## Papers

[DEIS78A] Deisch, Cecil W., *Simple Switching Control Method Changes Power Converter Into a
Current Source*, PESC'78, IEEE Power Electronics Specialists
Conference, 1978, pp. 300-306.

A switching converter with an LC output filter behaves as a loose-tolerance voltage-controlled current source if each switch closure is ended when switch current reaches an adjustable threshold. This converter is then combined with an external feedback to produce a precise output voltage. By generating a fixed voltage with a current source in this manner, the converter has many advantages including continuous protection of the switches, stable and equal load sharing when several converters are operated in parallel, inherent overload protection, automatic switch symmetry correction, and fast system response. (AUTHOR ABSTRACT) Discusses instability in constant frequency current-threshold controlled circuits for duty ratios over 50%. Chaos is implied (raucous whine) although not mentioned. Cure is decreasing magnetizing inductance or adding a ramp. (JF) Bell Laboratories, Naperville, IL. 7 pages, 10 figures, no tables, no equations, 3 references.

[MARO78A] Marotto, F.R., *Snap-back Repellers Imply Chaos in R^2*, J. Math. Anal. Appl.
63, 199-223, 1978.

Not abstracted in this hypertext. (JF)

[BAIL80A] Baillieul, J., R. W.
Brockett, and R. B. Washburn, *Chaotic Motion in Nonlinear
Feedback Systems*, IEEE Transactions on Circuits and Systems, Vol. 27, No.
11, pp. 990-997, Nov 1980.

New criteria are found which imply the existence of chaos in the mathematics literature, and in fact our methods apply to a class of systems which do not satisfy the hypotheses of the usual theorems on chaos in R^n. The results are stated in such a way as to preserve the flavor of many well-known frequency-domain stability techniques. The results provide easily verifiable criteria for the existence of chaos in systems which are of dimensions greater than one. (AUTHOR ABSTRACT) Scientific Systems Inc., Cambridge, MA (Baillieul and Washburn). Division of Applied Science, Harvard University and Scientific Systems (Brockett). 8 pages, 4 figures, no tables, 17 numbered equations, 5 references. (Highly mathematical)

[BROC84A] Brockett, Roger W., and Jonathan R. Wood, *Understanding Power Converter Chaotic Behavior Mechanisms in Protective and Abnormal Modes*, POWERCON'11, Eleventh Annual International Power Electronics Conference, April 1984, pp. E-4 1 to 15.

Mathematical Chaos has received much attention in the fields of pure and applied mathematics in recent years. The intent of this paper is to show mechanisms which seem to give rise to its occurrence in electronic power converters, and to illustrate its effect on the performance and component stresses of such converters. (AUTHOR ABSTRACT) (Describes the audio aspects of a chaotic power converter as "...typically producing an audible noise somewhat like the sound of frying bacon." [JF]) 15 pages, 16 figures, no tables, no numbered equations, 1 FORTRAN listing, 15 references. Updated and expanded in: [WOOD89A]

[MATS87A] Matsumoto, Takashi, *Chaos in Electronic
Circuits*, Proceedings IEEE, Vol. 75, No. 8, pp. 1033-1057, Aug. 1987.

Not abstracted in this hypertext. (JF)

[HAMI88A] Hamill, David C., *Subharmonics and Chaos in a Controlled Switched-Mode Power
Converter*, IEEE Transactions on Circuits and Systems, CAS-35, No. 8,
August, 1988, pp. 1059-1061.

A difference equation is derived for the output current, at successive switching events, of a simple switching-regulator dc/dc converter employing a pulsewidth modulator (PWM). Wideband feedback control of the nonlinear circuit leads to a one-dimension return map of zigzag form from which a stability criterion is found. Operation in the unstable region is described and verified by numerical simulation. The system exhibits a "noisy" bifurcation, chaos and subharmonics. (AUTHOR ABSTRACT) Department of Electronic and Electrical Engineering, University of Surrey, Guildford, U.K. 3 pages, 4 figures, no tables, 5+ equations, 14 references.

[DEAN89A] Deane, Jonathan H.B., and
David C. Hamill, *Instability, Subharmonics, and Chaos in
Power Electronic Systems*, PESC'89, IEEE Power Electronics Specialists
Conference, 1989, pp. 34-42. (Reprinted with minor changes in PELS'90)

The principles of chaos theory are applied to a variety of nonlinear power electronic circuits. With the onset of instability, the phenomena of subharmonics, quasi-periodicity, and chaos are predicted and observed. Examples treated are diodes with charge storage (with application to resonant converters); a ferroresonant circuit; a controlled thyristor rectifier circuit; and a buck converter controlled by pulse-width modulation. (AUTHOR ABSTRACT) Department of Electronic and Electrical Engineering, University of Surrey, Guildford, Surrey, UK. 9 pages, 18 figures, 1 table, 4 equations, 29 references.

