Volume 136. number 7.8 PHYSICS LETTERS A 17 April 1989 NUMERICAL STUDIES OF THE NOISY SINE CIRCLE MAP Maria MARKO~OVA and P. MARKO~ institute of Measurement and Measuring Technique ofEPRC, Slovak Academy of Sciences, DflbravskO cesta 9, CS-842 19 Bratislava, Czechoslovakia Received 27 June 1988; revised manuscript received 25 October 1988; accepted for publication 30 January 1989 Communicated by AR. Bishop The influence of additive noise on the period-locked regime of thesine circle map is studied numerically. Two critical values of noise, separating locked, transient and chaotic regimes have been found. In this Letter we study the role of additive noise As long as the map f 0(x, Q) is smooth and inver- on the sine circle map, tible, only regimes (i) and (ii) arise. Chaotic be- haviour occurs only for such ( 2, k) which cause f0 (x, ±1 =f,(x,, Q) Q) to be noninvertible, that is for k> I for the map =x,, +Q—(k/2it) sin(2itx,,)+ae,, , (1) (1). If k belongs to <0, 1>, then for any rational winding number W=P/Q there exists a nonzero in- respectively, because of the periodicity of the map terval of frequencies Q, z~QP/Q(k), in which the map (1), (1) has a periodical regime X,±Q=X,, (mod P) x,,~ =J~,(x,,, Q) (mod I) . (1’) (mode locking). These periodic intervals exhibit the so-called Arnold tongues (AT) in the (Q, k) space. Here e,, is a random noise variable, uniformly dis- The width of AT is zero for k~0and increases with tributed on the interval <— ~, ~>. a measures the k. For k> I different ATs begin to overlap, which strength of the external noise, and Qe <0, 1>, k~O yields the chaotic behaviour of the system. are parameters. For a 0 some special problems have been studied The circle map (1) serves as a good one-dimen- sional model for the study of the behaviour of dis- in refs. [5,6]. In ref. [5] the map (1) with Q= 0, k> 1 and with e,, being Gaussian has been treated sipative dynamical systems with two competing fre- both analytically and numerically. If k exceeds some quencies (e.g. the nonlinear damped oscillator driven critical value k~> 1, the existence of anomalous dif- by a periodical external force). Such systems were fusion of x,7 has been proved. Wiesenfeld and Satija studied in refs. [1—3]. For a=0, the properties of (1) are commonly [6] studied the circle map (I) for k< 1 in the pres- ence of a small random noise additive term and cal- known [4]. Let us briefly summarise the basic re- culated the power spectra for the locked and un- suits of ref. [4]: locked state. In accordance with ref. [7] they found For large n the winding number W, that the zero-frequency noise is highly suppressed W = urn [f (x, Q) —x] /n, (2) when the system is mode locked, and enhanced, when the dynamics of the system is unlocked. characterises the behaviour of (1). There are three In this Letter we study in detail the action of the possible regimes: additive noise on the periodic regime of the circle (i) periodic: x,+Q=x,,+P, then W=P/Q is map (1) and discuss a possible mechanism of turn- rational; ing the periodic regime into the chaotic one. To mea- (ii) quasiperiodic: W is irrational; sure the influence of noise quantitatively, we intro- (iii) chaotic. duce a new parameter, “the probability of occurrence 0375-960l/89/$ 03.50 © Elsevier Science Publishers B.V. 369 (North-Holland Physics Publishing Division)