IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
707
Border–Collision Bifurcations in the Buck Converter
Guohui Yuan, Soumitro Banerjee, Edward Ott, and James A. Yorke
Abstract—Interesting bifurcation phenomena are observed for
the current feedback-controlled buck converter. We demonstrate
that most of these bifurcations can be categorized as “border–collision bifurcations.” A method of predicting the local
bifurcation structure through the construction of a normal form
is applied. This method applies to many power electronic circuits
as well as other piecewise smooth systems.
Index Terms—Chaos, dc–dc power conversion.
I. INTRODUCTION
A
NOMALOUS bifurcation phenomena have been observed recently for some power electronic circuits,
both experimentally and numerically [1]–[6], [16], [18], [19].
Bifurcation behaviors, other than the familiar period doubling
bifurcation and the saddle-node bifurcation [Fig. 1(a)], are
often seen in the bifurcation diagrams of power electronic
circuits [e.g., Fig. 1(b)]. The goal of this paper is to explain
why such bifurcations occur.
The distinctive feature of all power electronic circuits working under closed loop control is that they have switches that
are turned ON or OFF through state feedback. Two sets of
differential equations are needed to describe the dynamics of
such a power electronic circuit for the ON and OFF periods,
respectively. Generally dynamical systems can be treated as
maps by the method of surface of section. For power electronic
circuits, we may sample the state variables in synchronism
in
with the clock pulses to obtain a “stroboscopic” map
discrete time [17]. The behavior of the system depends on the
mathematical properties of this map. In this paper, we consider
the current feedback-controlled buck converter, as illustrated
in Fig. 2. As shown in Fig. 3, the ON and OFF phases are
with the externally
determined by comparing the voltage
. We show
generated reference triangular wave voltage
later that the corresponding map is piecewise smooth.
imagine that there
For a general piecewise smooth map
exists a curve in the state space which divides it into two
regions (see Fig. 4); in region A the map has one functional
has a different functional
form and in region B the map
Manuscript received November 27, 1996; revised November 23, 1997. This
work was supported by the Department of Energy (Mathematical, Information,
and Computational Sciences Division, High Performance Computing and
Communications Program) and by the National Science Foundation (Divisions
of Mathematical Sciences and Physics), and also by the Office of Naval
Research under Grant N00014-96-1-1123. This paper was recommended by
Associate Editor T. Endo.
G. Yuan and J. A. Yorke are with the Institute for Physical Science and
Technology, University of Maryland at College Park, MD 20742 USA.
S. Banerjee is with the Department of Electrical Engineering, Indian
Institute of Technology, Kharagpur 721302, India.
E. Ott is with the Department of Electrical Engineering and the Department
of Physics, Institute for Systems Research, and Institute for Plasma Research,
University of Maryland at College Park, MD 20742 USA.
Publisher Item Identifier S 1057-7122(98)05300-8.
(a)
(b)
Fig. 1. (a)
doubling bifurcation for the logistic map
(b) A portion of the bifurcation diagram for the
buck converter discussed in this paper.
f (x)
=
Period
x(1 0 x).
form. We call such a curve a border. If the map is continuous
across the border but its derivative is discontinuous, then the
map is called piecewise smooth. Such maps are differentiable
on each side of the border, and on the border the map has two
sets of one-sided partial derivatives depending on whether the
border is approached from within A or within B.
The bifurcation phenomena of piecewise smooth systems
have been studied in [7]–[10]. As a parameter of the system is
varied through a critical value, an attracting periodic orbit may
cross a border and become unstable. Often the attracting periodic orbit bifurcates to a new attractor, which can be periodic
or chaotic. Such bifurcations are called border-collision bifurcations. Recently, a normal form theory for border-collision
bifurcations of two-dimensional (2-D) piecewise smooth maps
has been developed in [8] and in work (Yuan et al.) to be
reported elsewhere. It says that, by a change of coordinates,
any piecewise smooth map can be reduced to the normal form
(1) in some small neighborhood of the border-crossing periodic
point
1057–7122/98$10.00  1998 IEEE
for
(1)
for
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
Fig. 4. State space. Border is where the map
map is smooth in region A and region B.
