0 exists
in D+, Dp,
with their positions being, respectively,
0 = (0, 0, ~sr —6), Op = (0, 0, 6).
(5)Here, attention is focused only on one of the three setsof haotic parameters experimentally obtained by Endoand Chua I7]:

D,

( (gD,= 0.331,fpi = 18250 Hz, fpp = 25400 Hz,
ri = 45500/12500,
r2 = 5.027/5. 298,
6 = 2~ ( f pi —fp2) /(45 500 + 12 500),g)

leading to==0.614,= —1.242187, A2 = —0.038378,0.020119, B2 = —1.282425,&) = —Bo = ~ = —0.774565, P+ = 4. 836228.
AiA„A„,—A(A —A))(A —B2) + p~(A( —A) + AgB(A = 0, {8)= ( —1.241990, —2.932221, 1.649599).
giving (A„A,„A„)

In segment Dp, the real eigenvalue A] and the pair ofcomplex-conjugate eigenvalues A2 q are obtained by—A(A —A, )(A —B,) + P+(A —A, ) + A, B, A = 0, (9)1.242387, —0.641 1124 ~ 2.013 431i)
giving (A~, A23) = ( —The corresponding
eigenvectors (x, y, z) to the abovedeterminedeigenvalues
theinvariantmani folds
W'(0 ), W'(Op), E(Oo), and W"(0 ), and the extended
unstable manifold W'"(0+). Figure 2(a) schematically
depicts these invariant manifolds, where W'(0 ) andW" (0+) are planes, respectively, spanned by e' ande", or by the eigendirections e' and e', giving the lines
W'(0 ) 8 P and W"(0+) f1 g+. From Eqs. (5),(8), and (9), it is evident that the positions of 0+, Op,
and 0 depend only on the parameter 6, and that theeigenvalues (vectors) are not dependent on B. Hence, if
we only change 6, i.e. , the free-running angular frequency
fp2 of PLL 2, then W'(0 ), W'(Op), E(Op), W'(0 ),and W'"(0+) do not change their normal vectors ordirections, though they are shifted in the z direction by
the position changes of0+, and Op. Therefore,a critical parameter 6, is expected to exist under thefollowing situation: {i) The intersecting point p+ of P+
and W'(0+) is mapped onto the intersection of W'(0 )and g by the linear flow in Dp for 6 = 6, IFig. 2(a)]. If
6 4
the following situations are expected to occur depending on whether 6'
6, or 6 6, : (ii) for 6 8,. ,p+ is mapped to the half-space containing Op separated
6, ,by W (0 ), as shown in Fig. 2(b), or (iii) for 6
p+ is mapped to the opposite half-space separated byW'(0 ). Situation (i) leads to a set of two nonlinear(finite dimensional) equations derived from the solutionof Eq. (4) in Dp, where fp2 and the time interval T, inwhich P+ is mapped from g+ toare unknown.
Byusing the parameter values of fp~, g~ 2, and r~ 2 in Eq. (6),6, (i.e. , critical frequency fpz) for situations (i) can beobtained by solving these nonlinear equations, giving

0,6„()(6))(.

g,= 25 395.327, T = 0.787 206 .
(10)
From Eqs. (6) and (10), 6, = —0.774058. When 6 = 6,. ,fpp

the point mapped from p+ to the intersection of W'(0 )and P asymptotically goes tobecause A„. A,0. Thus, situation (i) indicates that a saddle connection
exists between 0+ andas shown in Fig. 2(a). Sucha connection can be considered as a homoclinic orbit if
0+ and 0 are identified. When 6 is slightly larger thanthe orbits from the neighborhood of p+ go to the0,g2(7)

The set of parameters in Eq. (7) corresponds to the real
eigenvalues
and A, of the linearized matrix of thevector fields at 0+ and 0 . These are obtained by solving. --and Bq = —(n /P~ + n~/P )P /2. Although h(z) canbe either sinusoidal or triangular depending on the typeof employed phase detector, only the triangular function(20 FaaRUARv 1995

0,6„

( (PHYSICAL REVIEW LETTERSVOLUME 74, NUMBER 820 FEBRUARY 1995

1.0-0.798

It-0.8140.010680.008

I0.01088

1.0

0.0-', o

1.0

0.0I

ee-0.008
5.4x10 5

60x10

0.0&,

in the original
tor aat
FIG. 4. A cross section of the attractor= 25400 Hz. (b) A cross section of the samerfd coor diinafdi an affine transforme
inattractor a tfor f&2 = 25400 Hz.