[KURO89A] Kuroe, Y. and S. Hayashi,
*Analysis of Bifurcation in Power Electronic Induction Motor
Drive System*, IEEE Power Electronics Specialists Conference Record, pp.
923-930, 1989.

Variable frequency induction motor drives are known to become unstable at certain operating conditions, which causes unusual vibrations in the systems. In this paper the instability phenomena in power electronic induction motor drive systems are investigated from the point of view of bifurcation theory. A method to determine bifurcation values of system parameters is discussed. It is shown that some kinds of bifurcations are observed in power electronic induction motor drive systems. The proposed method make it possible not only [to] determine instability regions of system parameters but also to investigate qualitative properties of the instability phenomena. (AUTHOR ABSTRACT) Short review of bifurcation theory showing three types of bifurcations depending on the solution to the system Jacobian matrix -- Saddle node bifurcation, period doubling or subharmonic bifurcation, and Hopf bifurcation. The characteristics of these types of bifurcation are discussed. (JF) Department of Electrical Engineering, Kobe University, Kobe, Japan. 8 pages, 12 figures, 55 equations, 11 references.

[WOOD89A] Wood, Jonathan, *Chaos: A Real Phenomenon in Power Electronics*, APEC'89,
IEEE Applied Power Electronics Conference, 1989, pp. 115-124.

Mathematical Chaos has received much attention in diverse disciplines in recent years. The phenomenon occurs readily in Power Electronics, though a comparatively small amount of helpful literature exists as yet. This paper is intended as an introductory tutorial on the subject, for the benefit of practicing engineers. In addition, previously unpublished experimental results will be given which validate predicted behavior presented in an earlier work. (AUTHOR ABSTRACT) MITRE Corp., Bedford, MA. 10 pages, 34 figures, no tables, no equations, one FORTRAN program, 20 references. Update of [BROC84A]

[DEAN90A] Deane, Jonathan H.B., and
David C. Hamill, *Instability, Subharmonics, and Chaos in
Power Electronic Systems*, IEEE Transactions on Power Electronics, Vol. 5,
No. 3, July 1990, pp. 260-268. PELS'90 (Reprint of PESC'89 paper with
minor changes.)

The concept of chaos is applied to a variety of nonlinear power electronic circuits. With the onset of instability, the phenomena of subharmonics, quasi-periodicity, and chaos are storage (with application to resonant converters); a ferroresonant circuit; a controlled thyristor rectifier circuit; and a buck converter controlled by pulse-width modulation (PWM). (AUTHOR ABSTRACT) Includes short glossary of terms. Good introductory paper. (JF) Department of Electronic and Electrical Engineering, University of Surrey, Guildford, Surrey, UK. 9 pages, 19 figures, no tables, 7 equations, 36 references.

[DEAN90B] Deane, Jonathan H. B., and
David C. Hamill, *Analysis, Simulation and Experimental Study
of Chaos in the Buck Converter*, PESC'90, IEEE Power Electronics
Specialist Conference, 1990, Vol II, pp. 491-498.

A buck dc-dc converter, whose output voltage is controlled by naturally-sampled, constant-frequency PWM, is operated in continuous conduction mode. Two versions are treated, a first-order and second-order circuit. Their behavior is modeled analytically and numerically. For certain values of the circuit parameters instability occurs. Strange phenomena of multiple pulsing, skipped cycles, subharmonics and chaos are predicted theoretically and observed experimentally, including a period-doubling route to chaos. There is good agreement between theory and experiment. (AUTHOR ABSTRACT) Department of Electrical Engineering, University of Surrey, Guildford, Great Britain. 8 pages, 12 figures, no tables, 28 equations, 7 references.

[KREI90B] Krein, Philip T., and
Richard M. Bass, *Types of Instability Encountered in Simple
Power Electronic Circuits: Unboundedness, Chattering, and Chaos*,
APEC'90, IEEE Applied Power Electronics Conference Proceedings, 1990, pp.
191-194.

Nonlinear characteristics of power electronic circuits bring about unusual instabilities. Three major types have been reported. Their characteristics and consequences are examined. Unbounded instability, the most familiar type, is characteristic of some circuit configurations of many converters. Chattering is a potentially destructive instability mode not predicted by some idealized models. Chaotic instability may be nondestructive. (AUTHOR ABSTRACT) Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois. 4 pages, 7 figures, no tables, 1 equations, 12 references.

