Chaos Analysis and Chaotic EMI Suppression of DC-DC Converters (Zhang/Chaos Analysis and Chaotic EMI Suppression of DC-DC Converters) || Discrete Subsystem Chaotic Point Process of

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6Discrete Subsystem Chaotic PointProcess of DC–DC Convertersand EMI Suppression

6.1 Introduction

The point process [1, 2] means that its process characteristics are decided by the eventtime. This physical phenomenon can be represented by the highly localized eventsdistributed randomly in the continuous system. Moreover, each localized event in themodel is expressed in terms of an idealized point, and this point is considered in thesame location with the event in a continuous system. For example, the time pointof a gamma photon produced by a radioactive source in radioactive decay and thespecial point traffic flow on an expressway can both be described by the point process.Two-dimensional events can also be analyzed by point process, such as the time andlocation sets about an earthquake, detection of blood flow in nuclear medicine, thedynamic parameters of physiological phenomenon in metabolic exchange, and so on.The research shows that the power electronic converter is a typical hybrid dynamic

system [3]. And it consists of two parts, one is the continuous subsystem described bybehavior and the other is the discrete subsystem described by the switching process.The behaviors of the discrete subsystem are decided by the pulse width modulation(PWM) controlled strategy of converter. The discrete subsystem after chaotization canbe partly regarded as a point process constituted by PWM pulse signals, the time pointcharacteristics of the switching value can be described and analyzed by chaotic pointprocess theory. In this chapter the spectrum of a converter and its statistical regularityare obtained, and the mechanism of converter chaos suppressing EMI (electromag-netic interference) is explored in depth. This chapter uses an AC–DC converter as anexample to analyze the spectral characteristics of fixed frequency PWM, periodicalPWM, and chaotic PWM.

Chaos Analysis and Chaotic EMI Suppression of DC-DC Converters, First Edition. Bo Zhang and Xuemei Wang.© 2015 John Wiley & Sons Singapore Pte Ltd. Published 2015 by John Wiley & Sons Singapore Pte Ltd.


110 Chaos Analysis and Chaotic EMI Suppression of DC-DC Converters

6.2 Description of Chaotic Point Process of DC–DC Converters

6.2.1 Model of Chaotic Point Process of DC–DC Converters

The power electronics based on hybrid system theory can be described by acontinuous-discrete subsystem, as shown in Figure 6.1 [4]. This system can beexpressed by the following equations

dxdt

= 𝜙(x, 𝜇) x ∈ X ⊂ Rn t ⊂ R 𝜇 ∈ Rl (6.1)

𝜇i+1 = G(𝜇i)|H(x, t) = 0 i = 0, 1, 2 · · · (6.2)

Equation 6.1 belongs to a continuous subsystem, and Equation 6.2 belongsto a discrete subsystem. The characters 𝜙 and G are nonlinear vectors, x is thecontinuous-time state, and 𝜇 is the discrete input or controls parameter. The twosubsystems are under the same feedback controlled loop, and have the followingcharacteristics.

1. When the condition H(x,t) is satisfied, the discrete subsystem will finish an itera-tion, and the 𝜇k and PWM pulse interval Tk can be calculated.

2. The states of the discrete subsystem as the input will be fed back to the continuoussubsystem.

It can clearly be seen that the PWM pulse interval Tk can reflect the duty circleand frequency change of switching variation. When the PWM frequency is fixed, Tk

is decided by the feedback current voltage and sawtooth wave. While it is chaoticPWM, Tk will be decided by the feedback current voltage and sawtooth wave whose

ɸ ʃ

μ

G

x

T

H t

Discrete subsystem

Continuous subsystem

x

Figure 6.1 Structure of a power converter as a continuous-discrete system [5]. © [2003]

IEEE. Reprinted, with permission, from Baranovski, A. L., Schwarz, W., Chaotic and random

point processes: analysis, design, and applications to switching systems. IEEE Trans. Circuitsand Systems I: Fundamental Theory and Applications, 2003