Fig. 2. Circuit diagram for the buck converter with integrated load current
feedback.
F
is not differentiable. The
expression for the corresponding stroboscopic map
it is
determined numerically. We show that the resulting map is
piecewise smooth with a well-defined borderline. In Section
III, we discuss the bifurcation diagrams of the buck converter
map . We show that the bifurcations are border-collision
bifurcations. In Section IV, we discuss how to construct the
normal form from a given piecewise smooth map. We show
that the bifurcations of the buck map can be predicted from
its normal forms. In Section V, we conclude that most of the
bifurcations seen are border-collision bifurcations. As long as
we know which periodic orbit collides with the border, and the
trace and the determinant of the Jacobian of that periodic orbit
before and after the collision, we can predict the accompanying
bifurcation phenomena from the normal form. We believe that
border collision bifurcations should be very common in a large
variety of power electronic circuits.
II. THE BUCK CONVERTER
Fig. 3. Switching logic of the buck converter. The voltage vio is compared
with a reference voltage vref ; which has a triangular waveform with period
T . The switch is OFF at the beginning. It turns ON when vio drops below vref
and turns OFF again only at times T ; 2T ; 3T ; 1 1 1. The dashed lines indicate
the extreme values of vref .
As the parameter is varied, local bifurcations depend only on
and
appearing in (1). We show in
the values of
this paper that many of the bifurcations of the buck converter
are border-collision bifurcations. For each border-collision
we reconstruct the
bifurcation of the buck converter map
corresponding normal form by calculating at the bifurcation
the trace and the determinant of the border-crossing periodic
orbit on each side of the border (see Section IV). We then plot
the bifurcation diagrams for the reconstructed normal map and
find that they agree with those obtained from the original buck
converter map .
This paper is organized as follows. In Section II, we
describe the circuit under consideration, the current feedbackcontrolled buck converter. Since there is no closed form
A buck converter is a power electronic circuit that converts a
dc voltage to a lower dc voltage, as shown in Fig. 2. (Note: an
output capacitor is usually present in practical circuits. Since
the objective of this paper is to demonstrate the occurrence of
border collision bifurcations, and it is easier to demonstrate it
in two dimensions, we reduce the system to two dimensions by
removing the output capacitor. When the capacitor is added,
the system still remains piecewise smooth and exhibits border
collisions—but at different parameter values.) There are two
sets of differential equations corresponding to the ON and OFF
states of the controlled switch, respectively. Define
where
circuit. Then
are parameters of the buck
(2)
(3)
where the current and voltage
when the switch is ON and
are shown in Fig. 3, and
when it is OFF. The
YUAN et al.: BORDER–COLLISION BIFURCATIONS IN THE BUCK CONVERTER
709
voltage
is compared with the reference voltage
which
has a triangular waveform. The period of the triangular wave is
the switch is ON, and if
the
denoted . If
switch is OFF. This logic, however, allows multiple switchings
in a single triangular wave cycle. This can be undesirable
in a practical circuit. In most circuits, therefore, a “latch” is
included, which prevents multiple switchings. In that case the
first drops below
but turns OFF
switch turns ON when
only at the end of the triangular wave cycle. In this paper, we
have considered the switching logic including this latch.