[DEAN91A] Deane, J. H. B., *Chaotic Behavior in Current-Mode Controlled DC/DC
Converters*, Inst. Elect. Eng. Electron. Lett., vol 27, no. 13, pp.
1172-1173, June 1991

Not abstracted in this hypertext (JF)

[DEAN92A] Deane, J. H. B., *Chaos in a Current-Mode Controlled Boost DC-DC Converter*,
IEEE Trans. Circuits syst., vol. 39, no. 8, Aug. 1992.

Not abstracted in this hypertext (JF)

[HAMI92A] Hamill, David C., Jonathan
H. B. Deane, and David J. Jefferies, *Modeling of Chaotic
DC-DC Converters by Iterated Nonlinear Mappings*, IEEE Transactions on
Power Electronics, Vol. 7, No. 1, January 1992, pp. 25-36. PELS'92

In parameter ranges where conventional methods break down, dc-dc converters may be described by iterated mappings, a nonlinear discrete modeling technique. The underlying principles are explained and are applied to the example of a PWM-controlled buck converter. Stable behavior and bifurcations to chaos are predicted by numerical evaluation of the governing mapping and are confirmed by experiment. (AUTHOR ABSTRACT) Department of Electronic and Electrical Engineering, University of Surrey, Guildford, U.K. 12 pages, 13 figures, 1 table, 16 equations, 12 references.

[TSE94A] Tse, C. K., *Flip Bifurcation and Chaos in Three-State Boost Switching
Regulators*, IEEE Transactions on Circuits and Systems I: Theory and
Applications, Vol.CAS-41, No. 1, pp. 16-23, Jan. 1994.

A first-order iterative map that describes the dynamics of a simple feedback boost switching regulator operating in the discontinuous current mode is derived. Analysis of this map shows that flip bifurcations occur at certain values of the feedback factor. Results from computer simulations and experiments reveal that the system exhibits a typical period-doubling route to chaos under the particular operation condition studied in this paper. It is found that a special kind of randomness, arising from the ability of the system to change its configuration in more than on predefined pattern, constitutes a unique feature of the chaotic dynamics of switched-mode converter circuits. (AUTHOR ABSTRACT) Department of Electronics Engineering, Hong Kong Polytechnic, Hong Kong. 8 pages, 22 figures, 2 tables, 32 equations, 10 references.

[TSE94B] Tse, C. K., *Chaos from a Buck Switching Regulator Operating in
Discontinuous Mode*, International Journal of Circuit Theory and
Application, vol 22, no 4, pp. 263-278, July/August 1994.

Not abstracted in this hypertext. (JF)

[HAMI95A] Hamill, D. C., *Power Electronics: A Field Rich in Nonlinear Dynamics*,
presented at the Int. Workshop Nonlinear Dynamics Electronic Syst., Dublin,
Ireland, July 1995.

This review paper starts by setting out the aims and applications of power electronics, and continues with a brief history and a list of the important power semiconductor devices. The related areas of ac machines and power systems are also briefly visited. The development of nonlinear dynamics in electronic circuits is reviewed. Then a typical power converter, a controlled buck dc-dc converter, is modelled by the conventional method of averaging and liniarization (which predicts stability), and by a nonlinear map based method, which reveals bifurcations, subharmonics and chaos. The numerical problems caused by the discontinuities in the state equations of power electronics are discussed. Finally, some possible future applications are considered. (AUTHOR ABSTRACT) Contains 72 references, including many on chaos. (JF)

[PODD95A] Poddar, Gautam, Krishnendu
Chakrabarty, and Soumitro Banerjee, *Experimental Control of
Chaotic Behavior of a Buck Converter*, IEEE Transactions on Circuits and
Systems, Vol. 43, No. 8, August 1995, pp. 502-504. CAS

This letter reports a method for control of chaos in the dc-dc buck converter. The method differs from existing ones and is particularly useful for piecewise linear systems with switching nonlinearity. (AUTHOR ABSTRACT) Department of Electrical Engineering, Institute of Technology, Kharagpur, India. 3 pages, 4 figures, no tables, no equations, 4 references.

[TSE95A] Tse, C. K. and W. C. Y. Chan,
*Chaos From a Current-Programmed Cuk Converter*,
International Journal of Circuit Theory and Application, vol 23, no 3, pp.
217-225, May/June 1995.

Not abstracted in this hypertext. (JF)

[ZAFR95A] Zafrany, I. and Ben-Yaakov,
S., *A Chaos Model of Subharmonic Oscillations in Current Mode
Boost Converters*. IEEE Power Electronics Specialists Conference, PESC-95,
1111-1117, Atlanta 1995.