Discrete Subsystem Chaotic Point Process of DC–DC Converters 111

frequency changes in chaos. The different production and means of control ofthe sawtooth wave will make the PWM drive pulse work in a fixed-frequency orchaotic state.Whether it is fixed-frequency PWM or chaotic PWM, their PWM pulse frequencies

can all be expressed by the following equation

f = fs + Δf = vs + kvn vn = 𝜑(vn−1) (6.3)

where vs is the fixed voltage and it is producing fixed nominal switching frequencyfs. vn is the spread frequency signal voltage generated by chaotic mapping 𝜑, whichwill decide chaotic switching frequency deviationΔf. And k is the chaotic modulationdegree.It can be seen from Equation 6.3 that the frequency or interval Tk and arrival time 𝜏k

of the chaotic PWM drive pulse signal are all chaotic variations decided by 𝜑(vf, t).The point process of chaotic PWM decided by the discrete subsystem is shown inFigure 6.2, and it can be expressed by the following formula

𝜉(t) =∞∑

k=1A𝛿(t − 𝜏k−1) Ω = {𝜏k ∶ k ≥ 0} (6.4)

where 𝜏k−1 is the arrival time point of the driving pulse, A is the driving pulse ampli-tude, and Ω is the chaotic point process sets of the discrete subsystem of the chaoticPWM converter. The spectrum character of the chaotic PWM converter is decided bythe driving pulse, namely, the chaotic point process.

6.2.2 Statistical Characteristics of the Chaotic Point Processin Converter

The interval Tk of the periodical and chaotic PWM pulse presents periodic variationand chaotic variation respectively [6]. The starting time 𝜏k of number k driving waveis the accumulation of the interval Tk, and it meets

𝜏k = 𝜏k−1 + TK−1 (6.5)

A

τ τ τ

Tk+1

q(t)Tk

Figure 6.2 Chaotic point process of discrete subsystems in power converters



If 𝜏0 = 0, it can obtain 𝜏k =k∑

i=1Ti, so the starting time 𝜏k of the number k PWM

pulse represents periodicity and chaos with different spread frequency signals.Thus, the joint probability density function of sequence 𝜏k is

f𝜏1,···𝜏k(x1, · · · , xk) =

𝜕kF(𝜏1 < x1, · · · 𝜏k < xk)𝜕x1 · · · 𝜕xk

= f𝜏(x1)k−1∏i=1

𝛿(xi+1 − 𝜙(i+1)(x1)) (6.6)

The characteristic function about 𝜏k is

Θ𝜏1,···,𝜏K(𝜔1, · · · , 𝜔k) = E

(exp

(j

k∑i=1

𝜔i ⋅ 𝜏i

))

= ∫x· · ·∫

(exp

(j

k∑i=1

𝜔i ⋅ xi

))f𝜏1,···,𝜏K

(x1, · · · , xk) (6.7)

According to sequence 𝜏k (k= 0, 1, … ), a continuous-time process N(t) can bedefined as follows

N(t) = max{k ∶ 𝜏k ≤ t} (6.8)

where N(t) represents the PWM pulse number during interval [0, t) it is called thecounting process or renewal process of the PWM driving pulse process 𝜉(t). Thecounting process N(t) of the PWM driving pulse process 𝜉(t) is shown in Figure 6.3.The discrete probability distribution Fk(T) of N(t) is

Fk(T) = P(N(t) = k) = F𝜏(t, k) − F𝜏(t, k + 1) (6.9)

F𝜏(t, k) in Equation 6.9 is the arrival time distribution of the number k chaotic PWMdrive pulse.

t

t

0123

k+1k

k+1

N

τk-1

τk-1

τ2τ1τ0

τ2τ1τ0

τk τk+1

τk+1τk

Figure 6.3 Counting process N(t) of the PWM drive pulse



6.3 Spectral Quantification Analysis of the PWM Pulse Process

6.3.1 Spectral Quantification Analysis of the Periodic PWM Pulse

In Equation 6.3, letΔf= fAsin(2𝜋fmt) be the sinusoid periodic spread spectrum signal,then the PWM switching frequency is [4, 6]

f = fs + fA sin(2𝜋fmt) (6.10)

where fm = 1/Tm is the frequency of the spread sinusoid signal, fA is the maximumfrequency deviation of the spread spectrum PWM driving pulse.