Remark: In the situation without the latch, chattering or
,
multiple switching occurs if, after the switching at
That is,
we have that (3) gives
(a)
In simulating the system we take the following approach
[11]. Since the differential equations are linear during both
the ON and OFF periods, we can solve them analytically. The
solutions are
(b)
Fig. 5. Possible types of grazing. (a) vio intersects vref at (n + 1)T . (b)
is tangent to vref at some time between nT and (n + 1)T .
vio
(4)
where
There is no closed form of the two-dimensional (2-D)
map of the buck circuit since the switching instants are
obtained through transcendental equations [12], but we can
very accurately
determine the switching time, denoted as
by numerical methods. Assume that the switch is OFF between
and
and is ON between
and
time
. We then use the OFF solution to calculate
and
and
. (Of course sometimes
solution and the switch remains OFF.) Then
there is no
and
we use the ON solution to obtain
from
and . We therefore have a
which maps
to
.
2-D map and call it
a piecewise smooth map, and where is the
Is the map
borderline? To answer these questions we have to go back to
at time
Fig. 3. We see that for some initial conditions
the voltage
does not fall below the reference voltage
for
. The switch remains OFF during this
falls below
time period. For other initial conditions,
at
which is between
and
and the switch
.
is turned ON. The switch is turned OFF again at time
The borderline conditions are those whose orbits graze the
or before. Recall in Fig. 3
triangular wave at time
that the triangular wave has maximum and minimum values of
and . In the parameter range we considered in this paper,
always exceeds . Therefore, border collisions in this
circuit result from grazing the peaks of the triangular wave
and there are two ways in which grazing can occur. These
are schematically illustrated in Fig. 5. The situation shown in
at the grazing
Fig. 5(b) never occurs because
(a)
(b)
(c)
Fig. 6. It is possible for two trajectories to arrive at vn+1 = V2 ; one
determined by the OFF equations as in (a) and one determined by the ON
equations as in (b) or (c). Only (a) is a border-collision case.
point.
. We note that if it could occur it would result in a
discontinuity in the map .) Thus our considerations to follow
are restricted to border-collision bifurcations accompanying
orbits of the type shown in Fig. 5(a). Considering the orbit of
for the OFF
Fig. 5(a) and setting
solution, we have the borderline equation
It is a straight line in the
plane. By construction, if
is on the border, then
is on the line
. Thus a period- orbit that crosses the border has one
. The converse is not true, however;
of its points on
does
that is, having one of its points on the line
not imply this period- orbit is on the border. For example, in
Fig. 6, only case (a) corresponds to a border collision. (Later
in the normal form analysis, we will consider the th iterate
of a border-crossing period- orbit, i.e., we will examine the
.) Clearly, the only discontinuity of
fixed point of the map
the system comes from switching, and it is first order, meaning
and
are discontinuous on
that the partial derivatives of
the border. The map itself is smooth on each side of the border
and varies continuously across the border. To see this, let us
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
(a)
(b)
(c)
(d)
Fig. 7. Bifurcation diagrams as Vin is decreased. All parameters aside from Vin are fixed. R1 = 10 k
; R2 = 220 ; T = 392 s; Rl = 20 ; Rr = 1 ,
C = 20 nF, L = 11.6 mH, V1 = 1.04 V, and V2 = 3.8 V. The horizontal line vio = 3.8 is the border. The state variable vio is plotted vertically. The
border-collision bifurcations are labeled (1)–(5). (b)–(d) are successive blow-ups of (a). The inset in (b) is a magnification of the bifurcation at Vin = 7.93.
consider an orbit near grazing in Fig. 5(a), so that it switches
. The time interval
to
on just before time
that the orbit spends in the ON state is small and
continuously approaches zero as the orbit approaches grazing
from below. (In contrast, the situation in Fig. 5(b) leads to a
discontinuous map .)
III. THE BORDER-COLLISION BIFURCATIONS
OF THE BUCK CONVERTER
Fig. 7(a) is a bifurcation diagram with
as the bifurcation
is decreased from 40 to 0. For each
value,
parameter.
we choose the initial condition to be the last point on the
trajectory at the previous parameter value and iterate the buck
map 1000 times without plotting. We then plot the next 200
vertically. All parameters aside from
values of the voltage
are fixed;
10 k
220
392 s
20
1
20 nF,
11.6 mH,
1.04
3.8 V. These parameter values are within the
V, and
domain of a working buck converter. Fig. 7(a) contains five
border-collision bifurcations.
