AD-A276 404

ARMY RESEARcH LABORATORY •

Remarks on a Resultof L.A.V. Carvalho

N. P. BhatiaUNIVERSITY OF MARYLAND

W. 0. EgerlandU.S. ARMY RESEARCH LABORATORY

ARL-TR-351 February 1994

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Remarks on a Result of L.A.V. Carvalho 4B592-321-13-3013

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N. P. Bhafia and W. 0. Egerland

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U.S. Army Research LaboratoryATTN: AMSRL-Cl-SAberdeen Proving Ground, MD 21005-5067

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U.S. Army Research LaboratoryATTN: AMSRL-OP-CI-B (Tech Lib) ARL-TR-351Aberdeen Proving Ground, MD 21005-5066

11. SUPPLEMENTARY NOTESThe first author is Professor of Mathematics at the University of Maryland, Baltimore County. His collaboradjon with thesecond author spans a period of 10 years. The Point of Contact for this report is Malcolm Taylor, U.S. Army ResearchLaboratory, ATT'N: AMSRL-CI-S, Aberdeen Proving Ground, MD 21005-5067.

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13. ABSTRACT (Maximum 200 words)

The report comments on two papers in the Journal of Mathematical Analysis and Applications and contains two newpropositions that supplement results that were previously published by the authors in BRL-TR-2702, BRL-TR-2762, andARO-Report 87-01. The note was submitted for publication in the Journal of Mathematical Analysis and Applications andis presently under review.

14. SUBJECT TERMS 15. NUMBER OF PAGES

periodic orbits, periodic loops, infinite loops, Sarkovskii's Theorem, periodic functions, 9

loops 16. PRICE CODE

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Preface.

The note "Remarks on a Result of L. A. V. Carvalho" was submitted forpublication in the Journal of Mathematical Analysis and Applications and ispresently under review. It is published here since the new Proposition A andB supplement results that appeared in the Technical Reports BRL-TR-2702,BRL-TR-2762, and in ARO-Report 87-1.We began to collaborate under the US Army Summer Faculty Research andEngineering Program in 1983. At that time "Chaos" was in its first bloomand we knew very little about it. We decided ab ovo to find out what hadbeen proved in the case of a continuous function of one real variable and, moreimportantly, to analyze the proof techniques that were used under the mereassumption of continuity. Beyond this "narrow focus" no program was es-tablished. The usual long preliminary process of getting to the essential corein a new subject was considerably shortened by Targonski's Studia Math-ematica Skript 6 Topics in Iteration Theory. The book had been acquiredby the BRL-Library at the recommendation of the BRL-mathematician R.E. Shear. After we had completed the "required reading" we noticed thatcertain proofs by contradiction could be replaced by constructive proofs. Inparticular, reasoning with predecessors, especially of fixed points, became aviable proof strategy. In time and in the context of periodic loops and ele-mentary periodic orbits the notions of L(oo) and E(oo) as infinite preorbitswere formalized and found their "rightful" places in the Sarkovskii ordering,places, where previously had been only "etc. dots". These and other resultswere presented at Army Conferences in 1984 (Rensselaer), 1986 (Cornell),1987 (West Point), 1988 (Colorado), 1991 (ARO-Durham) and informallydiscussed at other scientific meetings (Marseille-Luminy (1989), College Park(1991)).We wish to thank Messrs. H. L. Reed, S. S. Wolff and A. B. Cooper fortheir support of our endeavors in the "early years" and Messrs. W. H. Mer-magen and M. A. Hirschberg for their sponsorship in the "later years". Wealso take this opportunity to thank Messrs. Mermagen and Hirschberg fortheir patience during the writing of our paper "New Proof and Extension ofSarkovskii's Theorem" which went through a time-consuming monotone se-quence of improvements in an attempt to achieve a "best possible" measureof lucidity.

1 June 1993 N. P. BhatiaW. 0. Egerland

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TABLE OF CONTENTS

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Note

Remarks on a Result of L. A. V. Carvalho

N. P. BHATIA

University of Maryland, Catonsville, MD 21228

AND

W. O. EGERLAND

U. S. Army Research Laboratory, Aberdeen, MD 21005

In [1] Carvalho introduced the notion of a periodic n-step orbit for acontinuous function f : R? - R. By definition, f has a periodic n-step orbit(X0 , X1 ,',x,.'i-) if x, = x0< < X1 < < X,-1, where Xk+1 = f(xk), k =0, 1,... , n - 1. He proves that if f has a periodic n-step orbit, then f hasa periodic (n - 1)-step orbit. Invernizzi obtains the same result in a recentnote [2] via Miranda's theorem of 1940.

