Asymmetric Chua’s Circuit:
Implementation,
Modeling and Analysis
M.T aslakov and O. Yordanov
email: oleg.yordanov@gmail.com
Institute of Electronics, Bulgarian Academy of Sciences
M. Mateev seminar 2015 – p. 1/29
Outline
Introduction: Lorenz’s 1963 paper.
Examples of simple continuous 3D chaotic
systems.
Asymmetric Chua’s circuit: experimental
implementation
Model of the asymmetric nonlinearity (Chua’s
diode).
Qualitative analysis: fixed points, stability, etc
Numerical implementation
Thinks to do.
M. Mateev seminar 2015 – p. 2/29
Lorenz’s 1963 paper
◮ Motivated to show that the “old meteo school”
are wrong to employ linear regressions.
◮ In 1960: Lorenz’s first 12-variable dynamical
model with aperiodic solutions obtained on
Royal McBee LGP-30 computer (internal
memory of 4096 32-bit words). Presented on
a conference in Tokyo.
◮ Lorenz, Edward N., 1963: Deterministic
Nonperiodic Flow, J. Atmos. Sci., 20,
130–141. Just a three-variable dynamical
system! By around yesterday cited 14521
times! (12130)
M. Mateev seminar 2015 – p. 3/29
Lorenz system: quasi-hyperbolic regime
ẋ = −σ (x − y),
ẏ = rx − y − xz,
ż = −bz + xy,
where σ, r, and b are parameters.
Lorenz system:
Lorenz system:
=10.0, b=8/3, r=28.0
40
=10.0, b=8/3, r=28.0
50
30
40
20
z(t)
30
x(t) 10
y(t)
z(t)
20
0
10
-10
0
-20
-15
-10
-20
30
32
34
36
38
40
t
42
44
46
48
50
30
20
-5
x(
t)
10
0
0
5
10
-10
15
-20
20
y(
t)
-30
dV
= divF~ V = − (σ + b + 1) V
The system is dissipative:
dt
M. Mateev seminar 2015 – p. 4/29
Lorenz system: nonhyperbolic regime
ẋ = −σ (x − y),
Lorenz system:
300
ẏ = rx − y − xz,
ż = −bz + xy
Lorenz system:
=10.0, b=8/3, r=210.0
=10.0, b=8/3, r=210.0
400
200
300
z(t)
100
x(t)
y(t)
z(t)
200
100
0
-50
-100
0
0
26
28
30
t
32
34
x(
t)
50
100
0
-50
-100
-150
50
200
150
100
y
(t
)
M. Mateev seminar 2015 – p. 5/29
Rössler system: spiral regime
ẋ = −y − z,
ẏ = x + ay,
ż = b + z(x − c)
Rossler system: a=0.2, b=0.2, c=6.1
Rossler System
30
26
24
20
22
20
10
a=0.2
18
b=0.2
16
0
z(t)
x(t)
y(t)
z(t)
-10
c=6.1
14
12
10
8
6
-20
4
2
10
0
-10
-30
5
-5
0
0
-40
0
10
20
30
40
t
50
60
70
80
x(
-5
5
t)
-10
10
15
y(
t)
-15
dV
= divF~ V = (a + x − c) V
dt
Rössler, O. E. (1976) An equation for continuous chaos, Phys. Lett A,
57, 397.
M. Mateev seminar 2015 – p. 6/29
Rössler system: funnel regime
ẋ = −y − z,
ẏ = x + ay,
ż = b + z(x − c)
Rossler system
Rossler system
80
a=0.2
b=0.2
c=13
60
60
a=0.2
0.015
40
50
b=0.2
20
c=13
40
0
z(t)
0.003
-20
-40
10
0
-10
10
t)
-60
y(
z(t)
20
20
-0.015
x(t)
y(t)
30
-20
-80
0
0
10
20
30
40
t
50
60
70
-20
-10
0
10
20
30
-30
80
x(t)
M. Mateev seminar 2015 – p. 7/29
J. Sprott’s zoo of 3D chaotic attractors
Maximum quadratic nonlinearities, total of 30 parameters
X
X
dxi
bi,j xj +
ci,jk xj xk ,
= ai +
dt
j
j,k
Computer search.
Simplification: consider max 6 terms
Each parameter varies within [−5, 5] with a step 0.1, which gives
1020 cases.
Of them 108 were randomly chosen for examination
19 chaotic systems were found.
J.C. Sprott, (1994) Some simple chaotic flows, Phys. Rev E, 50, R647.