7.9
The first bifurcation seen in Fig. 7 occurs at
and is an ordinary period-doubling bifurcation of the type
commonly seen in smooth maps (it is not a border-collision
bifurcation). This is not so obvious from Fig. 7(b) in which the
bifurcation does not seem to be smooth. However, we find that,
upon magnification [as shown in the inset of Fig. 7(b)], the
bifurcation diagram displays the typical pitchfork structure of
a period doubling bifurcation of a smooth map (e.g., Fig. 1(a)
at
3). Furthermore, calculation of the eigenvalues of
evaluated at the period one orbit
the Jacobian matrix of
confirms the above observation. (I.e., appropriate to a perioddoubling bifurcation, one of the eigenvalues approaches 1 as
approaches the bifurcation point
7.9 from below.)
The five border collisions occur at approximately
8.41, 9.97, 15.45, 16.64, and 30.72, and we label them in Fig. 7
as (1)–(5), respectively. As previously discussed, a simple way
to identify a border crossing period orbit is to check if it has
3.8. For these five borderone point on the line
collision bifurcations, the border crossing orbit has a period of
2, 4, 3, 6, and 4, respectively. The detailed structures of these
five bifurcations are shown in Fig. 8. The left sides are the
bifurcation diagrams obtained from the circuit model, focusing
3.8, and the right sides
on the small region around
are the bifurcation diagrams obtained from the corresponding
normal form. The waveform plots at each bifurcation point
are shown in Fig. 9.
Border Collision (1): At
8.41, we have a
1)
transition from a period-2 attractor to a period-2 attractor. That
is, a stable period-2 attractor crosses the border, and remains
stable. From both the bifurcation diagram in Fig. 8(a) and (b)
and the waveform plot in Fig. 9 we see that one point of
the period-2 orbit is on the border. In this case the border
crossing shows up as a discontinuous change of the slope in
the
versus
bifurcation diagram [see Fig. 8(a)]. Since
YUAN et al.: BORDER–COLLISION BIFURCATIONS IN THE BUCK CONVERTER
711
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 8. Detailed structures of the five border-collision bifurcations. The left sides are bifurcation diagrams obtained from the circuit model, focusing on the
(a), (b)
region around the border; the right sides are bifurcation diagrams obtained from the corresponding normal form. The bifurcations occur at Vin
8.41, (c), (d) 9.97, and (e), (f) 15.45, respectively. The parameters of the normal form are: (1) “1
1,” A
0.611, and B
0.818; (2) “1
4-piece
chaos,” A
0.998, and B
1.204; (3) “nothing
1,” A
2.137, and B
0.929; (4) “1
1-piece chaos,” A
0.995, and B
2.614;
(5) “nothing
2-piece chaos,” A
5.913, and B
1.017. The values of A and B are less than 1002 in all cases.
=
!
=0
=
!
=0
=
in our normal form we will be examining border crossing of
period-1 orbits, to compare with the normal form result, we
consider the second iterate of the buck converter map. For the
the above period-2 manifests itself as
second iterate map
two period-1 attractors and we examine the normal form of
near the fixed point of
. With this in mind, we refer
1” border collision.
to (1) as a “1
Piece Chaos Border Collision Bifurcation (2):
2)
10.0 there
Examining Fig. 7(b), we observe that at
appears to be a sudden change in which the period-2 attractor is
replaced by a four-band chaotic attractor. In fact, the change
is not really sudden, but only very rapid. The real situation
is illustrated schematically in Fig. 11, in which the size of
interval
has been greatly exaggerated.
the
What happens is that the period-2 orbit experiences an ordinary
. (We have
period doubling to a period-4 orbit at
by numerically determining
verified the period doubling at
=0
!
!
=
=0
=
=
!
