Periodic Modeling and Analysis of Bifurcation Dynamics for Switching Converters
Chau-Chung Song and Yu-Kai Chen
Department of Aeronautical Engineering, National Formosa University
Yunlin 632, Taiwan, R.O.C.
E-mail: { ccsong, ykchen } @nfu.edu.tw
Der-Cherng Liaw
Department of Electrical Engineering, National Chiao Tung University
Hsinchu 300, Taiwan, R.O.C.
E-mail: [email protected]
Abstract— In this paper, the nonlinear dynamics of PWM- type converters is studied. The sampled-data approach is applied to the periodic modeling and stability analysis of switching converters. The period-one and period-two modeling for the converter dynamics is proposed to predict the occurrence of the corresponding bifurcations and used to analyze the stability of converter circuits. The example study of buck converters is presented to demonstrate the feasibility of the methodological approach proposed in this paper. Applying to buck converters, the period-doubling bifurcation is found to exhibit as the input voltage varies, which might produce a series of period-doubling bifurcations and then result in a chaotic motion. Furthermore, the system stability of buck dynamics is studied and evaluated.
Index Terms— periodic modeling, system stability, sampled- data approach, switching converters, buck converters, period- doubling bifurcation.
I. I NTRODUCTION
PWM type DC converters may be treated as a class of nonlinear and time-varying dynamical systems, which they are consisted of switches that cause the topological structure of the converter to vary with time, energy storage components and diodes with a nonlinear voltage-current characteristics. In recent years, lots of effort has been devoted to the analysis of nonlinear phenomena in power electronics [1]-[2]. Numerical and experimental results have shown that high frequency PWM dc-dc converters topologies might exhibit nonlinear and/or chaotic evolution [3]-[5] [9]-[12].
Among existing studies, most of them are devoted to deriving linear models of dc-dc converters which can be fitted into the framework of conventional linear system theory. The most popular approach is the so-called “averaging model,”
of where an average is taken across the state equations of all the different operational modes occurring within a cycle.
A linear model is then obtained by subsequent linearization for small perturbations. This heuristic technique has been shown to be theoretically sound [6]; However, it is hard to be used to predict the subharmonic instability while the system parameter varies. When considering behavior operating beyond the normal stability bound, a different approach must be employed to fulfill the design. An alternative is the so- called “discrete-time modeling.” [7]-[12] Instead of converting a switched-mode circuit into a continuous one, the system is described in terms of a sampled-data system. Such a nonlinear
form of discrete mapping from one sampling time instant to the next consecutive one. The variation of the state vector within a cycle can then be described by a functional map that may be very complicated.
In this paper, the sampled-data approach is applied to the periodic modeling and stability analysis of switching convert- ers. In section II, the period-one and period-two modeling for the converter dynamics is proposed to predict the occurrence of the corresponding bifurcations and analyze the stability of converter circuits. It is followed that an example study of buck converters is presented to demonstrate the feasibility of the methodological approach proposed in section III. The nu- merical simulations are used to demonstrate the main results.
Finally, the concluding remarks are presented.
II. S AMPLED -D ATA M ODEL OF S WITCHING C ONVERTERS
A general sampled-data modeling of PWM dc-dc converters proposed by Fang [10] is briefly recalled below. Suppose the PWM dc-dc converters operate in a continuous conduction mode (CCM) with fixed-frequency control. The switch and the diode are assumed to be ideal with no forward voltage.
Denote f s and T , the switching frequency and the correspond- ing period, respectively. For CCM operation, the switch is generally on and off once in a cycle. A general block diagram of PWM converters is shown in Fig. 1. It can be applied to most PWM converters such as buck, boost, buck-boost and Cuk types of converters. In this block model, A ´ 1 , A 2 ∈ IR n ×n , B 1 , B 2 ∈ IR n ×1 , C, E 1 , E 2 ∈ IR 1×n , and D ∈ IR are constant matrices, where n denotes the dimension of system, typically given by the number of energy storage elements in the power stage. Here, x ∈ IR n and y ∈ IR are the state and the measured output, respectively. Moreover, v s and v o
denote the source and output voltages, respectively, while the reference voltage v r is a constant value in most applications.
