Chaos in Cross-Coupled BVP Oscillatorsi01
v1g(v1)

LG1

G2i02
v2
LCC
r1g(v2)

r2

Tetsushi Ueta and Hiroshi Kawakami
Tokushima University, Tokushima, Japan

Chaos in Cross-Coupled BVP Oscillators – p.1/47

Nonlinear coupled oscillators✎ power lines system✎ neural networks✎ biological activities

Chaos in Cross-Coupled BVP Oscillators – p.2/47

Nonlinear coupled oscillators✎ power lines system✎ neural networks✎ biological activities
Decomposition a complex nonlinear dynamics intounit oscillators and their connections

Chaos in Cross-Coupled BVP Oscillators – p.2/47

Nonlinear coupled oscillators✎ power lines system✎ neural networks✎ biological activities


Decomposition a complex nonlinear dynamics intounit oscillators and their connections

⇓a reduced dynamical system with symmetry✎ synchronization✎ global/local bifurcations
Chaos in Cross-Coupled BVP Oscillators – p.2/47

BVP oscillator
v-portC✎ 2 dimensionalsystemLi-portg(v)r✎ 2 interface ports✎ 2 essential parameters✎ any mutual coupling model can be constructed✎ Coupled by resisters ⇒ diffusive coupling

Chaos in Cross-Coupled BVP Oscillators – p.3/47

Bifurcations in a BVP oscillator
2d1.5(c)k(e)(a)

1Gh1h20.5(d)(b)
Oscillatory

000.5

1γ

1.5

2
Chaos in Cross-Coupled BVP Oscillators – p.4/47

Bifurcations in a BVP oscillator
1.5

unstable
limit cycles

1C+0.5yO
0-0.5

stablelimit cycle

C−-1-1.5-1.5-1-0.500.5

11.5x
Chaos in Cross-Coupled BVP Oscillators – p.5/47

Resistively coupled BVP oscillators
R
v2

v1
L

r1

Cg(v1)L

Cg(v2)

r2

Chaos in Cross-Coupled BVP Oscillators – p.6/47

Resistively coupled BVP oscillators
RL

r1

C

v1

v2

v1
g(v1)

L

r2

Cg(v2)

g(v1)

v2

L
C

R
r1

L

C

g(v2)

r2

Chaos in Cross-Coupled BVP Oscillators – p.6/47

Resistively coupled BVP oscillators
RL

g(v1)

CL

g(v2)

C

v1

R
r1

CR

L

Cg(v2)r2v2

L
Cg(v1)

v2

L

r1

r2

r1g(v1)v1v2

v1

LCg(v2)r2

Chaos in Cross-Coupled BVP Oscillators – p.6/47

Resistively coupled BVP oscillators
RLg(v1)

C

v1

v2

v1
L

g(v2)

Cg(v1)C
r1i01
v1g(v1)

v1

v2

L
C

R
r1

L

r2

L

R

r2

r1

v2

LC

g(v2)

g(v1)

L

G1

g(v2)

C

r2

G2i02
v2
L

C

C
r1

g(v2)

r2

Chaos in Cross-Coupled BVP Oscillators – p.6/47

v -v coupled BVP oscillators
Rv2

v1
L

r1

Cg(v1)

LCg(v2)

r2✎ T. Ueta, et al., Strange attractor in resistively
coupled BVP oscillators, In Proc. 2001 Int. Conf. on
Progress in Nonlinear Science, Russia, July 2001,
International Journal of Bifurcation and Chaos (toappear)

Chaos in Cross-Coupled BVP Oscillators – p.7/47

v -i coupled BVP oscillators
v1
g(v1)

v2

LC

R
r1

LCg(v2)

r2✎ T. Ueta, et al., Bifurcation and Chaos in
Asymmetrically Coupled BVP Oscillators, ISCAS
2002, Scottsdale, Arizona, Int. J. Bifurcation andChaos, Vol. 13, No. 5, 2003.