=0
that one of the eigenvalues of the Jacobian matrix of the two
times iterated map
evaluated on an element of the periodapproaches
from below.)
2 orbit approaches 1 as
the border-collision bifurcation of the period-4
At
orbit results in what appears in Fig. 7(b) to be four bands of
chaos. [Note that a border-collision bifurcation does not occur
because it corresponds to the situation shown in
at
Fig. 6(b).] The orbit cyclically visits each of the four bands in
turn. A blow up of one of the four bands [Fig. 8(c) and (d)]
reveals, however, that it actually consists of four tiny subbands.
in the vicinity of
To summarize, a slight change of
causes a rapid change from a period-2 orbit to chaos.
The appearance of this in Fig. 7 is as a vertical displacement
of the attracting orbit. We refer to this as an “almost vertical
bifurcation” (although, as indicated in Fig. 11, it is actually
two bifurcations, one after the other in rapid succession).
Considering the fourth iterate of the map, we refer to the
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
(g)
(h)
(i)
(j)
Fig. 8. (Continued.) Detailed structures of the five border-collision bifurcations. The left sides are bifurcation diagrams obtained from the circuit model,
focusing on the region around the border; the right sides are bifurcation diagrams obtained from the corresponding normal form. The bifurcations occur at
Vin
(g), (h) 16.64 and (i), (j) 30.72, respectively. The parameters of the normal form are: (1) “1
1,” A
0.611, and B
0.818; (2) “1
4-piece
chaos,” A
0.998, and B
1.204; (3) “nothing
1,” A
2.137, and B
0.929; (4) “1
1-piece chaos,” A
0.995, and B
2.614;
(5) “nothing
2-piece chaos,” A
5.913, and B
1.017. The values of A and B are less than 1002 in all cases.
=
=
!
=0
=
!
=0
=
(a)
=0
!
!
(b)
(d)
=
=0
=
!
=0
(c)
(e)
Fig. 9. Waveform plots at each border-collision bifurcation point. The dots indicate the places where grazing occurs, or in other words, where the border
collision bifurcation occurs. Vin
(a) 8.41, (b) 10.05, (c) 15.60, (d) 16.60, and (e) 31.0.
=
bifurcation at
as a “1
4 piece chaos ” bifurcation
(where the four bands referred to are the four tiny subbands).
1 Border-Collision Bifurcation (3) At
3) Nothing
15.45, we have a bifurcation from a chaotic attractor to a
period-3 attractor [see Fig. 7(c)]. Since there is a jump from
the chaotic attractor to the period-3 attractor, we call it a
1” bifurcation (the 1 results by considering
).
“nothing
This bifurcation is hysteretic. That is, when we make the
(as opposed to
bifurcation diagram by slowly increasing
which was the way we made Fig. 7), then we
decreasing
YUAN et al.: BORDER–COLLISION BIFURCATIONS IN THE BUCK CONVERTER
Fig. 10. A saddle-node bifurcation of the map F . We follow the attracting
period-3 orbit (the node) and the period-3 saddle as we decrease Vin from 16.
They annihilate each other on the border at approximately 15.45. The saddle
is not shown in Fig. 7(c) because there we show only attractors.
Fig. 11. Schematic picture of an almost vertical bifurcation. Vd and Vb
are the bifurcation points of the period-doubling and the successive border-collision bifurcations, respectively. Vx appears to be a border point but
is not.
observe that the large chaotic attractor persists somewhat past
bifurcation point (3) and ends suddenly in a crisis bifurcation
[13] at some point (3 ). In the range between (3) and (3 ) both
attractors coexist, and which one an orbit approaches depends
on its initial condition. This phenomenon and the relation of
the border-collision bifurcation to the crisis will be further
discussed elsewhere. The period-3 attractor is created at the
bifurcation point (3) by a saddle-node bifurcation, as shown in
Fig. 10. The period-3 saddle and the period-3 node are created
on the border, which is different from ordinary saddle-node
bifurcations of the type occurring in smooth systems. We see in
is decreased, the attracting periodFig. 8(e) and (f) that, as
3 orbit (the node) collides with the border at bifurcation point
(3). To the left, there is no local attractor about the collision
point to which points previously following the period-3 orbit
might go. However, globally there is another large preexisting
chaotic attractor and every initial point is attracted to it. Thus
we have a discontinuous jump from period-3 to a wide-band
decreases through bifurcation point (3).