Chaos concepts formulated in a discrete form were applied to examine instability conditions in a current mode PWM Boost converter under open and closed outer-loop conditions. A simple expression for the maximum duty cycle for subharmonic-free operations was developed and applied to asses the effects of the outer loop on subharmonic oscillations in the converter under study. (AUTHOR ABSTRACT) (Copy Available on the Web)

[CHAK96A] Chakrabarty, Krishnendu;
Goutam Poddar, and Soumitro Banerjee, *Bifurcation Behavior of
the Buck Converter*, IEEE Transactions on Power Electronics, Vol. 11, No.
3, May 1996, pp. 439-447.

The dc-dc buck converter, a widely used chopper circuit, exhibits subharmonics and chaos if current feedback is used. This paper investigates the dependence of the system behavior on its parameters. The bifurcation phenomena and a mapping of the parameter space have been presented. This knowledge is vital for designing practical circuits.

[CHAN96A] Chan, W. C. Y., and C. K.
Tse, *A Universal Bifurcation Path Conjectured From
Current-Programmed Switching Converters*, in Proc. Int. Sym. Nonlinear
Theory Appl, Rochi, Japan, Oct. 1996, pp. 121-124.

Not abstracted in this hypertext. (JF)

[CHAN96B] Chan, W. C. Y., and C. K.
Tse, *Studies of Routes to Chaos Current-Programmed DC/DC
Converters*, IEEE Power Electronics Specialist Conference, Record, Baveno,
Italy, June 1996, pp. 789-795.

This paper reports two distinct types of bifurcation paths exhibited by current-programmed dc/dc/ converters, which are distinguished by the ability or inability to undergo repeated period-doublings. In particular, this paper focuses on the open-loop converters, i.e., ones that do not contain an output feedback loop. It is shown that for the case of the boost converter, the two different types of bifurcation paths can be viewed as part of another bifurcation in which a quasi-periodic sequence transmutes into a period-doubling sequence. For the buck converter, however, period-doubling routes to chaos are found not possible, and in this case, the bifurcation patterns essentially comprise quasi-periodic regions with windows of periodicity. With the range of one bifurcation parameter fixed, the bifurcation diagrams contain more scaled-down replicas, as another bifurcation parameter increases, which are arranged in a manner similar to the period-doubling cascade.

[DEAN96A] Deane, J. H. B., and D. C.
Hamill, *Improvement of Power Supply EMC by Chaos*,
Electronics Letters, Vol. 32 No. 12, p. 1045 June 1996.

Not abstracted in this hypertext. (JF)

[DIBE96B] di Bernardo, M., F.
Garofalo, L. Glielmo, and F. Vasca, *Quasi-Periodic Behaviors
in Dc/Dc/ Converters*, IEEE Power Electronics Specialists Conference
Record, pp. 1376-1381, 1996.

Quasi-periodicity and nonlinear phenomena in open and closed loop boost DC/DC converters are analyzed. A new discrete time nonlinear mapping, the impact map, which models closed loop DC/DC converters is presented and compared with the usual stroboscopic map. Also, it is shown that the converter can exhibit complicated periodic behavior and quasi-periodicity both in open loop, when simulated by a suitable external sinusoidal disturbance, and in closed loop with an appropriate choice of the state feedback gains. Simulation results show that in the closed loop case Naimarck-Sacker bifurcation and self-sustained tori can be obtained. (AUTHOR ABSTRACT) Department of Engineering Mathematics, University of Bristol (di Bernardo), Dipartimento di Informatica e Sistemistica, Università di Napoli Federico II (Glielmo, Garofalo, Vasca). (6 pages, 6 figures, 27 equations, 4 references.

[FOSS96A] Fossas, E., and G. Olivar,
*Study of Chaos in the Buck Converter*, IEEE Transactions
on Circuits and Systems Part I, Vol. 43, No. 1, pp. 12-25 January 1996.

Not abstracted in this hypertext. (JF)

[JALA96A] Jalai, S., I. Dobson, R. H.
Lasseter, and G. Venkataramanan, *Switching Time Bifurcations
in a Thyristor Controlled Reactor*, IEEE Trans. Circuits Syst. I, vol. 43,
pp. 209-218, March 1996

Not abstracted in this hypertext. (JF)

[TSE96A] Tse, C. K., S. C. Fung, and M.
W. Kwan, *Experimental Confirmation of Chaos in a
Current-Programmed Cuk Converter*, IEEE Transactions on Circuits and
Systems I: Theory and Applications, Vol.CAS-43, No. 7, pp. 605-608, Jul.
1996.