If Δf ={

fA0

nTm ≤ t < nTm + Tm∕2nTm + Tm∕2 ≤ t < (n + 1)Tm

, the periodic spread spectrum signal is

square wave, so the PWM switching frequency is

f =

{fs + fA

fs

nTm ≤ t < nTm + Tm∕2nTm + Tm∕2 ≤ t < (n + 1)Tm

(6.11)

If Δf = fA(tmodTm)Tm

, the periodic spread spectrum signal is a sawtooth, so the PWM

switching frequency is

f = fs +fA(tmodTm)

Tm(6.12)

The instantaneous frequency nth degree harmonic with the PWM driving pulse ofperiodical spread frequency is

finst,n = 1

2𝜋

d𝜃n

dt= n[fs + fA sin(2𝜋fmt)] (6.13)

where 𝜃n is the phrase of the nth degree harmonic, thus

𝜃n = 2𝜋n∫t

0

[fs + fA sin(2𝜋fmt)]dt = 2𝜋nfst − n𝛽 cos(2𝜋fmt) + n𝛽 (6.14)

where, 𝛽 = fAfmis defined as the parameter of the periodic spread spectrum.

Using a Fourier sequence to describe the PWM pulse process 𝜉(t), and substituting𝜃n into 𝜉(t), one can obtain

𝜉(t) =∞∑

n→−∞Cnej𝜃n

=∞∑

n→−∞Cnej(2𝜋nfst+n𝛽) {cos [n𝛽 cos (2𝜋fmt)

]− j sin

[n𝛽 cos (2𝜋fmt)

]}(6.15)

Using the Jacobi equation [7], cos and sin can be expressed as

cos[n𝛽 sin(2𝜋fmt)] = J0(n𝛽) +∞∑

k=even

2Jk(n𝛽) cos 2𝜋kfmt



and

sin[n𝛽 sin(2𝜋fmt)] =∞∑

k=odd

2Jk(n𝛽) sin(2𝜋kfmt)

respectively.Therefore, we get

G(t) =∞∑

n→−∞Cn

{ej2𝜋nfst ⋅ J0 (n𝛽) +

∞∑k=1

Jk(n𝛽)

×

[ej2𝜋t(kfm+nfs) + (−1)k

∞∑k=odd

Jk(n𝛽)ej2𝜋t(nfs−kfm)

]}(6.16)

Here Jk(⋅) is a k-order Bessel function, and it can be expressed by the followingsequence

Jk(n𝛽) =(

n𝛽2

)k⎡⎢⎢⎢⎢⎣1

k!−

(n𝛽2

)2

1!(k + 1)!+

(n𝛽2

)4

2!(k + 2)!−

(n𝛽2

)6

3!(k + 3)!+ · · ·

⎤⎥⎥⎥⎥⎦(6.17)

It has two characters as follows

Jk(n𝛽) ≈ 0, k > n𝛽 + 1 (6.18)

∞∑k=−∞

Jk2(n𝛽) = 1,∀n𝛽 (6.19)

The spectrum amplitude 𝜉(f, 𝛽) of driving pulse 𝜉(t) at frequency point f is

𝜉(f , 𝛽) =∞∑

n=−∞Cn

{J0 (n𝛽) 𝛿(f − nfs) +

∞∑k=1

Jk(n𝛽)[𝛿(f − nfs − kfm

)+ (−1)k𝛿

(f − nfs + kfm

)]}(6.20)

So it can be obtained that the power spectrum S𝜉(𝜔) in the periodic spread frequencyPWM pulse process is

S𝜉(𝜔) = limT→∞

1

T|𝜉(f , 𝛽)|2

= 2

||||||∞∑

n=1Cn

{J0 (n𝛽) 𝛿(f − nfs) +

∞∑k=1

Jk(n𝛽)[𝛿(f − nfs − kfm

)+(−1)k𝛿(f − nfs + kfm)

]}||||||(6.21)



If 𝛽 = 0 and J0(0)= 1 in Equations 6.18 and 6.19, the normal fixed-frequency PWMpulse spectrum can be expressed by

𝜉(f , 𝛽) =∞∑

n=−∞Cn𝛿(f − nfs) (6.22)

And the power spectrum density is

S(f ) = 2

∞∑n=1

|Cn|2𝛿(f − nfs) (6.23)

The power spectrum only contains the harmonic frequency of integer frequency fs.About the periodic spread spectrum PWM, the higher or lower sideband centered

in integer multiples fs is brought by Jk(n𝛽) based on Equations 6.17 and 6.18. Thus,the bandwidth Bn of the n-order harmonic is

Bn = 2(n𝛽 + 1)fm (6.24)

