Bifurcation and Chaos
in the Extended BVP Oscillator

T. Ueta and H. Kawakami
Tokushima University, Japan
Bifurcation and Chaosin the Extended BVP Oscillator – p.1/31

Coupled oscillators✎ practical industrial applications models✎ biological activities

Bifurcation and Chaosin the Extended BVP Oscillator – p.2/31

Coupled oscillators✎ practical industrial applications models✎ biological activities


Decomposite a complex nonlinear dynamics
into unit oscillators and their connections



Bifurcation and Chaosin the Extended BVP Oscillator – p.2/31

Coupled oscillators✎ practical industrial applications models✎ biological activities


Decomposite a complex nonlinear dynamics
into unit oscillators and their connections

⇓a reduced dynamical system with symmetry✎ synchronization✎ global/local bifurcations
Bifurcation and Chaosin the Extended BVP Oscillator – p.2/31

Previous studies
Resistively coupled BVP oscillators✎ Symmetrical connected oscillators have muchvariety of synchronization modes of limit
cyclesBifurcation and Chaosin the Extended BVP Oscillator – p.3/31

Previous studies
Resistively coupled BVP oscillators✎ Symmetrical connected oscillators have muchvariety of synchronization modes of limit
cycles✎ but do not have chaos within reasonable
parameter range.

Bifurcation and Chaosin the Extended BVP Oscillator – p.3/31

Previous studies
Resistively coupled BVP oscillators✎ Symmetrical connected oscillators have muchvariety of synchronization modes of limit
cycles✎ but do not have chaos within reasonable
parameter range.
✎ since symmetrical properties rather build“mild” dynamics.

Bifurcation and Chaosin the Extended BVP Oscillator – p.3/31

Previous studies
Resistively coupled BVP oscillators✎ Symmetrical connected oscillators have muchvariety of synchronization modes of limit
cycles✎ but do not have chaos within reasonable
parameter range.
✎ since symmetrical properties rather build“mild” dynamics.
✎ asymmetrical coupling induces Chaos !

Bifurcation and Chaosin the Extended BVP Oscillator – p.3/31

Single BVP oscillator
LCv

g(v)

RE

Bifurcation and Chaosin the Extended BVP Oscillator – p.4/31

Single BVP oscillator
LCvg(v)

RE

Circuit equation:
dv= −i − g(v)Cdtdi
= v − ri + ELdt
Bifurcation and Chaosin the Extended BVP Oscillator – p.4/31

Bifucations in single BVP
2d1.5(c)

k(e)
(a)

1G

h1

h2

0.5(d)(b)
Oscillatory

000.5

1

γ

1.5

2

➠
Bifurcation and Chaosin the Extended BVP Oscillator – p.5/31

Resistively coupled BVP oscillators
Rv2

v1
L

r1

Cg(v1)

LCg(v2)

r2

Bifurcation and Chaosin the Extended BVP Oscillator – p.6/31

Resistively coupled BVP oscillators
RL

r1

C

v1

v2

v1g(v1)

L

r2

Cg(v2)

g(v1)

v2

LCR
r1

LCg(v2)

r2

Bifurcation and Chaosin the Extended BVP Oscillator – p.6/31

Resistively coupled BVP oscillators
RL

C

v1

v2

v1
g(v1)

Lg(v2)

Cg(v1)C
r1

Rg(v1)

L

R

r2

r1

v2

Lg(v2)

r2

R

v2

L

v1

C

L

C
r1

Cg(v2)

r2

Bifurcation and Chaosin the Extended BVP Oscillator – p.6/31

Resistively coupled BVP oscillators
RLC

v1

v2

v1
g(v1)

Lg(v2)Cg(v1)C
r1

Rg(v1)L
r1

v1g(v2)

r2

v2

L

v1CR

Cg(v1)

LR

r2

r1v2

Lg(v2)

C

r2

v2

LC

R
r1

LCg(v2)

r2

Bifurcation and Chaosin the Extended BVP Oscillator – p.6/31

v-v coupled BVP oscillators
Rv2

v1
L

r1

Cg(v1)L

Cg(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.

