Binary signals: a note on the prime period of a point

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1 Binary signals: a note on the prime period of a point Serban E. Vlad Oradea City Hall Abstract The nice x : R! f0; 1g n functions from the asynchronous systems theory are called signals. The periodicity of a point of the orbit of the signal x is de ned and we give a note on the existence of the prime period. Keywords and phrases: binary signal, period, prime period. MSC (2008): 94A12 The asynchronous systems are the models of the digital electrical circuits and the nice functions representing their inputs and states are called signals. Such systems are generated by Boolean functions that iterate like the dynamical systems, but the iterations happen on some coordinates only, not on all the coordinates (unlike the dynamical systems). In order to study their periodicity, we need to study the periodicity of the (values of the) signals rst. Our present aim is to de ne and to characterize the periodicity and the prime period of a point of the orbit of a signal. De nition 1 The set B = f0; 1g is a eld relative to 0 0 ; 0 0 ; the modulo 2 sum and the product: A linear space structure is induced on B n ; n 1: Notation 2 A : R! B is the notation of the characteristic function of the set A R : 8t 2 R; 1; if t 2 A; A (t) = 0; otherwise : De nition 3 The continuous time signals are the functions x : R! B n of the form 8t 2 R; x(t) = ( 1;t0 )(t) x(t 0 ) [t0 ;t 1 )(t) ::: x(t k ) [tk ;t k+1 )(t) ::: (1) 1

2 where 2 B n and t k 2 R; k 2 N is strictly increasing and unbounded from above: Their set is denoted by S (n) : is usually denoted by x( 1+ 0) and is called the initial value of x. De nition 4 The left limit x(t the function 8t 2 R; 0) of x(t) from (1) is by de nition x(t 0) = ( 1;t0 ](t)x(t 0 ) (t0 ;t 1 ](t):::x(t k ) (tk ;t k+1 ](t)::: (2) Remark 5 The de nition of x(t 0) does not depend on the choice of (t k ) that is not unique in (1); for any t 0 2 R; the existence of x(t 0 0) is used in applications under the form 9" > 0; 8 2 (t 0 "; t 0 ); x() = x(t 0 0): De nition 6 The set Or(x) = fx(t)jt 2 Rg is called the orbit of x. Notation 7 For x 2 S (n) and 2 Or(x); we denote T x = ftjt 2 R; x(t) = g: (3) De nition 8 The point 2 Or(x) is called a periodic point of x 2 S (n) or of Or(x) if T > 0; t 0 2 R exist such that ( 1; t 0 ] T x x( 1+0); (4) 8t 2 T x \ [t 0 ; 1); ft + zt jz 2 Zg \ [t 0 ; 1) T x : (5) In this case T is called the period of and the least T like above is called the prime period of. Theorem 9 Let x 2 S (n) ; = x( t 0 ; t 1 2 R having the property that 1 + 0); T > 0 and the points t 0 < t 1 < t 0 + T; (6) ( 1; t 0 )[[t 1 ; t 0 +T )[[t 1 +T; t 0 +2T )[[t 1 +2T; t 0 +3T )[::: = T x (7) hold: a) For any t 0 2 [t 1 T; t 0 ); the properties (4), (5) are ful lled and for any t 0 =2 [t 1 T; t 0 ); at least one of the properties (4), (5) is false. b) For any T 0 > 0; t 00 2 R such that ( 1; t 00 ] T x x( 1+0); (8) 8t 2 T x \ [t 00 ; 1); ft + zt 0 jz 2 Zg \ [t 00 ; 1) T x ; (9) we have T 0 T and t 00 2 [t 1 T 0 ; t 0 ): 2