The associated switching decision for voltage mode control is determined by the external signal h(t) generally called
“ramp function”. Under such control scheme, we have the
duty cycle α := d ∗ n /T , where star denotes the equilibrium
value. In general, the operating condition switches back and
forth between states S 1 and S 2 once a clock cycle. The circuit
works in subsystem S 1 for y(t) > h(t), while operating in
subsystem S 2 when y(t) ≤ h(t).
⎩ ⎨
⎧
= +
=
⎩ ⎨
⎧
= +
=
:
:
2 2 2 2
1 1 1 1
x E v
u B x A S x
x E v
u B x A S x
o o
&
& y = Cx + D u
PWM Control:
Switch to S1 at t =nT Switch to S2 at t=nT+dn
) (t h
v o
⎥ ⎦
⎢ ⎤
⎣
= ⎡
r sv u v
Fig. 1. Block diagram of switching converters.
In this paper, the study of power converters is focus on the CCM operation. The system then have the following two stages depending on whether the controlled switch is on or off:
S 1 :
x ˙ = A 1 x + B 1 u
v o = E 1 x. (1)
S 2 :
x ˙ = A 2 x + B 2 u
v o = E 2 x. (2)
In the trailing-edge design, the controlled switch is ON at state S 1 , while the controlled switch is off in state S 1 in the leading-edge design.
A. Modeling of Period-one Dynamics
Consider t ∈ [nT, (n + 1)T ). Let x n = x(nT ) and v on = v o (nT ). In general, the switching frequency of a PWM-type dc-dc converter is sufficiently large such that the variations in v s and v r within the cycle are small and can be neglected.
Take v s and v r to be constant within the cycle and denoted as v sn and v rn , respectively. In the (n+1)-th cycle, let the signals y(t) and h(t) intersects at t = nT + d n , i.e., the constraint of system dynamics is y(nT + d n ) = h(nT + d n ), i.e., Cx n + Du n = h(d n ). The system equations become
x
n+1= f(x
n, u
n, d
n)
= e
A2(T −dn)(e
A1dnx
n+
dn0
e
A1(dn−σ)dσB
1u
n)
+
Tdn
e
A2(dn−σ)dσB
2u
n(3)
g (x
n, u
n, d
n) = y(nT + d
n) − h(nT + d
n) = 0
= C(e
A1dnx
n+
dn0
e
A1(dn−σ)dσB
1u
n)
+Du
n− h(d
n) (4)
v
on= Ex
n. (5)
In the PWM converters, the steady state operating condition is a periodic solution, but not an equilibrium point as predicted from the averaging method. A T -periodic solution for system in Fig. 1 corresponds to a fixed point for the sampled-data
model. Let the fixed point of the sampled-data dynamics (3)- (5) be (x n , u n , d n ) = (x 0 (0), u, d), where u = [v s , v r ] T . Then we have
x 0 (0) = f (x 0 (0), u, d) (6)
g(x 0 (0), u, d) = 0. (7)
Since the sampled-data dynamics (3)-(5) are nonlinear and constrained, we suppose that
∂g
∂d n
= C ˙ x 0 (d − ) − ˙h(d) = 0.
Then denote “hat” as the small perturbation of system state.
The linearization equations of the model (3)-(5) at the fixed point (x n , u n , d n ) = (x 0 (0), u, d) gives
ˆ
x n +1 ≈ Φˆx n + Γˆ u n (8)
v on = E ˆ x n , (9)
where Φ = ∂f
∂x
n− ∂f
∂d
n( ∂g
∂d
n)
−1∂g
∂x
n|
(xn,un,dn)=(x0(0),u,d)(10) Γ = ∂f
∂u
n− ∂f
∂d
n( ∂g
∂d
n)
−1∂g
∂u
n|
(xn,un,dn)=(x0(0),u,d). (11)
B. Modeling of Period-two Dynamics
In order to understand the bifurcation and chaotic phenom- ena, the period-two model merged from the period-doubling bifurcation of period-one model is proposed and analyzed.