This letter presents experimental evidence for the chaotic behavior in a fourth-order Cuk converter under current-programmed control which has been studied and simulated in TSE95A. (AUTHOR ABSTRACT) Department of Electronics Engineering, Hong Kong Polytechnic, Hong Kong. 4 pages, 6 figures, no tables, 2 references.

[ASTO97A] Aston, P. J., J. H. B.
Deane, and D. C. Hamill, *Targeting in Systems with
Discontinuities, with Applications to Power Electronics*, IEEE Transactions
on Circuits and Systems Part I, Vol. 44, No. 10, pp. 1034-1039, October
1997.

Not abstracted in this hypertext. (JF)

[BANE97A] Banerjee, S., E. Ott, *Anomalous Bifurcations in dc-dc Converters: Borderline Collisions
in Piecewise Smooth Maps*, PESC97, Record of the 28th Annual IEEE Power
Electronics Specialists Conference, June 22-27, 1997.

Recently, non-standard bifurcations have been reported in power electronic dc-dc converters. We show that sampled data models with stroboscopic sampling yield piecewise smooth maps and that most of the observed "anomalous" bifurcations fall into a recently discovered class called "border collision bifurcations." We offer analytical explanation of the dynamics of three converter topologies.

[CHAN97A] Chan, C. Y., and Chi K.
Tse, *Study of Bifurcations in Current-Programmed DC/DC Boost
Converters: From Quasi Periodicity to Period-Doubling,* IEEE Transactions
on Circuits and Systems - I: Fundamental Theory and Applications: Vol. 44, No.
12, December 1997.

This paper studies the bifurcation paths exhibited by a
simple second-order dc/dc boost converter under current-programmed control with
and without voltage feedback. Previous work in this area has reported two
distinct types of bifurcation paths, namely via *regions of quasi-periodic
orbits* and *period-doubling*. This paper demonstrates that the two
different types of bifurcation paths can, in fact, be viewed as part of another
bifurcation in which the quasi-periodic sequence transmutes into the
period-doubling sequence, and that such a bifurcation is observed regardless of
the presence of the outer voltage feedback loop as long as a suitable set of
bifurcation parameters is chosen. the describing iterative map is derived in
closed form and is used to develop the main results via a series of computer
experiments. The characteristic multipliers are calculated and the first onset
of flip-bifurcation is predicted. Computer simulation based on an exact
piecewise switched model confirms the predicted bifurcations. The exhibition of
quasi-periodic orbits is confirmed by computation of the Lyapunov exponent.
Finally, a series of return maps are generated to provide and alternative
viewpoint to the reported bifurcations in terms of a transmutation from a
tent-like map to a logistic-like map. *Index Terms.* Bifurcation chaos,
boost converter, current-programmed control. (AUTHOR ABSTRACT)

[CHAN97B] Chan, C. Y., and C. K. Tse,
*On the Form of Feedback Function That Can Lead to Chaos in
Discontinuous-Mode DC/DC Converters*, IEEE Power Electronics Specialist
Conference, Record, June 1997, pp. 1317-1322.

In this paper we study the conditions for the dc/dc boost and buck switching regulators operating in discontinuous conduction mode to possibly undergo a period-doubling route to chaos. The essential mathematical tool is the Schwarzian derivative. The result helps determine the form of feedback control scheme for which a period-doubling route to chaos may exist. Results from the calculations reveal that these two types of converters share the same set of conditions. Specifically, any monotonically decreasing control function with a sufficiently large small-signal gain may give rise to a period-doubling cascade in both the boost and the buck dc/dc switching regulators. The analysis in principle can be applied to other types of converters operating in discontinuous conduction mode. (AUTHOR ABSTRACT), Department of Electronic Engineering, Hong Kong Polytechnic University, Hong Kong, China. 6 pages, 3 figures, 1 table, 23 equations, 6 references.

[CHAU97A] Chau, K. T., J. H. Chen, C.
C. Chan, *Dynamic Bifurcation in Dc Drives*, IEEE Power
Electronics Specialists Conference Record, pp. 1330- 1336, 1997.

Dynamic bifurcation as well as chaotic behavior in a fixed-frequency current-mode controlled dc chopper-fed dc motor drive system is presented. The key is to derive an iterative map that describes the nonlinear dynamics of the system operating in the continuous conduction mode. It illustrates that different bifurcation diagrams can be obtained by varying different system parameters. Analytical modeling of period-1 and hence period-p orbits as well as their stability analysis using the characteristic multipliers are also presented. Hence, those stable ranges of various system parameters can be determined. Moreover, chaotic behavior is quantified by evaluating the Lyapunov exponents. The proposed approach is so general that it can readily be applied to other current-mode dc drives. (AUTHOR ABSTRACT) 7 pages, 19 figures, 38 equations, 10 references.