The power spectrums of the fixed carrier and sinusoidal modulation carrier areshown in Figure 6.4. The dashed line stands for the fixed carrier power spectrum, andthe active line stands for the sinusoidal modulation carrier power spectrum. We canfind that the amplitude of the frequency spectrum spreads to more frequency pointsafter the sinusoidal modulation.From Equation 6.23, the PWM driving pulse power spectrum focuses on the integer

frequency points of fs, which can be proved by the fixed PWM driving pulse waves.The fixed PWM driving pulse waves and their spectrum distribution are shown inFigures 6.5a and 6.6a respectively.The waves of the fixed-frequency PWM drive pulse are shown in Figure 6.5a. In

sinusoidal spread spectrum (SPSP), the PWM driving frequency value is periodi-cally changing during the sinusoidal signal cycle of 0.1ms, and the varying pattern isdecided by sinusoidal SPSP signal as shown in Figure 6.5b. The state of the sawtoothwave SPSP is shown in Figure 6.5c. The PWM pulse frequency changes from small tolarge during every sawtooth SPSP signal cycle of 0.1ms, and it changes synchronously

Fixed carrier

Sinusoidal modulation carrier

3fs 4fs2fsfs

S( f)

f (Hz)

Figure 6.4 Carrier frequency modulation [8].© [2002] IEEE. Reprinted, with permission,

from Tse, K. K, Chung, H.S-H, Hui, S. Y., and So, H. C., A comparative study of carrier-

frequency modulation techniques for conducted E MI suppression in PWM converters. IEEETrans. Industry Electronics, 2002



(a) (b)

(d)(c)

10

15

20

25

5

09.1 9.2 9.3

10

15

20

25

5

09.1 9.2 9.3 9.4

10

15

20

25

5

09.29.159.19.059 9.25 9.3

10

15

20

5

05.25.155.1 5.25 5.3

Figure 6.5 PWM drive waveform with (a) fixed frequency PWM, (b) sinusoidal signal

spread spectrum, (c) sawtooth signal spread spectrum, and (d) square signal spread spectrum

with the sawtooth wave SPSP signal. As shown in Figure 6.5d, the frequency valuevariation of the metal oxide semiconductor field effect transistor (MOSFET) drivepulse is decided by the square SPSP signal voltage during the square signal cycleevery 0.1ms by using square wave modulation. Therefore, the effectiveness of PWMfrequency control strategy Equation 6.3 has been proved by Figure 6.5.PWM driving pulse SPSPs consist of infinitely many discrete spectrums, and are

decided by 𝛽 and fm. Although the periodic spread frequency can reduce the peakvalue of PWM pulse SPSP, it will still obtain a discrete SPSP and the power focuseson special frequency points centered by nfs ± kfm, not spread completely. For the threeperiodic SPSP methods, the spectrum of sawtooth SPSP is smoother, sinusoidal SPSPtakes second place, and square SPSP is the most discrete type as shown in Figure 6.6.



100

10

100 m

10 m

100 u

00 200 K 400 K 600 K 800 K 1 M 1.2 M

1 m

1

2010

100 m

10 m

100 u0 200 K 400 K 600 K 800 K 1 M

1 m

1

100

10

100 m

10 m

100 u

10 u

1 m

1

(a)

(b)

(c)1 M0 200 K 400 K 600 K 800 K

100

10

100 m

10 m

100 u

1 m

1

(d)0 200 K 400 K 600 K 800 K 1 M 1.2 M

Figure 6.6 Spectrum of the PWM drive waveform with (a) fixed-frequency PWM, (b) sinu-

soidal signal SPSP, (c) sawtooth wave signal SPSP, and (d) square signal SPSP



6.3.2 Spectral Quantification Analysis of PWM Chaotic SPSP

From Figure 6.2, the PWM pulse interval Tk controlled by chaotic SPSP can bedescribed by the following chaotic mapping 𝜑

Tk+1 = 𝜑(Tk) (6.25)

The driving pulse starting time 𝜏k of number K is a chaotic sequence, so

𝜏k+1 = T1 + 𝜑(T1) + 𝜑(2)(T1) + · · · + 𝜑(K)(T1) = 𝜙(k)(T1) (6.26)