The aim of this study is to seek the unified methodology to analyze the nonlinear behaviors of PWM converters. Suppose the operating condition switches back and forth between S 1 and S 2 once per clock cycle, then the switching times d 1n and d 2n denote the times that the control voltage crossed the ramp in the first and second period, respectively. From the sampled- data dynamics (3)-(5) of PWM converters, the sampled-data model of buck converters for period-two orbit, that is, the sampled-data period is 2T , becomes
x
n+1= f(x
n, u
n, d
1n, d
2n)
= e
A2(T −d2n)(e
A1d2nx
n+
d2n0
e
A1(d2n−σ)dσB
1u
n)
+
Td2n
e
A2(d2n−σ)dσB
2u
n(12) where
x
n= f(x
n, u
n, d
1n)
= e
A2(T −d1n)(e
A1d1nx
n+
d1n0
e
A1(d1n−σ)dσB
1u
n)
+
Td1n
e
A2(d1n−σ)dσB
2u
n.
The constraint equations of sampled-data model become
g
1(x
n, u
n, d
1n, d
2n)
= y(2nT + d
1n) − h(2nT + d
1n) = 0
= C(e
A1d1nx
n+
d1n0
e
A1(d1n−σ)dσB
1u
n) + Du
n− h(d
1n) (13)
g
2(x
n, u
n, d
1n, d
2n)
= y((2n + 1)T + d
2n) − h((2n + 1)T + d
2n) = 0
= C(e
A1d2nx
n+
d2n0
e
A1(d2n−σ)dσB
1u
n) + Du
n− h(d
2n). (14)
Next, the stability for the period-two orbit is analyzed. The fixed point of sampled-data model is x 0 = (x(0), u, d 1 , d 2 ).
From the linearized model (8)-(9) and (10)-(11), the following results are obtained. Let ˆ u n = V I . The Jacobian matrix of linearized model is
Φ = ∂f
∂x
n− ∂f
∂d
n( ∂g
∂d
n)
−1∂g
∂x
n|
eq0=(x0(0),u,d1,d2)(15) and
Γ = ∂f
∂u
n− ∂f
∂d
n( ∂g
∂d
n)
−1∂g
∂u
n|
eq0=(x0(0),u,d1,d2)(16) where eq 0 represents the equilibrium point of system operat- ing. According to the methodology above, the modeling for period-n dynamics may be derived by extending the the period- two model. Furthermore, the stability of nonlinear sampled- data systems will be analyzed and observed.
III. E XAMPLE S TUDY FOR B UCK C ONVERTERS
A general sampled-data modeling of PWM dc-dc converters proposed by Fang and Abed [9][10]. Suppose the PWM dc-dc converters operate in a continuous conduction mode (CCM) with fixed-frequency control. Denote f s and T , the switching frequency and the corresponding period, respec- tively. Our goal is to describe the bifurcation behavior of Buck converters while the input voltage is varied and treated as system parameter to analyze the system stability from the numerical simulations. The output voltage is controlled by closed-loop PWM using natural sampling; the feedback loop has intentionally been left uncompensated in order to encourage chaotic operation.
time VU
VL vramp
Comparator A vramp
VI
vo L
C R
Vref vcon
High: closed Low: open
T
Fig. 2. Idealized buck converter circuits.
A. Sampled-data Model of Buck Converters
The following analysis assumes all components to be ideal.