~,~a one-dimensional (1D map numerana ytractor, we analyzed
from the attractor r oically obtaine d b y a projectionp~——0.814, —0.798] depicted in Fig. 4(a).
ininterval 1
I canma I
crest is that the numerically obtained 1D mapb e we 11 fitted using a piecewise linear discoi . ). The
rescaled to [0.0, 0. 1] (Fig.
w
ic I wastion, in w hichreason why such piecewise linearity occurs is now un—5 b res ectively, show
er inve
1Dfornd 25396 Hz.
the resultantmaps
fo2 = 25400 andI:
I:[0~&5.

finer Fig. 5(b)]. Such rapid growth in the complexity of
the 1D map near foz (i.
( e.., 6 ) coincides with the rapid
increase o f th e m aximum Liapunov exponent as fo2 ecreases to fo2 [ ].
of aIn summary, thisis Letter reports the discovery o
new class of chaotic dynamics arising from t e orfiip homoclinic orbit generated by a p ractical nonlinear,
mutually coup 1e d PLL circuit, while also providing anoccurs in a margina outa
explanation as to w h y chaos'n [7]. Because the system's govermngof lock con d'stione uation and the resultant geometry of the vector e are
e too lead to novel results whensimplistic, its use it expec ted
applied to systems in other fields, e.g. , a two--g enerator
deled b the swing equations [9], or twoorinteracting bar magnets confined to a plane (seethe simplified overdamped case).
S1ncere gratitude is exten d e d to Professor S. Oishi
Waseda University, for theirand Pro fessorr K.. Horiuchi,on o, Meiji
ssor T.. Endo,valuable comments, and also to Professor

i-o,1342a)

0.0,0.0

FIG. 5.

One-dimensional
map from the attractor
orbit-Ilip homoclinic orbit for various fo2 values.
25400 Hz and (b) fop = 25396 Hz.

nearthe(a) f&» =a,U niversi
rsityy, Professor H. Kokubu, K y oto University;
Waseda Universi it y,' andProfessor T.. Matsumoto,aroCornell University forr their
Professor S.. H.. Strogatz,corn leted
e e aat theb ene fi cia 1 discussions. This work wass comp"workshoworaris op "VariousApproach to Complexx Sy stems,Studies
A v
Institute forr Advance
held at the international(Kyoto-Osaka-Nara, Japan) in 1994.

E,[1] L. P. Shil'nikov, Sov. Math Dokl.1. 66, 163 1965).
enic e, and J. A. Feroe, SIAM J. Appl.
[2] J. W. Evans, N.. Fenichel,Math. 42, 219 (1982).
[3] E. Yanagida, J. Diff. Eq. 66, 243 (1987).
~[4] H. Kokubu and H. Oka (to be published).
[5] A. J. Homburg, H. Kokubu, and M.. Krupap(to beubli shed).

and T. Matsumoto, in Structure and[6] K. Iori, E. Yanagi iuaa, an'Bifurcations of Dynamicalal 5 stems, edited bby . s i i,Sy,'ms Vol. 11 (WorSeries in Dynamical System.
93 p.. 82.
Scientific, Singapore,[7] T. Endo and L. O. Chua, IEEE Trans. , CAS Vol. 37, o.

Advanced

19,[8] M. Kisaka, H. Kokubu, and H. Oka, J. Dynam.
9] F. M. A. Salam, J. E. Marsden, and P. P.
~AS Vol. 31, No. 8, 673 (1984).

[10] S. H. Strogatz, Nonlinear Dynamics an
Wesley, Massachusetts,1994), p. 286.

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