Then, the joint probability density function of the chaotic sequence is

f𝜏1,···𝜏k(x1, · · · , xk) =

𝜕kF(𝜏1 < x1, · · · 𝜏k < xk)𝜕x1 · · · 𝜕xk

= f𝜏(x1) ⋅k−1∏i=1

𝛿(xi+1 − 𝜙(i+1)(x1)

)(6.27)

The characteristic function of corresponding 𝜏k is

Θ𝜏1,···,𝜏K(𝜔1, · · · , 𝜔k) = E

(exp

(j

k∑i=1

𝜔i ⋅ 𝜏i

))

= ∫x· · ·∫

(exp

(j

k∑i=1

𝜔i ⋅ xi

))f𝜏1,···,𝜏K

(x1, · · · , xk) (6.28)

The probability density function f𝜏(x) of 𝜏k is an invariant distribution ofchaotic mapping 𝜑, and the convergence of invariant distribution is decided by theFrobenius–Perron operator [9]. Here, the Frobenius–Perron operator expressed

by

(𝜌 (y) =

∑xi=f−1(y)

𝜌(xi)f ′(xi)

)is linear and in infinite dimensions. The invariant

distribution is the fixed points of Frobenius–Perron operator or the eigenfunctionwith the eigenvalue 1, and it can be obtained by iterating the Frobenius–Perronoperator from the initial density f0(x)

fk+1(x) = 𝜌fk(x) (6.29)

For any fixed time T, it satisfies 𝜏N(T) ≤ T ≤ 𝜏N(T)+1. N(T) is a counted process ofthe chaotic PWMpulse driving process 𝜉(t) defined by Equation 6.8. Then, the Fourierspectrum of the chaotic PWM driving process 𝜉(t) is

𝜉T(j𝜔) =N(t)∑i=1

Ae−j𝜔𝜏i−1 (6.30)

Thus, the power spectrum density S𝜉(𝜔) of the chaotic SPSP PWM pulse processis [5]




1

TE(|𝜉T (j𝜔)|2) (6.31)

|𝜉T(j𝜔)|2 = N(T)∑i=1

N(T)∑k=1

Ae−j𝜔𝜏i(Ae−j𝜔𝜏k) =N(T)∑k=1

A2 + 2

N(T)−1∑i=1

N(T)−1∑k=1

A2 cos(𝜔(𝜏i+k − 𝜏i))

(6.32)The mathematical expectation can be obtained as follows

E(|𝜉T(j𝜔)|2) = NA2 + 2

∞∑L=1

PN(T)(T)N(T)−1∑

i=1

N(T)−1∑k=1

A2 cos(𝜔(𝜏i+k − 𝜏i)) (6.33)

Here, N = E(N(T)) and PN(T)(T) is the probability of event {𝜏N(T) ≤ T ≤ 𝜏N(T)+1}.Defining 𝜀 = T − 𝜏N(T), T is the fixed time as mentioned before, and 𝜏N(T) is the

arrival time of the number N(T) chaotic driving pulse. Thus, the probability densityfunction of 𝜀 is

p𝜀(x) =∞∑

k=1Fk(T)p𝜏k

(T − x) (6.34)

where p𝜏kis the invariant distribution of starting time 𝜏k, which is the number k chaotic

PWM driving pulse, and Fk(T) is the discrete probability distribution of N(T).When T→∞, the probability density function of 𝜀 is

p𝜀(x) =1

E(Tk)∫∞

xp𝜏k

(y)dy (6.35)

According to Wald’s law, one can get

E(T) = E(𝜀 + 𝜏N(T)) = E(𝜀) + E(𝜏N(T)) ≈ 𝜀′ + N ⋅ E(Tk) (6.36)

For the reason that0 ≤ 𝜀′ = E(𝜀) << NE(TK)

So there isE(T) ≈ NE(Tk) (6.37)

By Equations 6.33 and 6.28, the power spectrum density of the chaotic PWM pulseprocess in Equation 6.26 can be expressed as follows


1

NE(TK)A2E

(N + 2

∞∑L=1

PN(T)(T)N(T)−1∑

i=1

N(T)−1∑k=1

cos(𝜔(𝜏i+k − 𝜏i))

)(6.38)

because limT→∞

limN(T)→∞

PN(T)(T) = 1.