In operation, the output voltage v(t) is applied to the nonin- verting input of an amplifier with Gain A; a constant voltage V ref is applied to the inverting input; the output of the amplifier, which will be referred to as the control voltage, v con
is, therefore, v con (t) := A(v(t) −V ref ). This voltage is applied to the inverting input of the comparator; the noninverting input is fed by an independently generated ramp voltage, v ramp (t), which periodically and linearly rises from a lower voltage, V L
to an upper voltage, V U in a time T , and then instantaneously
returns to V L , where 0 ≤ V L < V U . The ramp voltage can be expressed as h(t) = v ramp (t) = V L + (V
U−V T
L) [t mod T ], where [t mod T ] equals to the remainder of t/T . The output of the comparator is used to determine the state of switch S, in such a way that S is ON only when v con < v ramp ; that is S changes state when v con = v ramp , referred to as the switching condition. Duty-factor control is therefore achieved by natural sampling, as in many commercial PWM chips.
Due to the fact that the discontinuous conduction mode does not take place, the converter can be represented by a piecewise- linear vector field, described by two subsystems of differential equations as follows:
System 1: v con (t) > v ramp (t) d
dt
i(t) v(t)
= A 1 x + B 1 u. (17) System 2: v con (t) < v ramp (t)
d dt
i(t) v(t)
= A 2 x + B 2 u. (18) Moreover, the output equation is
v con (t) = A(v(t) − V ref ) = Cx + Du v o (t) = [0 1]
i(t) v(t)
= Ex.
The condition of switching time d is v con (t) | t =d = v ramp (t) | t =d .
According to the block diagram of PWM converters, we let x = [i(t) v(t)] T and u = [V F V I V ref ] T . The following matrices are then defined:
A 1 = A 2 =
0 − L 1 C 1 − RC 1
=: J B 1 =
1
L 0 0
0 0 0
, B 2 =
0 L 1 0
0 0 0
C = [0 A], D = [0 0 − A]
E 1 = E 2 = [0 1] =: E
where v is the voltage across capacitance C, i is the current through inductance L, R is the load resistance, V F is the voltage across the diode, and V I is the input voltage, assumed constant except at the switching instants.
B. Stability Analysis for Period-one Orbit
Suppose the operating condition switches back and forth between S 1 and S 2 once per clock cycle and the switching time represents d n . From the sampled-data dynamics (3)-(5) of PWM converters, the sampled-data model of buck converters for period-one orbit, that is, the sampled-data period is T , becomes
x
n+1= e
JT(x
n+
dn0
e
−JσdσB
1u
n+
Tdn
e
−JσdσB
2u
n) (19) where
e
JT= e
−kTcos wT +
wksin wT −
Lw1sin wT
Cw1
sin wT cos wT −
wksin wT
.
Here, k := 2RC 1 and w := k
4R
2C
L − 1.
The constraint equation of sampled-data model becomes g (x
n, u
n, d
n)
= C(e
Jdnx
n+
dn0
e
J(dn−σ)dσB
1u
n) + Du
n− h(d
n). (20)
Suppose the diode is ideal, that is, V F = 0. The conditions of fixed point x 0 become
(e J T − I 2×2 )x n + e J T
T d
ne −Jσ dσB 2 u n = 0, (21) while the constraint equation (20) holds.
First, the bifurcation analysis is studied by numerical sim- ulation, while treating the input voltage V I as bifurcation parameter. From Eqs. (20) and (21), we can obtain the re- lationship between the switching time d n (or duty cycle α) and the system parameter V I . The numerical simulations are performed with the following parameter values (see Fig. 2):
L = 20mH, C = 47μF , R = 22Ω, A = 8.4, V ref = 11.3V , V L = 3.8V , V U = 8.2V , and T = 400μsec. Fig. 3 shows the duty cycle α obtained when V I is varied from 12 to 60.
The initial conditions (i 0 , v 0 ), when V I varies in this range, increase from (0.536, 11.758) (which corresponds to α = 0.01) to (0.654, 12.175) (which corresponds to α = 0.79).