Chaos in Cross-Coupled BVP Oscillators – p.8/47

Application. . .
v1g(v1)

v2

LCR
r1

LCg(v2)r2

Chaos in Cross-Coupled BVP Oscillators – p.9/47

Application. . .
v1g(v1)

v2

LCR
r1

LCg(v2)

r2

removing g(v2 ), letting r2 = ∞, R = 0,

Chaos in Cross-Coupled BVP Oscillators – p.9/47

Application. . .
v1
g(v1)

v2

LCLR
r1

Cg(v2)

r2

removing g(v2 ), letting r2 = ∞, R = 0,
Lg(v1)

v2

v1CCrChaos in Cross-Coupled BVP Oscillators – p.9/47

Cross-coupled BVP oscillatorsi01
v1g(v1)

L

G1

G2

i02
v2
LCC
r1g(v2)r2

Chaos in Cross-Coupled BVP Oscillators – p.10/47

Preceding researches✎ O. Papy and H. Kawakami, “SymmetricalProperties and Bifurcations of the Periodic Solutions
for a Hybridly Coupled Oscillator,” IEICE Trans.,
Vol. E78-A, No. 12, pp.1816–1821, 1995.
✎ O. Papy and H. Kawakami, “Symmetry Breakingand Recovering in a System of n Hybridly CoupledOscillators,” IEICE Trans. Vol. E79-A, No. 10, pp.
1581–1586, 1996.

⇒ focused on only synchronization phenomena of limit
cyclesChaos in Cross-Coupled BVP Oscillators – p.11/47

In this talk...
Revisit hybridly coupled BVP oscillators inimplementation point of view
✎ Local bifurcations✎ Torus doubling and breakdown✎ Circuit implementation✎ Chaos via period-doubling

Chaos in Cross-Coupled BVP Oscillators – p.12/47

Circuit equation

i01
v1
g(v1)

L

G1

G2

i02
v2
LC

C
r1

r2

dv1
Cdt
di1g(v )
Ldt
dv2
Cdt
di1
Ldt
2

= −i1 − g(v1 ) − i01= v1 − r1 (i1 + i02 )= −i2 − g(v2 ) − i02= v1 − r1 (i1 + i01 )

i01 = G1 (v1 − r2 (i2 + i01 ))i02 = G2 (v2 − r1 (i1 + i02 )) .
Chaos in Cross-Coupled BVP Oscillators – p.13/47

dv1
C
dt
di1
L
dt
dv2
C
dt
di2
L
dt

G1
(v1 − r2 i2 )= −i1 − g(v1 ) −1 + G1 r2
G2 r1(v2 − r1 i1 )= v1 − r1 i1 −1 + G2 r1G2
(v2 − r1 i1 )= −i2 − g(v2 ) −1 + G2 r1
G1 r2(v1 − r2 i2 )= v2 − r2 i2 −1 + G1 r2

Chaos in Cross-Coupled BVP Oscillators – p.14/47

Variable transformation
vj
xj =arij
yj = , kj = rjarLδj = Gj, j = 1, 2.
Cr
1Lτ=√,t, γ = ab
CLCC,
Lr

C,
L

A nonlinear negative conductanceg(v) = −a tanh bv
Chaos in Cross-Coupled BVP Oscillators – p.15/47

Normalized ODEδ1ẋ1 = −y1 + tanh γx1 −(x1 − k2 y2 )
1 + δ1 k2


δ2 k1


 ẏ1 = x1 − k1 y1 −(x2 − k1 y1 )1 + δ2 k1
δ2=−y+tanh
γx−(x−ky)ẋ222211
1 + δ2 k1



δ1 k2


(x1 − k2 y2 ) ẏ2 = x2 − k2 y2 −1 + δ1 k2

x: voltage, y : current

Chaos in Cross-Coupled BVP Oscillators – p.16/47

Symmetry✎ Symmetry in the state space:

P : R4 → R4
(x1 , y1 , x2 , y2 ) 7→ (−x1 , −y1 , −x2 , −y2 ).
The ODE is invariant under this transformation.
✎ Symmetry in the parameter space:(k1 , k2 , δ1 , δ2) → (k2 , k1 , δ2, δ1 )Chaos in Cross-Coupled BVP Oscillators – p.17/47

Fixing parametersδ1ẋ1 = −y1 + tanh γx1 −(x1 − k2 y2 )

1 + δ1 k 2δ2 k 1 ẏ1 = x1 − k1 y1 −(x2 − k1 y1 )1 + δ2 k 1
δ2(x2 − k1 y1 )ẋ2 = −y2 + tanh γx2 −
1 + δ2 k 1δ1 k 2(x1 − k2 y2 ) ẏ2 = x2 − k2 y2 −1 + δ1 k 2γ ≈ 1.6✎ k1 = k2 = 0.75: identical oscillators case✎ free parameters case

Chaos in Cross-Coupled BVP Oscillators – p.18/47

Bifurcation in coupled identical oscillators 1.5
Torus and its doubling(A)

=

δ2G
δ11h1
δ2(B)
h2
0.5

h1NS(C)

h2
000.5

11.5

Chaos in Cross-Coupled BVP Oscillators – p.19/47

δ1 = δ2Phase portrait with different initial values in x1 -x2 plane.
21.5

10.50-0.5-1-1.5-2-2-1.5-1-0.5

00.5

11.5

2
Chaos in Cross-Coupled BVP Oscillators – p.20/47

xiTime response
1.5
10.5
0-0.5
-1-1.5150x1, x2155

160

165

170

175

180

185

190

195200txi(a) completely in-phase synchronization
1.5
10.5
0-0.5
-1-1.5
150

155

160

165

x1x2

170

175

180

185

190

195200t(b) completely anti-phase synchronization.

δ1 = δ2 = 1.0

Chaos in Cross-Coupled BVP Oscillators – p.21/47

Limit sets
✎ In area (A) and almost all (B): There exist stablesinks C + and C −✎ in-phase mode solution is disappeared by the tangentbifurcation.
✎ anti-phase mode solution is disappeared by theNeimark-Sacker bifurcaiton

Chaos in Cross-Coupled BVP Oscillators – p.22/47

Bifurcation diagram
1.5

Torus and its doubling(A)

=

δ2G
δ11

h1
δ2(B)
h2
0.5

h1NS(C)h2000.5

11.5

Chaos in Cross-Coupled BVP Oscillators – p.23/47

Classificationδmodesmall (weak coupling)in-phasemedium (moderate coupling) in-phase and anti-phaselarge (strong coupling)anti-phase

Chaos in Cross-Coupled BVP Oscillators – p.24/47

110.50.500yy

Torus: k1 = k2 = 0.75, δ1 = 1.4, δ2 =
1.2.

-0.5

-0.5

-1-1
-1

-0.5

0x

0.5

1-1

-0.5

0x

0.5

1x1 -y1 planex2 -y2 plane
As a result of compromise between in-phase an anti-phasemode
Chaos in Cross-Coupled BVP Oscillators – p.25/47

Torus: k1 = k2 = 0.75, δ1 = 1.4, δ2 =1.2. 1

0.5

0-0.5

-1-1-0.5

0x1 -x2 plane

0.5

1Chaos in Cross-Coupled BVP Oscillators – p.26/47

Poincaré mapping on x0 = 0.9
y1 -y2 plane

Chaos in Cross-Coupled BVP Oscillators – p.27/47

Torus doubling, k1 = k2 = 0.75. δ =1.4989.
y -y plane
1

2

Chaos in Cross-Coupled BVP Oscillators – p.28/47

Torus doubling, k1 = k2 = 0.75. δ =1.4989.
y -y plane
12

Chaos in Cross-Coupled BVP Oscillators – p.29/47

Poincaré mapping on x0 = 0.9, δ =1.4989.
x -y plane
22

Chaos in Cross-Coupled BVP Oscillators – p.30/47

Torus doubling

M. Sekikawa, T. Miyoshi, and N. Inaba. “SuccessiveTorus Doubling,” IEEE Trans. Circuits and SystemsI, Vol. 48, No. 1, pp. 28–34, Jan 2001.



