IN A PIECEWISE LINEAR CIRCLE MAP. YANG Wei-ming( #$B| ) and HAO Bai-lin(. Institute of Theoretical Physics, Academia Sinica,. Beijing, China. Received ...
Commun. in Theor. Phys. (Beijing,
China)
Vol.8,
NOul(1987)
1-15
HOW THE ARNOLD TONGUES BECOME SAUSAGES IN A PIECEWISE LINEAR CIRCLE MAP YANG Wei-ming( # $ B | ) and HAO B a i - l i n ( I n s t i t u t e of Theoretical Physics, Academia Sinica, Beijing, China R e c e i v e d J u l y 2 8 , 1986
Abstract
The width of mode-locking tongues shrinks to zero at well-defined points in a piecewise linear circle map. The dimension of the set of quasiperiodic points along its critical line appears to be zero as compared to the "universal" value D=0.87. Many properties of this map can be described analytically by rational fractions.
I. Introduction Maps of the unit circle into itself x
n+2=f(xn>Hxn+A-Bg0 for all x. It is called the forward map. Similarly, the left boundary is reached when the map is backward, i.e., f g (x)-p-x0.15451 B ^ O . 15451 B2p) "
A=l/2+B
(q-p)-th
A=l/(4-8B)-B^
2p-th
(q4p)
A=l/(4+8B)+B
(q-2p)-th
(2pg
How the Arnold Tongues Become in a Piecewise Linear Circle
Sausages Map
13
curves, a andfc.being integers. Therefore, we can fix the subcritical mode-locking structure by drawing these curves without calculating the tongue boundaries explicitly.
IV. Discussion A few words about the supercritical behavior may be in order. Although there exists multi-basin structure, period-doubling bifurcation of the q-cycle within a p/q tongue cannot occur. The stability of a periodic trajectory in the p/q tongue depends only on the value of B. If it is larger than the solution in between 1/4 and 1/2 of the equation (l-4B)(l+4B)'g_1=-l
(23)
no stable q, 2q, 3q, 4q... periodic trajectories, can exist. Above B=l/2 there is no region of stable periodic orbits in the parameter space. The properties of the piecewise linear map (2) are quite, dif-, ferent from that of the sine map. Although- the supercritical behavior of the piecewise linear map is too simple and specific to have more meaning than the sine map, we think the subcritical and critical behavior of map (2) may shed more light on general onedimensional circle mappings. f(x) This of a part
We have also studied a family of piecewise circle map with smoothed from C° to C1-function, to be denoted as fR(-x). family- satisfies all the conditions imposed in the definition general one-dimensional circle mapping. In ffl(x) the periodic gfi(x) is defined as follows: f (_l>*(4x-2k),
if x€[(k/2)-0.25+R, (k/2)+0..25-R) ,
[(_l)*C(-2/R)(x-(K/2)-0.25)2+l-2R), if x€[(k/2) +0.25-R, (k/2)+0.25+R)'. (24) It is found that the fractal dimension of the set of qUasiperiodic winding numbers along the critical line is neither zero nor 0.87. For example, numerical computation gives D=?0.346±0.003 when R=0.001. In the sine map, the Farey tree construction is thought to give a good description,of the self-similar nature of mode-locking
YANG Wei-ming
14
and HAO
Bai-lin
intervals. However, this construction is not suitable to explore the self-similarity of the mode-locking fractal in the piecewise linear map.(2) or the piecewise map f R (x). Eventually, we do not know much about the self-similarity structure of the complete devil's staircase in these maps. The subcritical behavior of f#(x) was found to be very similar to the piecewise linear case. The mode-locking tongues become very narrow somewhere though they do not shrink to a point. Every shrinking point given in,Sec.Ill corresponds to where the tongue gets narrower locally. Therefore, the nontrivial subcritical structure described in this paper is expected to have some general meaning. Recently, narrowing of Arnold tongues has been also found in commensurate-incommensurate transitions'- \ The conditions for the nontrivial structure we derived in Sec.Ill may be helpful to understand related phenomena. The nature of aperiodic orbits in the supercritical regime as well as the transition to them will be the subject of a separate paper.
Acknowledgements The authors are grateful to Prof. ZHU Zhao-xuan, as well as to DING Ming-zhou, ZENG Wan-zhen and LI Jing-lin for helpful discussions. Note added in proof.
See P. Alstrom, Commun. Math. Phys.,
104(1986)581 for a rigorous proof of D=0 in the piecewise linear case.
We thank Dr. Alstrom and Prof. Griffiths for discussions at
the STATPHYS 16 meeting in Boston.
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