3 Proof. a) Let t 0 2 [t 1 T; t 0 ): From we infer the truth of (4). Furthermore, we have ( 1; t 0 ] ( 1; t 0 ) T x ; T x \[t 0 ; 1) = [t 0 ; t 0 )[[t 1 ; t 0 +T )[[t 1 +T; t 0 +2T )[[t 1 +2T; t 0 +3T )[::: and we take an arbitrary t 2 T x \ [t 0 ; 1): If t 2 [t 0 ; t 0 ); then ft + zt jz 2 Zg \ [t 0 ; 1) = ft; t + T; t + 2T; :::g [t 0 ; t 0 ) [ [t 0 + T; t 0 + T ) [ [t 0 + 2T; t 0 + 2T ) [ ::: [t 0 ; t 0 ) [ [t 1 ; t 0 + T ) [ [t 1 + T; t 0 + 2T ) [ ::: T x : If 9k 1 0; t 2 [t 1 + k 1 T; t 0 + (k 1 + 1)T ); then there are two possibilities: Case t 2 [t 0 + (k 1 + 1)T; t 0 + (k 1 + 1)T ); when ft+zt jz 2 Zg\[t 0 ; 1) = ft+( k 1 1)T; t+( k 1 )T; t+( k 1 +1)T; :::g [t 0 ; t 0 ) [ [t 0 + T; t 0 + T ) [ [t 0 + 2T; t 0 + 2T ) [ ::: [t 0 ; t 0 ) [ [t 1 ; t 0 + T ) [ [t 1 + T; t 0 + 2T ) [ ::: T x ; Case t 2 [t 1 + k 1 T; t 0 + (k 1 + 1)T ); when ft+zt jz 2 Zg\[t 0 ; 1) = ft+( k 1 )T; t+( k 1 +1)T; t+( k 1 +2)T; :::g [t 1 ; t 0 + T ) [ [t 1 + T; t 0 + 2T ) [ [t 1 + 2T; t 0 + 3T ) [ ::: [t 1 ; t 0 + T ) [ [t 1 + T; t 0 + 2T ) [ [t 1 + 2T; t 0 + 3T ) [ ::: T x : We suppose now that t 0 =2 [t 1 T; t 0 ): If t 0 < t 1 T; we notice that maxft 0 ; t 0 T g < t 1 T and that for any t 2 [maxft 0 ; t 0 T g; t 1 T ); we have t 2 T x \ [t 0 ; 1) but t + T 2 ft + zt jz 2 Zg \ [t 0 ; 1) \ [t 0 ; t 1 ); thus t + T =2 T x and (5) is false. On the other hand if t 0 t 0 ; then x(t 0 ) 6= implies t 0 =2 T x and consequently (4) is false. b) The fact that t 00 2 [t 1 T 0 ; t 0 ) is proved similarly with the statement t 0 2 [t 1 T; t 0 ) from a): t 00 t 0 is in contradiction with (8) and t 00 < t 1 T 0 is in contradiction with (9). We suppose now against all reason that (8), (9) are true and T 0 < T: Let us note in the beginning that maxft 1 ; t 0 + T T 0 g < minft 0 + T; t 1 + T T 0 g 3

4 is true, since all of t 1 < t 0 + T; t 1 < t 1 + T T 0 ; t 0 + T T 0 < t 0 + T; t 0 + T T 0 < t 1 + T T 0 hold. We infer that any t 2 [maxft 1 ; t 0 + T T 0 g; minft 0 + T; t 1 + T T 0 g) ful lls t 2 [t 1 ; t 0 + T ) T x \ [t 00 ; 1) and t 0 +T maxft 1 +T 0 ; t 0 +T g t+t 0 < minft 0 +T +T 0 ; t 1 +T g t 1 +T in other words t+t 0 2 ft+zt 0 jz 2 Zg\[t 00 ; 1); but t+t 0 2 [t 0 +T; t 1 +T ); thus t + T 0 =2 T x ; contradiction with (9). We conclude that T 0 T: Lemma 10 We suppose that the point 2 Or(x) is periodic: T > 0; t 0 2 R exist such that (4), (5) hold. If for t 1 < t 2 we have [t 1 ; t 2 ) T x \ [t 0 ; 1); then 8k 1; [t 1 + kt; t 2 + kt ) T x : Proof. Let k 1 and t 2 [t 1 + kt; t 2 + kt ) be arbitrary. As t kt 2 [t 1 ; t 2 ) and from the hypothesis t kt 2 T x \ [t 0 ; 1); we have from (5) that t 2 ft kt + zt jz 2 Zg \ [t 0 ; 1) T x : Theorem 11 We ask that x is not constant and let the point = x( 1 + 0) be given, as well as T > 0; t 0 2 R such that (4), (5) hold. We de ne t 0 ; t 1 2 R by the requests 8t < t 0 ; x(t) = ; (10) x(t 0 ) 6= ; (11) t 1 < t 0 + T; (12) 8t 2 [t 1 ; t 0 + T ); x(t) = x(t 0 + T 0); (13) Then the following statements are true: x(t 1 0) 6= x(t 1 ): (14) t 1 T t 0 < t 0 < t 1 ; (15) ( 1; t 0 )[[t 1 ; t 0 +T )[[t 1 +T; t 0 +2T )[[t 1 +2T; t 0 +3T )[::: T x : (16) Proof. The fact that x is not constant assures the existence of t 0 as de ned by (10), (11). On the other hand t 1 as de ned by (12), (13), (14) exists itself, since if, against all reason, we would have 8t < t 0 + T; x(t) = x(t 0 + T 0); (17) then (10), (11), (17) would be contradictory. By the comparison between (10), (11), (13), (14) we infer t 0 t 1 : From (4), (10), (11) we get t 0 < t 0 : 4

