An Aspectof Oscillatory Conditions in Linear Systems
and Hopf Bifurcations in Nonlinear Systems
T. Ueta and H. Kawakami
Tokushima University, JapanNDES2004@ Évora – p. 1/35

Background
Oscillators — limit cycles in nonlinear circuits✎ generation: lost stability for equilibrium —Hopf bifurcation✎ required structural stability✎ design problem✏ realization: RC phase shift or LC loop
✏ design parameters: physical elements, wave
shape and frequency

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Oscillator in text books
VoVi +−A(jω)β(jω)

1. provide an amplifier A( jω) and a positive
feedback β( jω).
2. the trigger can be noise in the circuitry.
3. the output is applied into the input in phase.
4. no external force is needed anymore.
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Transfer function expression
The transfer function of the feedback system:A( jω)
H( jω) =1 − A( jω)β( jω)
If Aβ ≈ 1, ∃ω = ω0, then H( jω0 ) = ∞.

Barkhausen criterion

ReA( jω0 )β( jω0 ) = 1,ImA( jω0 )β( jω0 ) = 0NDES2004@ Évora – p. 4/35

Polar plots
Barkhausen criterion of Nyquist plot:Im H(j ω)ω = ω0Re H(j ω)−10ω = οο

ReA( jω0 )β( jω0) = 1,

ω=0ImA( jω0 )β( jω0) = 0
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Linear and nonlinear systems
Barkhausen criterion for linear systems gives a
harmonic oscillation — structurally unstable !
⇓Need a gimmick to suppress amplitude growing.
✎ To retain the stability in large: nonlinearity— anatural assumption: limiter, clamper,
thermistor✎ getting the structural stability

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Wien bridge oscillator
R2C2A( jω)
C1+v

R1−R3βR4Nonlinearity

R4 should be nonlinear to keep stable oscillation.
NDES2004@ Évora – p. 7/35

Wien bridge oscillator
R2Z2Z3C2R3+−R1C1R4Z4Z1

Z1 Z3 = Z2 Z4NDES2004@ Évora – p. 8/35

In this talk . . .
Design an oscillator — derive a condition for a given
linear circuit.
We show that the following factors are equivalent:✎ creation of zero impedance✎ a current flows without power source — virtualsource method✎ Barkhausen criterion✎ Hopf bifurcation

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Virtual source method
Assumption of a virtual source by using the phasor
method:✎ Adding a virtual voltage (current) source intothe system✎ zero impedance (admittance) makes a non-zerocurrent (voltage) without source

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Virtual voltage source✎ place a virtual voltage source E = E0e jωt in an
appropriate location in the circuit✎ compute the whole impedance Z✎ ZI = E shows “a condition to keep the current
non-zero is Z = 0.”

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Virtual current source✎ place a virtual current source I = I0 e jωt in an
appropriate location in the circuit✎ compute the whole admittance Y✎ YV = I shows “a condition to keep the voltage
non-zero is Y = 0.”






These virtual source methods are equivalent toHopf bifurcation analysis and Barkhausen criterion.

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Example—LCR circuit
adding a virtual current source
vI0e jωtCRLir1kR
Z1 = (r1 + jωL1 ) kjωCR(r1 + jωL1)=r1 + R − ω2 RL1C + jω(L1 + r1RC)
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LCR circuit
substitute 0 + j0 into the denominator of Z1
R + r1 − ω2 RL1C = 0
ω(L1 + Rr1C) = 0
since L1 > 0, C > 0, we haverR + r1
frequency
ω=RL1CL1
Rr1 = −
Hopf bifurcation set
C
R < 0 required.
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Hopf bifurcation analysis
Assume R−1 v = g(v) Hopf bifurcation analysisdv
C = −g(v) − i,dtdiL1 = v − ridtJacobian matrix:1  1−  −CRC .
J = r  1− L1
L1

where R−1 is a linear part of dg(v)/dt
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χ(µ) = det(A − µI) = 0
1R+r
χ(µ) = µ +=0+ rL1 µ +CR
CRL12!

Substitute Hopf bifurcation condition: µ = 0 + jω
into above equation.
From the real and imaginary part, we have:rR + r1
ω=RL1CL1
Rr1 = −CNDES2004@ Évora – p. 16/35

Barkhausen criterion for BVP
R2vC+Li−r

R3R4NDES2004@ Évora – p. 17/35

as a bridge
Z2Z3R2R3+−LCr
R4Z4Z1NDES2004@ Évora – p. 18/35

Voltage feedback ratio
for the positive feedback loop:
Z2R2 (1 − ω2 L1C + jωCr)β==
Z1 + Z2 r + R2 − ω2 L1CR2 + jωL1 + jωCrR2
Let R = R3 = R4 , and from ReAβ = 1, ImAβ = 0, wehave:r
R + r1L1, Rr1 = −ω=RL1CC⇒ showing the same result.