[DIBE97A] di Bernardo, M., F.
Garofalo, L. Glielmo, and F. Vasca, *Analysis of Chaotic Buck,
Boost and Buck-Boost Converters through Switching Maps*, IEEE Power
Electronics Specialists Conference Record, pp. 754-760, 1996.

This paper is concerned with the analysis and comparison of nonlinear phenomena in current and voltage controlled boost, buck and buck-boost DC/DC converters. The study is carried out by using three different discrete time maps; the stroboscopic map, the S-Switching map and the A-switching map, which are obtained by different samplings of the state variables. The construction of a general A-switching map is outlined. Under realistic assumptions on the switching frequency, simplified closed form maps are presented. The use of a certain Jacobian matrix for the analytical detection of some bifurcations is proposed. Routes to chaos exhibited by the three converters are discussed and compared via numerical experiments. (AUTHOR ABSTRACT) Department of Engineering Mathematics, University of Bristol (di Bernardo), Dipartimento di Informatica e Sistemistica, Università di Napoli Federico II (Glielmo), Facoltà di Ingegneria di Benevento, Università di Salerno (Garofalo, Vasca). (7 pages, 14 figures, 16 equations, 14 references.

[HAMI97A] Hamill, D. C., J. H. B.
Deane, and P. J. Aston, *Some Applications of Chaos in Power
Converters*, in Proc. IEE Colloquium on Update on New Power Electronic
Techniques, May 1997, pp.5/1-5/5.

Applications are beginning to be found for chaotic power converters. Using the peak current controlled boost converter as an example throughout, the paper re-views the theory of chaos, shows how it may be employed to improve the electromagnetic compatibility (EMC) of power supplies, and presents a recently developed targeting scheme that can make a chaotic converter jump rapidly between two stabilized modes of operation. (AUTHOR ABSTRACT)

[HUI97A] Hui, S. Y. R., S. Sathiakumar,
and Y. Shrivastava, *Progressive Change of Chaotic PWM
Patterns in DC-AC Random PWM Schemes Using Weighted Switching Decision*,
IEEE Power Electronics Specialists Conference Record, pp. 1454-1461, 1997.

This paper shows that random PWM (RPWM) schemes are in fact chaotic in nature and that the degree of chaos can be manipulated in order to improve spectral performance of RPWM method for DC-AC power conversion. An improved RPWM method which involves a weighted switching decision process for controlling the degree of chaos in RPWM is described. With appropriately added deterministic nature, progressive change of such chaotic behavior in RPWM schemes caused by the weighted process is demonstrated in a qualitative manner. (AUTHOR ABSTRACT) Department of Electronic Engineering, City University of Hong Kong (Hui), Department of Electrical Engineering, University of Sydney, Australia (Hui, Sathiakumar, Shrivastava). 8 pages, 16 figures, 25 references.

[PAVL97A] Pavljasevic, Stjepan, and
Dragan Maksimovic, *Subharmonic Oscillations in Converters
with Current-Mode Programming Under Large Parameter Variations*, IEEE Power
Electronics Specialists Conference Record, pp. 1323-1329, 1997.

This paper presents analysis of subharmonic oscillations under large parameter and large signal variations in a converter with current-mode programming. Onset of oscillations is investigated through analysis of period-doubling bifurcation. The analysis is based on a nonlinear discrete-time model derived under the assumption of piecewise linearity of the system but without the usual linear-ripple approximation. Results are confirmed by measurements on an experimental converter and by simulations. It is also shown that a model based on the linear-ripple approximation gives erroneous results both quantitatively and qualitatively. (AUTHOR ABSTRACT) Institute of Physics, Beograd, Yugoslavia (Pavljasevic), ECE Department, University of Colorado, Boulder, Colorado (Maksimovic). 7 pages, 9 figures, 28 equations, 13 references.

[TSE97A] Tse, C. K., *Bifurcation and Chaos from Autonomous Switching Converters:
Phenomena and Applications*, 1997 International Symposium on Nonlinear
Theory and Its Applications (NOLTA'97), Honolulu, U.S.A, Nov 29-Dec 2,
1997.