We wish to point out : (a) Periodic n-step orbits were already intro-duced under the name of n-periodic loops in [3]; (b) If f has an (n + 1)-periodic loop and n > 3, then f has two distinct n-periodic loops as shownin [3; Corollary 5.3]; (c) Carvalho's extension of Sarkovskii's order is onlyone of many corollaries to Theorem (SR), the principal result of [3]. Moreimportantly, we introduced in [3] the notion of an infinite loop. By defini-tion, f has an infinite loop if there exists x0 E R with an infinite preorbit(Xo, X-1,. . ,x-,,...) satisfying x0 < ... < x_, < x-(n,-) < ." < X- 2 < X-1

or x0 > .. > _- > ... > x- 2 > x-1 , where f(x_,) = X_(,-_).It follows then [3; Theorem 5.4] that if f has an infinite loop, then f hasn-periodic loops of all orders in at least two distinct copies for each n > 3.Furthermore, the notion of an infinite loop, as already shown in [3], is equiv-alent with the notion of turbulence introduced by Block and Coppel in [4],following a suggestion of Lasota and Yorke.

We conclude this note by proving two propositions. Proposition A givesa proof that an (n + 1)-periodic loop implies the existence of two distinctn-periodic loops if n > 3. This proof is different from the previous proof

[3; Corollary 5.3]. Proposition B states a four-point inequality that ensuresthe existence of an infinite loop and hence two distinct n-periodic loops foreach n > 3.

Proposition A. Let f : R --* R be continuous. If f has an (n + 1)-periodicloop, n > 3, then f has two distinct n-periodic loops.

Proof. For the proof we state first a lemma.Lemma F. Let f : R -- R be continuous and L1 , L2 ... , L, compact inter-vals such that

f(Li) D Li+,, i=1,2,...,n- 1

andf(L,) D L1 .

If Li fn Lj, i 3 j. is either empty or a singleton, then there is a point xo E L1with xi E Li+,, i = 1,2,... ,n - 1, and xo = x,. Such an xo has period n ifn is odd and period n or if n is even, but the period 1L is possible only if

22xi E Lj+j n L(!i+i+l), i = 0, 1, 2,..., 2 - 1.

Lemma F is proved in [5]. The proof of Proposition .,A. then follows for anyn > 3 from the construction exhibited in the following Fig. I for the specialcase n = 4, where xo, x 1 , X2 , x3 and x4 are the points of a 5-periodic loop, cois a fixed point, and c- 1 ,c- 2, and c-3 are predecessors of co.

L1L2 L3 L4

X5 XO C- 3 1 C-2M1 M2 M 3 M 4

Fig. 1. Any 5-periodic loop implies two 4-periodic loops.

The intervals L1 = [C- 3 , x1], L2 = [c- 2, x2], L3 = [c-1 , x3], L 4 = [co, x4],and M1 = [xI,c- 2], M2 = [x2, c- 1], M3 = [x3, c0], M 4 = [cO, x4] ensure theexistence of two distinct 4-periodic loops by Lemma F.Remark. Carvalho [1] and Invernizzi [2] prove only the existence of the 4-periodic loop defined by the L-intervals. We observe that the 4-periodic loopdefined by the M-intervals does not have a 4-periodic point in the interval

2

Proposition B. Let f : R --+ R be continuous. If

Xn+1 < Xn < XO < Xi

for some n > 2, then L(oc) holds, i.e.. f has an infinite loop.

Proof. The proof is based on Lemma 4.1 in [3] which states that if co isa fixed point of f with predecessors c- 1, c- 2 satisfying co< c-2 < c-1, thenL(oo) holds. We shall construct a six-point inequality that contains the as-sumptions of Lemma 4.1. We observe first that there is a fixed point co off satisfying x,, < co < x0, and we may clearly choose it so that the inter-val (co,xo] contains no other fixed points of f. Hence f(x) > x for eachx E (co, x0]. We note next that there is a successor Xk of xo. 1 < k < n - 1,such that zk+i < CO < xo < Xk. This inequality implies the existence ofanother fixed point do satisfying xo < do < xk, and we may assume thatf(x) < x in the interval (do, Xk]. Our final observation is that there must bean xj, 0 < 1 < k - 1, such that Xo < x, < do, x1+1 > Xk. This completes theconstruction

Xk+1 < CO < x1 < do < xk < X1+1,

which guarantees predecessors c- 1 and c- 2 in the intervals (do. Xk) and (xa, d0 ),respectively, with co < c- 2 < c-1 . Hence, by Lemma 4.1. L(oc) holds andthe proof is complete.

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REFERENCES

1. L. A. V. CARVALHO, On an Extension of Sarkovskii's Order, J. Math.Anal. Appl.138 (1989), 52-58.2. S. INVERNIZZI, On a Result by Carvalho Concerning Sarkovskii's Order,J. Math. Anal. Appl.168, Note, (1992), 284-285.3. N. P. BHATIA AND W. 0. EGERLAND, A Refinement of Sarkovskii'sTheorem, Proc. Amer. Math. Soc.102(1988), 965-972.4. L. S. BLOCK AND W. A. COPPEL, Stratification of Continuous Mapsof an Interval, Trans. Amer. Math. Soc.297(1986), 587-604.5, N. P. BHATIA AND W. 0. EGERLAND, New Proof and Extension ofSarkovskii's Theorem, to appear.

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