Bifurcation and Chaosin the Extended BVP Oscillator – p.7/31

v-i coupled BVP oscillators
v1g(v1)

v2

LCR
r1

LCg(v2)

r2✎ T. Ueta, et al., Bifurcation and Chaos in
Asymmetrically Coupled BVP Oscillators,
ISCAS 2002, Scottsdale, Arizona, Int. J.
Bifurcation and Chaos (to appear)

Bifurcation and Chaosin the Extended BVP Oscillator – p.8/31

In this talk. . .
Reducing the v-i coupled circuit
v1
g(v1)

v2

LCR
r1

LCg(v2)

r2

Bifurcation and Chaosin the Extended BVP Oscillator – p.9/31

In this talk. . .
Reducing the v-i coupled circuitv1g(v1)

v2

LCR
r1

LCg(v2)

r2

removing g(v2 ), letting r2 = ∞, R = 0,Bifurcation and Chaosin the Extended BVP Oscillator – p.9/31

In this talk. . .
Reducing the v-i coupled circuitv1g(v1)

v2

LC

R
r1

LCg(v2)

r2

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

v2

v1

CC
r

Bifurcation and Chaosin the Extended BVP Oscillator – p.9/31

Extended BVP oscillator
Lg(v1)

rv1
v2v1g(v1)

CC

C1

v2
L

C2r

vs
Extended BVP
ChuaThis circuit have already shown as one of all
combinations of L, C, r, g(v). [Chua et al, 1992]

Bifurcation and Chaosin the Extended BVP Oscillator – p.10/31

Extended BVP oscillator
Lg(v1)

r

v1
v2

v1

g(v1)

CC

C1

v2
L

C2r

vs
Extended BVP
ChuaThis circuit have already shown as one of all
combinations of L, C, r, g(v). [Chua et al, 1992]☞But not mentioned these dynamics.

Bifurcation and Chaosin the Extended BVP Oscillator – p.10/31

Extended BVP oscillator
Lg(v1)r

v1
v2

v1g(v1)

CC

C1v2
LC2r

vsExtended BVP
ChuaThis circuit have already shown as one of all
combinations of L, C, r, g(v). [Chua et al, 1992]☞But not mentioned these dynamics.
☞Naturally induced from coupled system!

Bifurcation and Chaosin the Extended BVP Oscillator – p.10/31

Circuit equation
dv

1

= −i − g(v1 )
C

dt


v2
dv2
= i−C
dtr


di
L= v1 − v2
dt

Bifurcation and Chaosin the Extended BVP Oscillator – p.11/31

Circuit equation
dv
1

= −i − g(v1 )
C

dt

v2dv2= i−C
dtr


di
L= v1 − v2dt√
g(v) = −atanhbv,
τ=t/LC,rr
1 LL, δ=γ = abrr Cr Cv2 Ci
v1 C, y=, z= .
x=a La L
aBifurcation and Chaosin the Extended BVP Oscillator – p.11/31

Normalized equationẋ = −z + tanh γxẏ = z − δyż = x − yEquivalently,...
x + α(x)ẍ + β(x, ẋ)ẋ + δx − tanh γx = 0where,2α(x) = δ − γ sech γxβ(x, ẋ) = 2 + 2γ 2 sech2 γx tanh γxẋ − δγ sech2 γxBifurcation and Chaosin the Extended BVP Oscillator – p.12/31

g(v) = −a tanh bv

γ-sensitivity:
1tanh(0.5x)
tanh(x)tanh(1.5x)

0.5
0-0.5-1-4-20

24

Bifurcation and Chaosin the Extended BVP Oscillator – p.13/31

Implementation of g(v)
Rc47K
2SK30AGR5B47K

The output of an inverter driving this circuit canrealize −a tanh bv.

Bifurcation and Chaosin the Extended BVP Oscillator – p.14/31

i [mA]

Experimental measurement
86420-2-4-6-8-10

Rc = 1000
Rc= 500
Rc = 200
Rc = 100
Rc = 0-50510v [V]γ can be controlled by Rc .

Bifurcation and Chaosin the Extended BVP Oscillator – p.15/31

Bifurcation diagram
2Oscillatory

(C)

d+G

1h1(A)

1.5Ih2ChaosG1G2
Non-oscillatory

1G2(D)

11.5+h0(B)

(E)
Pfdδ

Non-oscillatory

G30.5

Oscillatory

000.5

γ2➠

Bifurcation and Chaosin the Extended BVP Oscillator – p.16/31

Bifurcation diagram (magnified)
1.5(H)
1.4(C)

h2I(I)

1.2(J)

1d+δG

1.3(F)

G3(G)G2

1.1(E)
11

1.2

1.4

1.6

γ

1.8

2

Bifurcation and Chaosin the Extended BVP Oscillator – p.17/31

11

0.5

0.5

0.5

0-0.5z→1z→y→Double scroll

0-0.5-1-0.5-1-1-0.5

00.5

10-1-1-0.5

00.5

1-1-0.5

0x→y→x→x-yy-zx-z

0.5

1

Bifurcation and Chaosin the Extended BVP Oscillator – p.18/31

Perspectives

Bifurcation and Chaosin the Extended BVP Oscillator – p.19/31

Perspectives

Bifurcation and Chaosin the Extended BVP Oscillator – p.20/31

Laboratory experiments

By changing an serial resistance in g(v),r = 467[Ω].