M. Mateev seminar 2015 – p. 8/29
J. Sprott’s zoo of 19, 3D chaotic attractors
(A)
ẋ = y,
ẏ = −x + yz, ż = 1 − y 2
(B) ẋ = yz,
ẏ = r(x − y),
ż = 1 − xy
(C) ẋ = yz,
ẏ = x − y,
ż = 1 − x2
......
......
ẏ = x + z 2 ,
ż = 1 + y − 2ry
......
......
......
(N) ẋ = 2qy,
......
(S)
ẋ = −x − 4y, ẏ = x + z 2 ,
ż = 1 + x
(U) ẋ = y,
ẏ = z,
ż = y 2 − x − az
ẋ = y,
ẏ = z,
ż = xy 2 − x − az
(V)
M. Mateev seminar 2015 – p. 9/29
Sprott’s generalized B-system
ż = 1 − xy
ẋ = yz,
ẏ = r(x − y),
with two fixed points: f1 = (1, 1, 0) and f1 = (−1, −1, 0)
bmodel system R = 0.9
bmodel system R = 0.9
35
30
4
25
20
y(t)
z(t)
2
15
10
5
0
0
z(t)
x(t)
-5
-2
-10
-15
-4
-20
-25
-6
-6
-30
-4
-2
0
-35
0
20
40
60
80
x(
t)
t
1
2
4
2
0
6
-1
8
-2
10
y(
3
4
t)
-3
dV
= divF~ V = (−r) V
dt
M. Mateev seminar 2015 – p. 10/29
Sprott’s generalized N-system
ẏ = x + z 2 ,
ż = 1 + y − 2rz
ẋ = −2qy,
For q 6= 0 the system has a single fixed point:
f0 = (−1/4r2 , 0, 1/2r).
nmodel system q = 1.0, r = 0.95
nmodel system q = 1.0, r = 0.95
20
4
10
x(t)
y(t)
2
0
z(t)
z(t)
-10
-20
0
-2
-30
-4
10
-30
-40
-20
0
-10
20
40
60
80
100
x(
t)
-10
0
10
t
y(
t)
-20
dV
= divF~ V = (−2r) V
dt
M. Mateev seminar 2015 – p. 11/29
Sprott’s generalized N-system and B-system
Dimitrova E. S. and Yordanov O. I., “Statistics of
some low-dimensional chaotic flows”,
Intern. Journal of Chaos and Bifurcations, vol. 11,
No. 10, pp. 2675-2682, (2001).
M. Mateev seminar 2015 – p. 12/29
Piecewise linear chaotic systems
ẋ = y,
ẏ = z,
ż = −y − az + |x| − 1
Two fixed points: f−1 = (−1, 0, 0) and f1 = (1, 0, 0)
dV
= divF~ V = (−a) V
dt
M. Mateev seminar 2015 – p. 13/29
Chua’s oscillator
r
IC
b
Inl
R
Vb
Ca
Va
Rnl
L
Figure 1: Basic Chua’s circuit.
Rnl is a “negative” resistor, i.e a devise supplying energy to
the system in the form of a nonlinear current, Inl
M. Mateev seminar 2015 – p. 14/29
Chua’s system of differential equations
dVa
= R−1 (Vb − Va ) − g (Va )
Ca
dt
dVb
Cb
= −R−1 (Vb − Va ) + I
dt
dI
L = −rI − Vb .
dt
where Inl = g(Va ) . Apart from g(Va ), the system depends
on 5 parameters: Ca , Cb , R, L and r.
M. Mateev seminar 2015 – p. 15/29
Model for the nonlinear resistor
2.0m
g(V )
a
1.0m
V
1
0.0
V
V
2
V
-2
-1
-1.0m
-2.0m
-3.0m
-3
-2
-1
0
V
a
1
2
3
[volts]
Parametrization of the asymmetric Chua’s diode g(Va ).
M. Mateev seminar 2015 – p. 16/29
Model for the nonlinear resistor

V /R−2 + a−2





−V /R−1 + a−1
g (V ) = −V /R0



−V /R1 + a1



V /R2 + a2
for V ≤ −V−2
for −V−2 ≤ V ≤ −V−1
for −V−1 ≤ V ≤ V1
for V1 ≤ V ≤ V2
for V ≥ V2
(1)
The “resistors” Rq , q = ±1, ±2, are effective and used to
model the slopes of the respective V-I segments. The
a-constants are determined by the requirement of
continuity of g (V ).
M. Mateev seminar 2015 – p. 17/29
Fixed points
I = −Vb /r,
r
Vb =
Va
R+r
Defining equation
2.0m
g(V )
a
1.0m
V
1
0.0
V
V
2
V
-2
-1
-1.0m
Va
= g (Va )
−
(R + r)
-2.0m
-3.0m
-3
-2
-1
0
V
a
1
2
3
[volts]
M. Mateev seminar 2015 – p. 18/29
Fixed points
1.