Cf. an attractor via “swollen bifurcation”: ⇒ no
successive doubling is occurred.

⇒ New type of torus doubling ?

Chaos in Cross-Coupled BVP Oscillators – p.31/47

Chaos, δ = 1.522

Chaos in Cross-Coupled BVP Oscillators – p.32/47

Chaos, δ = 1.535

Chaos in Cross-Coupled BVP Oscillators – p.33/47

Asymmetrical case
δ1 = 0.377, δ2 = 2.696.
5

4.5

stablelimit cycle

4Gsinks

3.5

k2I
3oshaC

2.5

2stablelimit cycle

Pfh

1.5

10.25

0.3

0.35

0.4

k1

0.45

0.5
Chaos in Cross-Coupled BVP Oscillators – p.34/47

Phase portrait, k2 = 2.96, δ1 = 0.337,
δ2 = 2.696.
1y0.5

0-0.5

-1-1-0.5

00.5

1Chaos in Cross-Coupled BVP Oscillators – p.35/47

Phase portrait, k2 = 2.96, δ1 = 0.337,
δ2 = 2.696.
1y0.5

0-0.5

-1-1-0.5

00.5

1Chaos in Cross-Coupled BVP Oscillators – p.36/47

Phase portrait, k2 = 2.96, δ1 = 0.337,
δ2 = 2.696.
1

y0.5

0-0.5

-1-1-0.5

00.5

1Chaos in Cross-Coupled BVP Oscillators – p.37/47

Phase portrait, k2 = 2.96, δ1 = 0.337,
δ2 = 2.696.
1

y0.5

0-0.5

-1-1-0.5

00.5

1Chaos in Cross-Coupled BVP Oscillators – p.38/47

Phase portrait, k2 = 2.96, δ1 = 0.337,
δ2 = 2.696.
1y0.5

0-0.5

-1-1-0.5

00.5

1Chaos in Cross-Coupled BVP Oscillators – p.39/47

Phase portrait, k2 = 2.96, δ1 = 0.337,
δ2 = 2.696.
1y0.5

0-0.5

-1-1-0.5

00.5

1Chaos in Cross-Coupled BVP Oscillators – p.40/47

Lab. experiments

2000[Ω], G1 = 1/2000[f], G2
ranged from 216–500 [Ω].

r2= 1/250[f].

=r2 isChaos in Cross-Coupled BVP Oscillators – p.41/47

Lab. experiments

2000[Ω], G1 = 1/2000[f], G2
ranged from 216–500 [Ω].

r2
= 1/250[f].

=r2 isChaos in Cross-Coupled BVP Oscillators – p.42/47

Lab. experiments

2000[Ω], G1 = 1/2000[f], G2
ranged from 216–500 [Ω].

r2
= 1/250[f].

=
r2 is

Chaos in Cross-Coupled BVP Oscillators – p.43/47

Lab. experiments

2000[Ω], G1 = 1/2000[f], G2
ranged from 216–500 [Ω].

r2
= 1/250[f].

=r2 isChaos in Cross-Coupled BVP Oscillators – p.44/47

Lab. experiments

2000[Ω], G1 = 1/2000[f], G2
ranged from 216–500 [Ω].

r2
= 1/250[f].

=r2 isChaos in Cross-Coupled BVP Oscillators – p.45/47

Lab. experiments

2000[Ω], G1 = 1/2000[f], G2
ranged from 216–500 [Ω].

r2
= 1/250[f].

=r2 isChaos in Cross-Coupled BVP Oscillators – p.46/47

Conclusions
Cross-coupled BVP oscillators✎ Classification of synchronization mode
✎ Torus doubling and chaos✎ Period doubling cascade✎ Circuit realization

Chaos in Cross-Coupled BVP Oscillators – p.47/47