Next, the stability for the period-one orbit is analyzed and discussed. For a periodic-one orbit, it is known that the system is asymptotically stable if, given initial condition (i 0 , v 0 ) sufficiently close to the periodic orbit, the trajectory in the phase plane from (i 0 , v 0 ) asymptotically approached that of periodic orbit. The characteristic equation is used to determine whether a periodic orbit is stable or not. If all the eigenvalues of characteristic equation are inside the unit circle of complex plane, then the orbit is stable, while any eigenvalue outside unit circle suffices to render the orbit unstable.
The fixed point of sampled-data model is x 0 = (x(0), u, d).
From the linearized model (8)-(9) and (10)-(11), the following results are obtained. Let ˆ u n = V I as the bifurcation parameter.
The Jacobian matrix of linearized model is Φ =
∂f1∂in
∂f1
∂vn
∂f2
∂in
∂f2
∂vn
−
∂f1∂dn
∂f2
∂dn
( ∂g
∂d
n)
−1∂g
∂i
n∂g
∂v
n(22) at (x n , u n , d n ) = (x 0 (0), u, d) and Γ = [ ∂u ∂f
1n
, ∂u ∂f
2n
] T , since
∂g
∂u
n= 0 at (x n , u n , d n ) = (x 0 (0), u, d).
From the Jacobian matrix of linearized model, we can compute all the eigenvalues for any fixed points as the system parameter V I varies. Fig. 4 depicts the evolution of eigenvalues for period-one orbit in complex plane, when V I sweeps the range from 12 to 25. We observe first that the eigenvalues are complex conjugates that move on a circle of radius ≈ 0.82, and so the orbit is asymptotically stable. Near V I = 24, the eigenvalues become both real, and when V I reaches a certain value between V I = 24 and V I = 25, one of the eigenvalues becomes outside the unit circle, and so the periodic orbit be- comes unstable. After V I = 25, it remains unstable. The point which one of the eigenvalues goes through the unit circle is the so-call “bifurcation point” around V I ≈ 24.517. This implies the system undergoes the “period-doubling” bifurcation.
When V I < 24.517, the state trajectories of system converge to the stable period-one orbits, while V I > 24.517, those become the period-two or chaotic orbits. Fig. 5 shows the period-one orbit corresponding to V I = 24 < 24.517 in (t, v con )-axes with the ramp function. Fig. 6 shows the period- two orbit corresponding to V I = 25 > 24.517 in (t, v con )-axes with the ramp function.
C. Stability Analysis for Period-two Orbit
Furthermore, the period-two model is used to analyze ad evaluate the bifurcation and chaotic phenomena merged from the period-doubling bifurcation of period-one model. Here, let the switching times d 1n and d 2n denote the times that the control voltage crossed the ramp in the first and second period, respectively. From the sampled-data modeling (12)-(14) of PWM converters, the period-two model of buck converters, that is, the sampled-data period is 2T , becomes
x
n+1= e
J2T(x
n+
d1n0
e
−JσdσB
1u
n+
Td1n
e
−JσdσB
2u
n)
+e
JT(
d2n0
e
−JσdσB
1u
n+
Td2n
e
−JσdσB
2u
n) (23)
where
e
J2T= e
−k2Tcos w2T +
wksin w2T −
Lw1sin w2T
Cw1
sin w2T cos w2T −
wksin w2T
.
Suppose the diode is ideal, that is, V F = 0. The system (23) becomes
x
n+1= e
J2T(x
n+
Td1n
e
−JσdσB
1u
n) + e
JT Td2n
e
−JσdσB
2u
n. (24)
The constraint equations of sampled-data model become g
1(x
n, u
n, d
1n, d
2n) = Ce
Jd1nx
n+ Du
n− h(d
1n) = 0 (25) and
g
2(x
n, u
n, d
1n, d
2n) = Ce
Jd2nx
n1+ Du
n− h(d
2n) = 0 (26) where
x
n1=
i
1nv
1n= e
JTi
nv
n+
Tdn
e
−JσdσB
2u
n. (27)
The conditions of fixed point x 0 become
(e
J2T− I
2×2)x
n+ e
J2T Td2n
e
−JσdσB
2u
n+ e
JT Td1n
e
−JσdσB
2u
n= 0,
(28)
while the constraint equations (25) and (26) holds.