5 Case t 0 2 [t 1 T; t 0 ) In this situation t 0 + T 2 [t 1 ; t 0 + T ); (18) (4) = x(t 0 ) (5) = x(t 0 + T ) (13);(18) = x(t 1 ) (19) and from (10), (11), t 0 t 1 ; (19) we have that t 0 < t 1 : (15) is true. Let us note that (13), t 0 < t 1 ; (19) imply [t 1 ; t 0 + T ) T x \ [t 0 ; 1) and, from Lemma 10 together with (10), (11) we obtain the truth of (16). Case t 0 < t 1 T As t 1 T < t 0 we can write (10) = x(t 1 T ) (5) = x(t 1 ) (20) and the property of existence of the left limit of x in t 1 shows the existence of " > 0 with 8t 2 (t 1 "; t 1 ); x(t) = x(t 1 0): (21) We take " 0 2 (0; minft 1 T t 0 ; "g); thus for any t 2 (t 1 T " 0 ; t 1 T ) we have t + T 2 (t 1 " 0 ; t 1 ) (t 1 "; t 1 ); (22) and since we conclude t > t 1 T " 0 > t 0 ; (23) t < t 1 T < t 0 (24) (10);(24) = x(t) (5);(23) = x(t + T ) (21);(22) = x(t 1 0): (25) The statements (14), (20), (25) are contradictory, thus t 0 < t 1 impossible. T is Example 12 We take x 2 S (1) ; x(t) = ( 1;0) (t) [1;2) [3;5) [6;7) [8;10) [11;12) ::: In this example = 1; t 0 = 0; t 1 = 3; T = 5 is prime period and t 0 2 [ 2; 0): We note that T may be prime period without equality at (16) but if we have equality at (16) then, from Theorem 9, T is prime period. Remark 13 Theorems 9 and 11 refer to the case when the periodic point coincides with x( 1 + 0): The situation when 6= x( 1 + 0) is not di erent in principle. 5

6 References [1] D. V. Anosov, V. I. Arnold (Eds.) Dynamical systems I., Springer- Verlag, (Encyclopedia of Mathematical Sciences, Vol. 1) (1988). [2] D. K. Arrowsmith, C. M. Place, An introduction to dynamical systems, Cambridge University Press, (1990). [3] Michael Brin, Garrett Stuck, Introduction to dynamical systems, Cambridge University Press, (2002). [4] Robert L. Devaney, A rst course in chaotic dynamical systems. Theory and experiment, Perseus Books Publishing, (1992). [5] Robert W. Easton, Geometric methods in discrete dynamical systems, Oxford University Press, (1998). [6] Boris Hasselblatt, Anatole Katok, Handbook of dynamical systems, Volume 1, Elsevier, (2005). [7] Richard A. Holmgren, A rst course in discrete dynamical systems, Springer-Verlag, (1994). [8] Jurgen Jost, Dynamical systems. Examples of complex behaviour, Springer-Verlag, (2005). Serban Vlad - Department of Computers Oradea City Hall Piata Unirii, Nr. 1, , Oradea, Romania serban_e_vlad@yahoo.com 6