Using the Birkhooff Ergodic Theorem and Lebesque measurement [9, 10], the fur-ther obtained power spectrum density of the PWM pulse is

S𝜉(𝜔) =1

E(Tk)A2

∞∑k=0

∫ f𝜏1,···𝜏K(x1, · · · , xk)E

(cos𝜔

k∑j=1

xj

)

= 1

E(Tk)

∞∑k=0

A2Θ𝜏k(𝜔) (6.39)

where Θ𝜏k(𝜔) is the characteristic function of the chaotic PWM driving pulse arrival

time 𝜏k.From Equations 6.38 and 6.39, the spectrum of the chaotic SPSP controlled PWM

pulse which has a continuous chaotic SPSP driving pulse power spectrum is no longerfastened on the special frequency points. However, it contains the spectrum of every

PWM drive waveform

(V)

Time (ms)

(a)

(b)

12

15

18

21

3

6

9

02.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

0 200 K 400 K 600 K 800 K 1 M 1.2 M

1

10

10 m

100 u

10 u

1 u

1 m

m

Figure 6.7 (a) PWM driving wave and (b) PWM driving spectrum under chaotic

frequency-spreading



frequency point. On the other hand, the power no longer focuses on special frequencypoints, but is spread throughout the entire frequency. Therefore, the EMI suppressingeffect of chaotic SPSP is much better than periodic SPSP, and it has become an effec-tive way to solve converter EMI problems. The chaotic SPSP controlled PWM pulseand spectrums are shown in Figure 6.7; it can be seen that the spectrum is continuousand the peak is smooth.

6.4 Conclusions

Taking an AC–DC converter as an example to analyze the spectrum characteristicsof converter SPSP PWM, this chapter studied the PWM driving pulse. It has beenproved in theory that the EMI peak of the fixed frequency PWM is higher and cannotbe suppressed effectively because of the discrete spectrum and high frequency EMIfocused on the frequency multiplication area. By putting a PWM driving wave asa process or event flow, we can obtain conclusions based on Fourier spectrum andk order Bessel function analysis. First, although periodic PWM can change thespectrum distribution, it cannot change the characteristics of the discrete spectrumdistribution and limits the suppressing PWM effect. Second, using random pointprocess theory based on the principle of statistics has allowed analysis of thecharacteristic function of chaotic SPSP PWM and its probability density function.And the power spectrum density of chaotic SPSP PWM has been analyzed throughinvariant distribution calculations. Finally, it turns out that the spectrum of chaoticSPSP PWM is fully continuous. So it can suppress the converter EMI effectively andhas huge potential for suppressing EMI.

References[1] Snider, D.L. (1982) Stochastic Point Processes, People’s Education Press, Beijing.[2] Deng, Y. and Liang, Z. (1992) Stochastic point processes and applications, Science Press, Beijing

(in Chinese).

[3] Hu, Z., Zhang, B., Hu, S. et al. (2005) The study of DC-DC converter output controllability based

on switched linear system (in Chinese). Chin. J. Electron., 33(2), 370–373.[4] Yang, R., Zhang, B. and Qiu, D.Y. (2008) Chaotic point proce s s de scription of converter discrete

subsystem and EMI suppression. Acta Phys. Sinica, 57(3), 1389–1397.[5] Baranovski, A.L. and Schwarz, W. (2003) Chaotic and random point processes: analysis, design,

and applications to switching systems. IEEE Trans. Circuits Syst. I, 50(8), 1081–1088.[6] Yang, R., Zhang, B. and Qiu, D.Y. (2006) Quantification de scription of the chaotic PWM spectrum

of the switching converter. Acta Phys. Sinica, 55(11), 5667–5673.[7] Carlson, A.B. (1986) Communication Systems: An Introduction to Signal and Noise in Electrical

Communication, McGraw-Hill Press, New York.

[8] Tse, K.K., Chung, H.S.-H., Huo, S.Y. and So, H.C. (2002) A comparative study of carrier- fre-

quency modulation techniques for conducted E MI suppression in PWM converters. IEEE Trans.Ind. Electron., 49(3), 618–627.

[9] Ott, E. (2002) Chaos in Dynamical Systems, Cambridge University Press, London.

[10] Gao, P. (2005) Nonlinear Dynamics, NUDT Press, China (in Chinese).

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Chaos Analysis and Chaotic EMI Suppression of DC-DC Converters (Zhang/Chaos Analysis and Chaotic EMI Suppression of DC-DC Converters) || Discrete Subsystem Chaotic Point Process of