389 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 47, NO. 3, MARCH 2000 Transactions Briefs__________________________________________________________________ Bifurcations in One-Dimensional Piecewise Smooth Maps—Theory and Applications in Switching Circuits Soumitro Banerjee, M. S. Karthik, Guohui Yuan, and James A. Yorke Abstract—The dynamics of a number of switching circuits can be rep- resented by one-dimensional (1-D) piecewise smooth maps under discrete modeling. In this paper we develop the bifurcation theory of such maps and demonstrate the application of the theory in explaining the observed bifur- cations in two power electronic circuits. Index Terms—Chaos, nonlinear dynamics, power electronics. I. INTRODUCTION In the past few years it has been revealed that most feedback con- trolled switching circuits exhibit nonlinear phenomena and chaos over significant portions of the parameter space. It has been reported that, in addition to well-known nonlinear phenomena such as saddle-node bifurcation and period-doubling cascade, the feedback controlled switching circuits exhibit many atypical bifurcation phenomena. Generally the nonlinear dynamics of physical systems are analyzed by obtaining discrete models, mathematically known as maps. Power electronic circuits are particularly suitable for such discrete domain modeling because of the existence of periodic clock pulses in their con- trol logic. One can observe the state variables in synchronism with the clock pulses and can obtain a function that maps the vector of state variables from one clock instant to the next. We say a map is smooth if it has a continuous derivative. It is mono- tonic if there is no change in the sign of the derivative. It has been found that the discrete domain modeling of most switching circuits yield maps that are piecewise smooth and piecewise monotonic [1]. The analysis of the dynamics of such systems therefore requires a theoretical frame- work for the bifurcations that occur in such maps. In this paper we explore the dynamics of general piecewise smooth piecewise monotonic one dimensional (1-D) maps and apply the re- sults in explaining the nonlinear phenomena in two power electronic switching circuits. We will deal with the dynamics of the more general two-dimensional (2-D) maps in a subsequent paper [2]. II. THE PIECEWISE SMOOTH MAP Consider a 1-D map that maps the real line to itself and depends smoothly on a parameter (Fig. 1). A point on the real line divides it into two regions and . The map is piecewise smooth if 1) is continuous in and 2) is smooth in on each of the regions and , but its derivative is discontinuous at . In particular, the one-sided limits of the partial derivatives of must exist at the border . For piecewise mono- tonicity, we require that can change sign only at the border . Manuscript received July 20, 1998; revised December 8, 1998. This paper was recommended by Associate Editor H. Kawakami. S. Banerjee and M. S. Karthik are with the Department of Electrical Engi- neering, Indian Institute of Technology, Kharagpur 721302, India. G. H. Yuan and J. A. Yorke are with the Institute of Physical Science and Technology, University of Maryland, College Park, MD 20742 USA. Publisher Item Identifier S 1057-7122(00)02321-7. Let the map be given by for for (1) III. BIFURCATIONS IN THE SMOOTH REGIONS If a fixed point is in one of the regions or , the generic bifur- cations include the period doubling bifurcation and the saddle-node or tangent bifurcation. If at and and then there is a tangent bifurcation at . In one side of there is no fixed point while in the other side of there is one stable and one unstable fixed point, both on the same side of the border. The fixed points originating in a tangent bifurcation can collide with the border with a further change of parameter. The results of such border collision bifurcation will be outlined in the next section If then there is a period doubling bifurcation at . In the case of piecewise monotonic maps, the period doubling bifurcation has a spe- cial property. Lemma 1: Assume and are each monotonic functions. Then if a fixed point undergoes a period doubling bifurcation at and the double-period orbit bifurcates at then the periodic orbit must collide with the border for some in . Proof: Let the fixed point given by be in (the same argument will apply if it were in ). If the period doubling at occurs due to border collision (we will see in Section IV-B that this can happen), then the parameter range contains a border collision. If a smooth period doubling occurs at then (2) and (3) where is the second iterate of . Let be varied beyond such that Then for slightly greater than , both the fixed points of are on . Since is monotonic and its slope is negative, (4) 1057–7122/00$10.00 © 2000 IEEE