PHYSICAL REVIEW E VOLUME 48, NUMBER 3 SEPTEMBER 1993 Geometric mechanism for antimonotonicity in scalar maps with two critical points Silvina Ponce Dawson Laboratory for Plasma Research, University of Maryland, College Park, Maryland 20742 Celso Grebogi Laboratory for Plasma Research, Institute for Physical Science and Technology, and Department of Mathematics, University of Maryland, College Park, Maryland 20742 Huseyin Kodak Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124 (Received 21 April 1993) Concurrent creation and destruction of periodic orbits — antimonotonicity — for one-parameter scalar maps with at least two critical points are investigated. It is observed that if, for a parameter value, two critical points lie in an interval that is a chaotic attractor, then, generically, as the parameter is varied through any neighborhood of such a value, periodic orbits should be created and destroyed infinitely often. A general mechanism for this complicated dynamics for one-dimensional multimodal maps is proposed similar to the one of contact-making and contact-breaking homoclinic tangencies in two- dimensional dissipative maps. This subtle phenomenon is demonstrated in a detailed numerical study of a specific one-dimensional cubic map. PACS number(s): 05. 45. + b I. INTRODUCTION Bifurcations of periodic points of one-dimensional maps command a prominent place in theoretical and ex- perimental investigations of dynamical systems. For ex- ample, the one-parameter quadratic map x„+, =a — x„ has been studied as the quintessential example exhibiting one of the most common routes to chaos: period- doubling cascades [1]. Moreover, some of the important bifurcation behaviors of the quadratic map have been found to be universal in a large class of unimodal maps — maps with one critical point [2]. Despite the re- markable success of unimodal maps in modeling bifurca- tions in many applications [3], they also have inherent limitations. For example, as the parameter a in the quadratic map is increased, it has been shown that periodic orbits are only created but never destroyed [4]. Unlike the monotone bifurcation behavior of the quad- ratic map, creation and destruction of periodic orbits have been observed both numerically [5] and experimen- tally [6] in various nonlinear systems. For example, as depicted in Fig. 1, reversals of period-doubling cascades are indeed visible in a numerically computed bifurcation diagram of the Poincare map of the periodically forced oscillator of Van der Pol, which is prototypical model for many nonlinear oscillatory phenomena. The Poincare map of this nonlinear oscillator, as well as many others, can be captured by a degree-one circle map, or simply by a scalar multimodal map — a map with several critical points [7]. Indeed, this is one of the reasons why mul- timodal maps have been the center of different analytical and numerical studies [8]. In this paper we present a geometric mechanism for the creation and destruction of periodic orbits infinitely X -1. 45 5. 194 5.240 FIG. 1. Reversals of period-doubling cascades in the periodi- cally forced oscillator of Van der Pol x + 5(x — 1)x + x =40sin(cot). In this bifurcation diagram, the x coordinate of the Poincare map is plotted against the driving frequency co. often as a parameter is increased near certain common parameter values in generic chaotic multimodal scalar Inaps with at least two critical points. We call such con- current creation and destruction of periodic orbits an- timonotonicity [9]. To describe antimonotonicity for mul- timodal scalar maps more precisely, we proceed with some definitions. In a one-parameter scalar map x„+, =F(x„, a), a pa- rameter value cz=ao is called an orbit creation value if a periodic orbit does not exist for a&ao but exists for n)ao. Similarly, a=no is called an orbit destruction ualue if a certain periodic orbit exists for a&ao but it does not exist for a) ao. A map x„+, =F(x„, a) is said to be increasing (decreasing) monotone in an interval J of parameter values if, for aE J, periodic orbits are only 1063-651X/93/48(3)/1676(7)/$06. 00 48 1676 1993 The American Physical Society