chaotic attractor as
Piece Chaos Border-Collision Bifurcation (4): At
4)
16.64, we have a bifurcation from a period-3 attractor to
a chaotic attractor [see Fig. 7(c)]. The period-3 attractor loses
713
stability and experiences another almost vertical bifurcation
(i.e., there is a period doubling to a period-6 attractor, and
one of the resulting period-6 orbit points very soon after hits
the border and becomes unstable as
is slightly increased).
Fig. 8(g) and (h) reveals that each period-6 point bifurcates
1-piece chaos”
into a 1-piece chaos. We call it a “1
bifurcation.
Two-Piece Chaos Border-Collision Bifurca5) Nothing
30.72, we have a bifurcation from a chaotic
tion (5): At
attractor to another chaotic attractor. As for bifurcation (3), this
bifurcation is also hysteretic and is associated with a crisis
destroying the wide-band chaotic attractor at some point (5 )
slightly past (5). From Figs. 7(d) and 8(i) and (j) we see that
the attractor just to the right of bifurcation point (5) has four
branches, each of which actually consist of two smaller chaotic
bands [see Fig. 8(i) and (j)].
and
Windows: Fig. 1(a) for the smooth map
Fig. 1(b) for our piecewise smooth buck map differ in that
border collision bifurcations are present in Fig. 1(b). They
also, however, differ in another significant way. In particular,
in the chaotic range of the parameter , the map
exhibits a dense set of windows of periodicity, the widest of
3.83).
which is the period-3 window (in the region near
Each such window begins with a tangent bifurcation creating
a stable periodic orbit. This situation is generally thought to
be a common feature of smooth maps with chaotic attractors.
In contrast, in the case of the piecewise smooth buck map, we
numerically observe that the chaotic regions are apparently
“solid” in that they are not permeated by a dense set of
windows. We believe that this absence of windows may be
a common feature in a wide class of piecewise smooth maps.
IV. APPLICATION OF THE NORMAL FORM THEORY
In the normal form theory, any 2-D piecewise smooth map
can be reduced to the normal form (1) by a change of variables
0 the
(involving phase space and parameters). When
on the border, as shown in Fig. 12.
map has a fixed point
and
are two points one on each side of the border.
as the Jacobian of the map at point . Let
Define
and
be the trace and determinant of
. Then
(5)
is taken with
in region A.
where the limit
and
are the corresponding trace and
The numbers
with
in region B. If
is a
determinant as
and
are replaced by
point of a period orbit,
and
respectively. Border collision occurs if, say, when
the fixed point of the map is in region A, and when
the fixed point is in region B. Numerically we
and
to be the fixed points (assuming they exist)
choose
and
where
and
are close to
at parameter values
. Then
and
are also close to
the critical value
. Therefore, by calculating the trace and the determinant
and
we may obtain a very
of the Jacobian matrix at
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
good approximation of
and . The trace and the
determinant of the border-crossing periodic orbit for each of
the five border collisions discussed earlier are calculated. The
results are the following.
(1) Period-2
(2) Period-4
Fig. 12. Definition of A ; A ; B ; and B : P0 is a fixed point of the map
F (or F p ) on the border. PA and PB are arbitrary points very close to the
border, one on each side of the border. Define A = lim tr DF (PA ) and
A = lim det DF (PA ) as PA
P0 from within region A. The numbers
B and B are the corresponding trace and determinant as PB
P0 from
within region B.