F(0) : In the interval, V−1 ≤ Va ≤ V1 , F(0) = (0, 0, 0).
2.
F(−1) : In the interval, V−2 ≤ Va ≤ V−1 ,
F(−1) = I(−1) (−R − r, −r, 1).
3.
F(1) : In the interval, V1 ≤ Va ≤ V2 ,
F(1) = I(1) (R + r, +r, −1).
4.
F(−2) : In the interval, Va < −V−2 ,
F(−2) = I(−2) (−R − r, −r, 1).
5.
F(2) : In the interval, Va > V2 ,
F(2) = I(2) (R + r, r, −1).
M. Mateev seminar 2015 – p. 19/29
Conditions for existence
F(0) : The fixed point at zero always exists independent of
the values of the parameters. In the exceptional case of
R + r = R0 , all points in the interval are fixed.
A typical condition for existence, say for F(1) :
R1
R0 ≤ (R + r) ≤
1 − (1 − R1 /R0 )(V1 /V2 )
M. Mateev seminar 2015 – p. 20/29
Divergencies
1
1
1
dV
~
+
V
= divF V = −
+
r
dt
Ca R−2 Ca R Cb R + L
dV
1
1
1
~
V
+
= divF V = −
+
r
dt
C a R2 C a R C b R + L
M. Mateev seminar 2015 – p. 21/29
Re-scaled system
Va = V−1 x, Vb = V−1 y, I = (V−1 /R0 ) z, and T = (Ca R0 ) t
dx
= a (y − x) − ge (x)
dt
dy
= σc [−a (y − x) + z]
dt
dz
= −c [y + r̄z] ,
dt
where a = R0 /R, σc = Ca /Cb , c = Ca R02 /L, r̄ = r/R0 and
g (x) = (R0 /V−1 ) g(V−1 X)
e
M. Mateev seminar 2015 – p. 22/29
Eigenvalues
Complex Eigenvalues of the Chua's Unfolded system
0.25
0.5 K
0.85 K
0.20
R=1.5 K
1.1 K
Im ( )
0.15
λ0 - real number
λ1,2 = λr ± iλi ;
1.27 K
0.10
Other Parameters:
C =
1
0.05
L=
4.6 nF C =
2
8.5 mH
R =1.2 K
1
0.00
-0.010
-0.005
0.000
R =
L
69 nF
85
R = 3.3 K
2
0.005
0.010
Re( )
λr e
e
t + φ)
v1 (t) = v10 e t cos(λie
M. Mateev seminar 2015 – p. 23/29
Dynamics of the Chua’s circuit.
R = 1.27 KΩ
C1 = 4.6 nF
C2 = 69 nF
L = 8.5 mH
r = 85 Ω
R1 = 1.2 KΩ
R2 = 3.3 KΩ
V
[V]
V
[V]
1
2.5
2
2.0
I
L
[mA]
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0
1
2
3
4
5
time [ms]
λ = 0.0 ± i0.18;
M. Mateev seminar 2015 – p. 24/29
Chua’s circuit in the phase space
0
Z
R=1.27 K
-2
0.4
0.3
0.0
0.2
0.5
X
0.1
1.0
0.0
1.5
2.0
-0.1
Y
-0.2
M. Mateev seminar 2015 – p. 25/29
Dynamics of the Chua’s circuit.
V
[V]
V
[V]
1
R = 1.23 KΩ
C1 = 4.6 nF
C2 = 69 nF
L = 8.5 mH
RL = 85 Ω
R1 = 1.2 KΩ
R2 = 3.3 KΩ
2
2
I
L
[mA]
V
1f
I
1
Lf
V
0
2f
-V
2f
-1
I
Lf
-V
1f
-2
5
6
7
8
9
10
time [ms]
λ = 0.0047 ± i0.183;
M. Mateev seminar 2015 – p. 26/29
Chua’s circuit in the phase space
2
Z
R=1.23 K
0
-2
-2.0
-1.5
-1.0
-0.5
X
0.3
0.2
0.1
0.0
0.0
0.5
-0.1
1.0
1.5
-0.2
2.0
-0.3
Y
M. Mateev seminar 2015 – p. 27/29
Things to do
Breaking the system on five linear systems
and match the solutions in the passing
between the segments
Constructing the Poincaré map.
Computing the Lyapunov exponents.
Evaluating the statistical properties of the
signals.
M. Mateev seminar 2015 – p. 28/29
the thank-you-acknowledgment slide
The work is carried out with no support
from a funding agency!
Thank you for your attention!
M. Mateev seminar 2015 – p. 29/29