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Cautions✎ Controlling Hopf bifurcation by choosing
parameter values✎ No guarantee for global stability of the limitcycle✎ post-process: getting global stability, e.g.,
replacing R by a nonlinear conductor✎ Frequency and wave shape — simulation isneeded

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Application — extension of BVPv

CR

L1

L2

i1i2

r1

r2

virtual source method:
Z2 = Z1 k (r2 + jωL2 )R(r1 + r2) + r1 r2 − ω2 (r2RL1C + L2(r1RC + L1 )) = 02

R(L1 + L2) + r1 L2 + r2 L1 + r1r2RC − ω RL1 L2C = 0NDES2004@ Évora – p. 21/35

We have Hopf bifurcation curve H(r1 , r2), andr
(r1 + r2)R + r1r2
ω=
L1 L2 + r2RCL1 + r1RL2C
r2 → ∞r
R + r1
ω∞ =RL1C
r2 → 0r
Rr1
ω0 =L1 L2 + Rr1 L2CNDES2004@ Évora – p. 22/35

Slow oscillation (6 kHz)C = 0.022[µF], r1 = 500[Ω], L1 = 10[mH], L2 = 1[mH](a) 100 kΩ
NDES2004@ Évora – p. 23/35oscillation deadC = 0.022[µF], r1 = 500[Ω], L1 = 10[mH], L2 = 1[mH](a) 400 ΩNDES2004@ Évora – p. 24/35

revival

C = 0.022[µF], r1 = 500[Ω], L1 = 10[mH], L2 = 1[mH](a) 100 ΩNDES2004@ Évora – p. 25/35

Fast oscillation (34 kHz)C = 0.022[µF], r1 = 500[Ω], L1 = 10[mH], L2 = 1[mH](a) 100 ΩNDES2004@ Évora – p. 26/35

Hopf bifurcation set in k1-k2 plane
1.6
1.4

20

Slow-Oscillatory30

1.2
40k21500.8H−55
60

0.6

65

Quiescent

0.4
163

164

0.2

H+165

166

Fast-Oscillatory
000.2

0.4

k1

0.6

0.8

1NDES2004@ Évora – p. 27/35

Bifurcation of Hopf bifurcation set-10-10

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

-5510-10-55(a) α = 0.1.

(b) α = 0.14.

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

-55(c) α = 0.145.

10-10-551010(d) α = 0.15.
NDES2004@ Évora – p. 28/35

α = 0.181Oscillatory
Hk2

0.8
0.6

Quiescent

0.4
0.2
two dimensional

0.2

0.4

0.6

0.8

1k1
NDES2004@ Évora – p. 29/35

extended BVP circuit 1/5v
C2RC1Lr11
Z=Rkk jωL +krjωC1jωC2!!

NDES2004@ Évora – p. 30/35

extended BVP circuit 2/5
R(r − ω2C2 rL + jωL)
Z=R(1 + jωC2 r) + (1 + jωC1R)(r − ω2C2 rL + jωL)
The denominator ≡ 0 + j02

R + r − ω (C2r − C1R)L = 0(C1 + C2)rR + L − ω2C1C2 rRL = 0

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extended BVP circuit 3/5
Frequency:
ω=r

L + (C1 + C2 )rRC1C2rRL

Bifurcation set:
(L + (C1 + C2)rR)(C2r + C1R)L − (R + r)C1C2 rRL = 0⇒ gives Hopf bifurcation set in r-R plane.

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extended BVP circuit 4/5

r=q−C2 L − C12R2 ± −4C1C22 LR2 + (C2 L + C12R2 )22C 2R

In case that C1 = C2 = C:
LrR = − ,Cr = −RNDES2004@ Évora – p. 33/35

extended BVP circuit 5/5
2
Oscillatory

h1(C)

(A)

1.5I

h2

Chaos

G1G2
Non-oscillatory

1(E)

(D)

1

1.50(B)+hG2dδ

Non-oscillatory

G3

0.5

Oscillatory

000.5

γ2
NDES2004@ Évora – p. 34/35

REMARKS
Oscillator design:✎ Hopf bifurcation analysis, Barkhausen
criterion, virtual source method✎ Hopf bifurcation set: oscillation condition curveitself✎ Phasor method is easer to analyze the system

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