!
(3) Period-3
(4) Period-6
(5) Period-4
In addition, in all the cases above,
and
are less
than 10 . The result that the determinant is small can be
understood on the basis of an analytical estimate
(6)
where is the period of the border-crossing orbit (this result
for is derived in the Appendix). For example, for the period2.0 10 . Thus it is
4 orbit shown in Fig. 11, we have
and
to zero. In that case,
a good approximation to set
the normal form (1) reduces to a one-dimensional map
for
for
(7)
We plot the corresponding bifurcation diagrams of the reand obtain
constructed normal form map (7) by varying
the second column of Fig. 8. We see the same bifurcation
behavior as shown in the first column of Fig. 8. This is
consistent with our assertion that the local bifurcation structure
of a 2-D piecewise smooth map depends only on the values
and
of the corresponding border-crossing
of
periodic orbit.
We have also found that, for the buck converter, the local
bifurcation structure is independent of the parameter varied.
etc.)
That is, if we vary different parameters (e.g.,
through the same bifurcation point in the parameter space, we
get qualitatively the same bifurcation structure in the neighborhood of the critical parameter value. We now explain why.
(as
Again let us assume that the buck map has a fixed point
shown in Fig. 12) on the border at the bifurcation point. As
one of the parameters is varied and approaches its critical value
from both above and below, the fixed point orbit approaches
from within both regions A and B. Although different
!
paths may be taken in the process for different parameters
varied, the limit (5) should be the same. Therefore we have
the same normal form for different bifurcation parameters; as
a consequence, we have the same bifurcation structure locally.
Nusse and Yorke [10] studied the one-parameter family
are
of skew tent maps defined in (7), where
constants and is the bifurcation parameter. Their result is the
and
border-collision
following. For many choices of
0. The nature of these bifurcations
bifurcations occur at
. Let
2 be any positive integer.
depend only on
1 and
1, one may have a borderThen for
collision bifurcation from a fixed point attractor to an attracting
-piece chaotic attractor, or an period- orbit, or a
piece chaotic attractor, or a one-piece chaotic attractor as the
parameter crosses zero. Fig. 13 gives a heuristic picture of
the border-collision bifurcation of a skew tent map. The slopes
and
in the
on each side of the border correspond to
. If
1, the map has a
normal form. Assume
0. For
0, the eigenvalue of the
stable fixed point when
1, then the fixed point is
fixed point changes to . If
1, while the eigenvalue of the period-2
still stable; if
1, the fixed point is unstable but the period-2
orbit
orbit is stable, so we have a border-collision bifurcation from
a fixed point attractor to an attracting period-2 orbit; further,
1, the period-2 orbit is unstable, and depending on
if
and
we may have a large variety of borderthe values of
collision bifurcations, as discussed in [10]. It is interesting to
note that the “almost vertical bifurcation” previously discussed
is close to one, as is the case
corresponds to the case where
0,
for bifurcations (2) and (4). To see this we note that for
for
in these cases, there is a stable fixed point
which from (7) is
. Since
the map
is very large for bifurcations (2) and (4), we have
1, i.e.,
versus is almost vertical.
V. CONCLUSION
We have shown that many of the bifurcations of the buck
converter studied here are border-collision bifurcations. The
local bifurcation structure is independent of the parameter
varied. We have also reconstructed the normal form from the
eigenvalues of the border-crossing periodic orbits. We show
YUAN et al.: BORDER–COLLISION BIFURCATIONS IN THE BUCK CONVERTER
715
for our parameters. Thus the third term in (3) can be neglected.
Regarding (2) and (3) as a flow in
space, we can take
the divergence of this flow to obtain
(A3)
which is a constant independent of time . From (A3) it follows
space whose points are evolved using (2)
that areas in
.
and (3) contract exponentially with time as
2.7 from which we obtain (6).