First, the bifurcation analysis is studied by numerical sim- ulation, while treating the input voltage V I as bifurcation parameter. From Eqs. (28) which satisfies the constraints (25)- (26), the relationship between the switching times d 1n and d 2n (or duty cycle α 1 and α 2 ) and the system parameter V I
is derived. Fig. 7 shows the duty cycles α 1 and α 2 obtained when V I is varied from 24.517 to 30. There are two duty cycles existed in this interval and it is verified that the sampled-data model for period-two orbit is feasible to analyze the nonlinear behavior of PWM converters.
Next, the stability for the period-two orbit is analyzed to
illustrate the feasibility of the proposed model. The fixed point
of sampled-data model is x 0 = (x(0), u, d 1 , d 2 ). From the linearized model (15)-(16), the following results are obtained.
Let ˆ u n = V I as the bifurcation parameter. The Jacobian matrix of linearized model is
Φ =
∂f
1
∂i
n∂f
1∂v
n∂f
2∂i
n∂f
2∂v
n− ∂f
1
∂d
1n∂f
1∂d
2n∂f
2∂d
1n∂f
2∂d
2n× ∂g
1∂d
1n∂g
1∂d
2n∂g
2∂d
1n∂g
2∂d
2n−1
∂g
1∂x
1∂g
1∂x
2∂g
2∂x
1∂g
2∂x
2(29) at (x n , u n , d 1n , d 2n ) = (x 0 (0), u, d 1 , d 2 ) and Γ = [ ∂u ∂f
1n
∂f
2∂u
n] T , since ∂u ∂g
n
= [ ∂u ∂g
1n
, ∂u ∂g
1n
] T = [0, 0] T at (x n , u n , d 1n , d 2n ) = (x 0 (0), u, d 1 , d 2 ).
From the Jacobian matrix of linearized model, all the eigenvalues for any fixed points are calculated while the system parameter V I varies. Fig. 8 depicts the variations of eigenvalues for period-two orbit in complex plane, when V I
sweeps the range from 24.5 to 32. Near V I = 25, both eigenvalues enter the unit circle, yielding stable orbits. Next, they move on the circle of radius 0.679 until V I = 31 is reached, After V I = 31, one of the eigenvalues goes out of the unit circle and the stability is lost and not recovered. In order to obtain a clearer understanding of what is going on, one may construct a bifurcation diagram of system states by taking V I as bifurcation parameter. The bifurcation point of “period- doubling” is around V I ≈ 31.118. When 24.517 < V I <
31.118, the state trajectories of system converge to the period- two orbits, while V I > 31.118, those become the period-four or chaotic orbits.
D. Chaotic Phenomena of Buck Converters
In this section, based on the above discussions, we will investigate the bifurcation and chaotic phenomena by selecting input voltage V I as bifurcation parameter. Our simulation is based on the sampled-data model derived in previous sections.
From previous analysis, for higher values of V I , a cascade of period-doubling bifurcation follows until an apparently chaotic zone is reached. Fig. 9 introduces the bifurcation diagrams i n
as V I is varied from 15 to 40 with the other parameters fixed.
It is obvious that the bifurcation diagrams reveal repeated period-doublings to chaos. The system starts with a period- one behavior, which continues up to V I ≈ 24.517. Then it bifurcates into a period-two orbits up to V I ≈ 31.118. After that, a period-four behavior occurs quickly. Fig. 10 shows the chaotic orbit corresponding to V I = 35 in (t, v con )-axes with the ramp function. It is found that the trajectories are not periodic but bounded, like chaos.