Bifurcation and Chaosin the Extended BVP Oscillator – p.21/31

Jack-in-the-box phenomenonzz

2
1.5
10.5
0-0.5
-1-1.5-2-1.5-1.52
1.5
-1 1
0.5
-0.5 0-0.5-10-1.5x -2
0.5
-1.5

-1-0.5

0y

10.5

11.5

1.5

-1.5
-1-0.50x0.5-1-0.5

0y

10.5

1

1.5

1.5

Bifurcation and Chaosin the Extended BVP Oscillator – p.22/31

Jack-in-the-box phenomenon(2)
1x0.5
0-0.5
-1100

200300t

Bifurcation and Chaosin the Extended BVP Oscillator – p.23/31

Long transient of jack-in-the-box(x0 , y0 , z0 ) = (3, 5, 8)

Bifurcation and Chaosin the Extended BVP Oscillator – p.24/31

Long transient of jack-in-the-box(x0 , y0 , z0 ) = (3, 5, 8)

Bifurcation and Chaosin the Extended BVP Oscillator – p.24/31

Long transient of jack-in-the-box(x0 , y0 , z0 ) = (3.00001, 5, 8)

Bifurcation and Chaosin the Extended BVP Oscillator – p.24/31

Long transient of jack-in-the-box(x0 , y0 , z0 ) = (3.00001, 5, 8)

Bifurcation and Chaosin the Extended BVP Oscillator – p.24/31

Generation of jack-in-the-boxshrink

Wreck of a limit cycle

Ws

Bifurcation and Chaosin the Extended BVP Oscillator – p.25/31

Basin boundary in single BVP

LCvg(v)

R
E

Bifurcation and Chaosin the Extended BVP Oscillator – p.26/31

Basin boundary in single BVP

LCv

g(v)

R
E

Bifurcation and Chaosin the Extended BVP Oscillator – p.26/31

Basin boundary in single BVP

L
Cv

g(v)

R
E

Each basin is continuously separable.

Bifurcation and Chaosin the Extended BVP Oscillator – p.26/31

Basin boundary of a 2D map⇒
magnify
Fractal boundary (Mille-feuille structure) is also
separable.

Bifurcation and Chaosin the Extended BVP Oscillator – p.27/31

Lorenz system

 Bernd Krauskopf and Hinke Osinga
Stable manifold of the origin forms a “surface.”

Bifurcation and Chaosin the Extended BVP Oscillator – p.28/31

Basin boundary (x-y plane, with z = 0)
γ = 1.3

Bifurcation and Chaosin the Extended BVP Oscillator – p.29/31

Basin boundary (x-y plane, with z = 0)
γ = 1.238

Boundary is suddenly blurred.

Bifurcation and Chaosin the Extended BVP Oscillator – p.29/31

Basin boundary (x-y plane, with z = 0)
γ = 1.234

Bifurcation and Chaosin the Extended BVP Oscillator – p.29/31

Basin boundary (x-y plane, with z = 0)
γ = 1.2

not continuously separable.

Bifurcation and Chaosin the Extended BVP Oscillator – p.29/31

Fractal basin is not appeared !
γ = 1.4

Bifurcation and Chaosin the Extended BVP Oscillator – p.30/31

Fractal basin is not appeared !
γ = 1.234

Bifurcation and Chaosin the Extended BVP Oscillator – p.30/31

Fractal basin is not appeared !
γ = 1.238

Boundary is suddenly blurred.

Bifurcation and Chaosin the Extended BVP Oscillator – p.30/31

Fractal basin is not appeared !
γ = 1.2

God only knows which.

Bifurcation and Chaosin the Extended BVP Oscillator – p.30/31

Conclusions
The extended BVP oscillator✎ analyses of bifurcations and chaos✎ Jack-in-the-box phenomenon✎ Blurred basin boundary
Future problems✎ investigation of the stable manifold of theorigin✎ 3D structure of the basin boundary (stablemanifold)

Bifurcation and Chaosin the Extended BVP Oscillator – p.31/31