For our parameters,
ACKNOWLEDGMENT
The bifurcation diagrams and the eigenvalue computation
are made using Dynamics [14]. The authors gratefully acknowledge the use of the computer facilities provided by a
grant from the W. M. Keck Foundation.
(a)
REFERENCES
(b)
Fig. 13. One-dimensional piecewise linear map. (a) The slopes on each side
of the border correspond to A and B in the normal form, respectively. As is varied through zero, the stable fixed point crosses the border and suddenly
becomes unstable. (b) The second iterate of the map is plotted. When < 0,
the fixed point is the only periodic orbit. When > 0, an additional period-2
orbit emerges. (When 0, the second iterate is shaped like the “ < 0”
curve except that it passes (0, 0).)
=
that the bifurcations in the reconstructed maps agree with
those in the original buck system. The circuits described and
the bifurcation diagrams displayed in [1]–[6], [11], and [12]
make it likely that border-collision bifurcations are occurring
in their systems. We believe that the normal form theory will
be useful for analyzing many power electronic circuits which
are essentially piecewise smooth systems.
APPENDIX
Averaging (2) over time we have
(A1)
is the fraction of the time that the switch is ON.
where
Equation (A1) allows us to estimate the relative sizes of the
first and third terms in (3). The ratio of the average of these
terms is
(A2)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
Guohui Yuan was born in Harbin, China, in 1967.
He received the B.S. degree in physics from Tsinghua University, Beijing, China, in 1990. He
is currently working toward the Ph.D degree in
the Department of Physics of the University of
Maryland at College Park.
His research interests include theories of nonlinear dynamics and chaos and their applications.
Soumitro Banerjee was born in Calcutta, India, in
1960. He received the B.E. degree in electrical
engineering from Bengal Engineering College,
Calcutta, India, in 1981, and the M.Tech degree
in energy studies and the Ph.D. degree from the
Indian Institute of Technology (IIT), New Delhi,
India, in 1983 and 1987, respectively.
After receiving the master’s degree, he then
worked on power system applications of superconducting magnetic energy storage at IIT and
completed his doctoral work. He has served at IIT,
Kharagpur, India, as a Lecturer from 1985 to 1990 and as an Assistant
Professor from 1990 to the present. His current interests include nonlinear
dynamics and chaos in electrical circuits and fractal geometry as applied to
image compression.
Edward Ott received the B.S. degree in electrical engineering from Cooper
Union, New York, NY, and the M.S. and Ph.D. degrees in electrophysics from
Polytechnic Institute of New York, Brooklyn, NY.
Before joining the faculty of Cornell University, Ithaca, NY, as a Professor
of Electrical Engineering, he was a National Science Foundation Post-Doctoral
Fellow in the Department of Applied Mathematics and Theoretical Physics at
Cambridge University, U.K. In 1979, he joined the Departments of Physics
and Electrical Engineering of the University of Maryland at College Park
where he is Distinguished University Professor. His past research has been in
plasma physics and charged particle beams. More recently, his work has been
on fundamental issues and applications of chaos. Some of the issues he has
been concerned with include the fractal dimensions of chaotic attractors and
basin boundaries, transition to chaos, control of chaos, etc. He is the author
of over 250 journal papers and of the book Chaos in Dynamical Systems
(Cambridge, U.K.: Cambridge Univ., 1993).
James A. Yorke received the B.S. degree from Columbia University, New York, and the Ph.D. degree
in mathematics from the University of Maryland at
College Park (UMCP) in 1966.
He then began his academic career in the Department of Mathematics, UMCP, where he remains
today as a Distinguished University Professor and
as the current Director of the Institute for Physical
Science and Technology, UMCP. He is best known
to the general public for coining the term “chaos” in
a classic paper entitled “Period three implies chaos”
which he co-authored with T. Y. Li. He and his collaborators and his students
actively investigate the dynamics of chaotic processes. His interests also
include the applications of chaos theories in physical, biological, and chemical
system. He has co-authored three books.