A free-running Cuk Converter is studied in this paper. An autonomous state equation and its dimensionless form are derived. Analysis of this equation shows that the system loses stability via a supercritical Hopf bifurcation. The boundary of stability is derived and trajectories of motion studied. In the unstable region, the largest Lyapunov exponent is positive, verifying the chaotic operation. Experimental results verify the analytical results regarding stability and chaos. Finally, potential application areas are briefly discussed. (AUTHOR ABSTRACT)

[BANE98A] Banerjee, S., and K.
Chakrabarty, *Nonlinear Modeling and Bifurcations in the Boost
Converter*, IEEE Transactions on Power Electronics, Vol. 13, No. 2, March
1998, pp. 252-260.

Occurrence of nonlinear phenomena like subharmonics and chaos in power electronic circuits has been reported recently. In this paper, we investigate these phenomena in the current-mode-controlled boost converter. A nonlinear model in the form of a mapping from one point of observation to the next has been derived. The map has a closed form even when the parasitic elements are included. The bifurcation behavior of the boost converter has been investigated with the help of this discrete model. (AUTHOR ABSTRACT), 9 pages, 9 figures, 20 equations, 10 references.

[CHAN98A] Chan, W. C. Y., and C. K.
Tse, *Bifurcations in Current-Programmed DC/DC Buck Switching
Regulators - Conjecturing a Universal Bifurcation Path*, International
Journal of Circuit Theory and Applications, Vol. 26, No. 2, pp. 127-45, Mar-Apr
1998.

Not abstracted in this hypertext. (JF)

[CHAN98B] Chan, W. C. Y., and C. K.
Tse, *What Form of Control Function Can Drive
Discontinuous-Mode Boost Converter to Chaos Via Period-Doubling*,
International Journal of Circuit Theory and Applications, Vol. 26, No. 3,
May-June 1998.

Not abstracted in this hypertext. (JF)

[DIBE98A] di Bernardo, M., F.
Garofalo, L. Glielmo, and F. Vasca, *Switchings, Bifurcations
and Chaos in DC/DC Converters*, IEEE Transactions on Circuits and Systems
Part I, Vol. 45, No 2, pp 133-141, February 1998.

Not abstracted in this hypertext. (JF)

[SUTO98A] Suto, Z, I. Nagy, and E.
Masada, *Avoiding Chaotic Processes in Current Control of AC
Drive*, IEEE Power Electronics Specialists Conference Record, pp. 255-261,
1998.

Not abstracted in this hypertext. (JF)

[YUAN98A] Yuan, G. H., S. Banerjee,
E. Ott, and J. A. Yorke, *Border Collisions Bifurcation in the
Buck Converter*, IEEE Transactions on Circuits and Systems Part I, Vol 45,
No. 7, pp. 707-716, July 1998.

Not abstracted in this hypertext. (JF)

[BATL99A] Batlle, C., E. Fossas, and
G. Olivar, *Stabilization of Periodic Orbits of the Buck
Converter by Time-Delayed Feedback*, International Journal of Circuit
Theory and Applications, Vol. 27, pp. 617-631, 1999.

Not abstracted in this hypertext. (JF)

[KOUS99A] Kousaka, T., T. Ueta, and
H. Kawakami, *Bifurcations in Switched Nonlinear Dynamical
Systems*, IEEE Transactions on Circuits and Systems Part II, Vol. 46, No.
7, pp. 878-885, July 1999.

Not abstracted in this hypertext. (JF)

[IU00A] Iu, H. H.C., and C. K. Tse, *Instability and Bifurcation in Parallel-Connected Buck Converters
Under a Master-Slave Current-Sharing Scheme*, IEEE Power Electronics
Specialists Conference Record, 2000.

Not abstracted in this hypertext. (JF)

[TSE00A] Tse, C. K., Y. M. Lai, and H.
H. C. Iu, *Hopf Bifurcation and Chaos in Free Running
Current-Controlled Cuk Converters*, IEEE Transactions on Circuits and
Systems I, IEEE Transactions on Circuits and Systems I, Vol. 47, No. 4, April
2000. pp. 448-457.

An autonomous free-running C'uk converter is studied in this paper. Analysis of the describing nonlinear state equations shows that the system loses stability via a supercritical Hopf bifurcation. The boundary of stability is derived and local trajectories of motion studied. Cycle-by-cycle simulations of the actual system reveal the typical bifurcation from a stable equilibrium state to chaos, via limit cycles, and quasi-periodic orbits. Experimental measurements confirm the bifurcation scenarios. The occurrence of such kinds of bifurcation in autonomous dc/dc converters has been rarely known in power electronics [AUTHOR ABSTRACT]. 10 pages, 12 figures, 36 equations, 16 references. Hong Kong Polytechnic University, Kowloon, Hong Kong, China.