The bifurcation diagram is presented above while V I is varied with the other parameters remaining fixed. It is found that the structure of the bifurcation diagrams depends strongly on the choice of the parameters. If the bifurcation diagrams for all possible choice of parameters are analyzed and presented, the bifurcation space will be described and discussed for all the system parameters. Such parameter bifurcation space may be very practical from an engineering point of view to design
the system parameters of converter circuits and to satisfy the requirements of the particular output behavior.
IV. C ONCLUSIONS
Applying the sampled-data approach, the periodic modeling and stability analysis of switching converters are studied. The period-one and period-two modeling for the converter dynam- ics is utilized to predict the occurrence of the corresponding bifurcations and analyze the stability of converter circuits.
According to the methodological approach proposed in this paper, the nonlinear dynamics of buck converters is analyzed.
The period-doubling bifurcation is found to exhibit as system parameter varies, which might produce a series of period- doubling bifurcations and then result in a chaotic motion.
It has been found that the system does undergo through the serial period-doubling bifurcations which lead to a step-wise transition from period-one to chaos phenomena. Moreover, there are quite wide regions in the parameter space in which the converter behaves chaotically. The understanding of a novel methodological approach opens up new possibilities of operating regimes that can help to optimize design of converters.
A CKNOWLEDGMENTS
This research was supported by the National Science Coun- cil, Taiwan, R.O.C., under Grant: NSC 98-2625-M-150-001 and the Chung-Shan Institute of Science and Technology, Ministry of National Defense, Taiwan, R.O.C., under Grants:
XB97514P and XB98102P.
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10 15 20 25 30 35 40 45 50 55 60 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Input Voltage(V I)
Duty Cycle (α)
Duty Cycle α for period−one orbit
Fig. 3. Variation of the duty cycle for the period-one orbit.
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5 0 0.5 1 1.5
real part
imag part
Eigenvalues of period−one: V I∈ [12, 25]
VI=12
VI=12 VI=25 VI=25
Unit Circle
Vbif=24.517
eigenvalues
Fig. 4. Variation of the eigenvalues for the period-one orbit.
0.0482 0.0482 0.0484 0.0486 0.0488 0.049 0.0492 0.0494 0.0496 0.0498 0.05 3
4 5 6 7 8 9 10
time (sec) vcon & vramp
Period−one orbit in the (t,vcon) space, VI=24
vcon(t) vramp(t)
Fig. 5. Period-one orbit in the (t, v
con) space with the ramp function, V
I= 24.
0.0482 0.0482 0.0484 0.0486 0.0488 0.049 0.0492 0.0494 0.0496 0.0498 0.05 3
4 5 6 7 8 9 10
time (sec) vcon & vramp
Period−two orbit in the (t,vcon) space, VI=25
vcon(t) vramp(t)
20 22 24 26 28 30 32 34 36 38 40
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Input Voltage(V I) Duty Cycles: α1 & α2
Duty Cycle for period−two orbit
Vbif=24.517
α1 α2
Fig. 7. Variation of the duty cycle for the period-two orbit.
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5 0 0.5 1 1.5
real part
imag part
Eigenvalues of period−two: V I∈ [24.5, 32]
VI=24.5 VI=24.5
VI=32 VI=32
Unit Circle
Vbif=31.118
eigenvalues
Fig. 8. Variation of the eigenvalues for the period-two orbit.
15 20 25 30 35 40
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
VI in
Bifurcation Diagram with VI∈ [15, 40]
Fig. 9. Bifurcation diagram i
nvs V
Iwith V
I∈ [15 40].
0.0422 0.043 0.044 0.045 0.046 0.047 0.048 0.049 0.05
3 4 5 6 7 8 9 10 11 12
time (sec) vcon & vramp
Chaos orbit in the (t,vcon) space, VI=35
vcon(t) vramp(t)