[BANEXXA] Banerjee, S., C. Grebogi,
and E. Ott, *Border Collision Bifurcations: An Explanation of
the Observed Bifurcation Phenomena in the Boost Converter*, IEEE Trans
Power Electron. to be published.

[IUEXXA] Iue, H. H. C., and C. K. Tse,
*Study of Synchronization in Chaotic Autonomous DC/DC
Converters*, IEEE Transactions on Circuits and Systems I, Accepted.

## Books

[BANEXXA] Banerjee, S., and G. C.
Verghese (Editors), *Nonlinear Phenomena in Power Electronics:
Attractors, Bifurcations, Chaos and Nonlinear Control*, New York: IEEE
Press, to appear.

[HOLD86A] Holden, Arun V., (editor),
*Chaos*, Princeton University Press, Princeton, New
Jersey, 1986. 324 pages.

This volume sets out the basic applied mathematical and
numerical methods of chaotic dynamics and illustrates the wide range of
phenomena, inside and outside the laboratory, that can be treated as chaotic
processes. Topics include: what is the use of chaos?; a graphical zoo of
strange and peculiar attractors; one-dimensional iterative maps;
two-dimensional iterative maps; chaos in feedback systems; the Lorenz
equations; instabilities and chaos in lasers and optical resonators;
differential systems in ecology and epidemiology; oscillations and chaos in
cellular metabolism and physiological systems; periodically forced nonlinear
oscillators; chaotic cardiac rhythms; chaotic oscillations and bifurcations in
squid giant axons; quantifying chaos with Lyapunov exponents; estimating the
fractal dimensions and entropies of strange attractors; and how chaotic is the
universe? (JF)

[MEES86A] The chapter *Chaos in Feedback Systems* by A. Mees discusses chaos in
pulse-width-modulated (PWM) feedback systems where the existence of a snap-back
repeller is sufficient for chaos. The conditions are plotted on the familiar
saw-tooth amplitude-vs-time diagram. (JF)

Quote from Mees "For the engineer designing a feedback system, it would appear that chaos is always to be avoided: 'noisy' oscillations with little information content, or sudden unpredictable excursions of physical variables, are seldom likely to be desirable. Feedback tends to be used to stabilize systems, not to randomize them."

[PARK89A] Parker, T. S., and L. O. Chua, *Practical
Numerical Algorithms for Chaotic Systems*, Berlin, Germany:
Springer-Verlag. 1989

Not abstracted in this hypertext. (JF)

[RASH90A] Rashand, S. N., *Chaotic Dynamics of Nonlinear
Systems*, New York: Wiley, 1990

Not abstracted in this hypertext. (JF)

[PEIT92A] Peitgen, H., H. Jurgens, and D. Saupe, *Chaos and
Fractals - New Frontiers of Science*, Berlin, Germany: Springer-Verlag,
1992

"This book is written for everyone who, even without much knowledge of technical mathematics, wants to know the details of chaos theory and fractal geometry. This is not a textbook in the usual sense of the word, nor is it written in a 'popular scientific' style. Rather, it has been our desire to give the reader a broad view of the underlying notions behind fractals, chaos, and dynamics. In addition, we have wanted to show how fractals and chaos relate to each other and to man other aspects of mathematics as well as to natural phenomena. A third motif in the book is the inherent visual and imaginative beauty in the structures and shapes of fractals and chaos." [AUTHOR PREFACE]

Topics include: Introduction, causality principle, deterministic laws and chaos; the backbone of fractals, feedback and the iterator; classical fractals and self-similarity; limits and self-similarity; length, area, and dimension, measuring complexity and scaling properties; encoding images by simple transformations; the chaos game, how randomness creates deterministic shapes; recursive structures, growing of fractals and plants; Pascal's triangle, cellular automata and attractors; irregular shapes, randomness in fractal construction; deterministic chaos, sensitivity, mixing, and periodic points; order and chaos, period doubling and its chaotic mirror; strange attractors, the locus of chaos; Julia sets, fractal basin boundaries; the Mandelbrot set, ordering the Julia sets; a discussion of fractal image compression; multifractal measures; and a bibliography.

[HILB94A] Hilborn, R. C., *Chaos and Nonlinear
Dynamics*, London, U.K.: Oxford University Press, 1994

Not abstracted in this hypertext. (JF)

[KUZN96A] Kuznetsov, Y. A., *Elements of Applied Bifurcation Theory*, Berlin, Germany:
Springer-Verlag, 1996

Not abstracted in this hypertext. (JF)