Bifurcation and Chaos in AsymmetricallyCoupled BVP Oscillators

T. Ueta and H. KawakamiTokushima University, Japan

Brief history of BVP (Bohnhoffer van der Pol)oscillator

• A 2nd-dim sytem derived from Hodkin-Huxley(HH) equation.

• FitzHugh-Nagumo oscillator, extracting exci-tatory behavior of HH equation.

• Nonlinearity: only a cubic term is included.

1

Circuit realization BVP oscillator is a natural ex-tension of van der Pol oscillator. 日本語日本語

• evaluate internal impedance of a coil

• add a bias power source to remove symmetryof the origin

¶ ³

A. N. Bautin, “Qualitative investigation of aParticular Nonlinear System,” PPM, 1975. →a detail topological classification of BVP equa-tion

µ ´

2

Coupled BVP oscillators

Coupled symmetrical BVP oscillators systemhave been studied by using the group theory.

⇓No chaotic behavior is found in real circuitry

3

This presentation shows . . .

• asymmetrically coupled BVP oscillators

• circuit configuration and equations

• bifurcaiton phenomena of equilibria and limitcycles

• results of laboratory experiments

4

Single BVP Oscillator

g(v)L C

R

g(v)L

v

R E

Cdv

dt= −i− g(v)

Ldi

dt= v − ri + E

There exist two port to extract state variables vand i.

5

Nonlinear resisterwith 2SK30A FET:

+

−

2SK30AGR

47K

47K

47K

47K

100

741

g(v)

6

Measurement of the nonlinear resister

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

-10 -5 0 5 10

i [A

]→

v [V]→

lab. dataa tanh bx

error

g(v) = −a tanh bv with a = 6.89099 × 10−3, b =0.352356.

7

BVP equation

¶ ³

x = −y + tanh γx

y = x− ky.µ ´

· = d/dτ, x =

√√√√√C

Lv, y =

i

a

τ =1√LC

t, k = r

√√√√√C

L, γ = ab

√√√√√L

C

8

Bifurcation of Single BVP oscillator

0

0.5

1

1.5

2

0 0.5 1 1.5 2

k

γ

d

h1 h2

G

(b)

(a)

(c)

(d)

(e)

Oscillatory

9

An example flow in area (d)

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

y

x

stable

limit cycle

unstable

limit cycles

C +

C −

O

10

Circuit parameters

L = 10 [mH], C = 0.022 [µF]

⇓

γ = 1.6369909,

√√√√√L

C= 674.19986.

0

0.5

1

1.5

2

0 0.5 1 1.5 2

k

γ

d

h1 h2

G

(b)

(a)

(c)

(d)

(e)

Oscillatory

11

Resistively coupled BVP oscillators

g(v1)L

v1

C

r1

g(v2)L

v2

C

r2

R

g(v1)L

v1

C

r1

g(v2)L

v2

C

r2

R

g(v1)Lv1

C

r1

g(v2)L

v2

C

r2

R R

g(v1)Lv1

C

r1

g(v2)L

v2

C

r2

R

12

Asymmetrically coupled BVP oscillators

g(v1)L

v1

C

r1

g(v2)L

v2

C

r2

R

13

Circuit equations

Cdv1

dt= −i1−g(v1)

Ldi1dt

= v1−r1i1+Gr1

1+Gr1(r1i1 − v2)

Cdv2

dt= −i2−g(v2)+

G

1+Gr1(r1i1−v2)

Ldi2dt

= v2−r2i2

14

Normalized equations

xj =vj

a

√√√√√C

L, yj =

ija

, kj =rj

√√√√√C

L, j=1,2.

τ =1√LC

t, γ=ab

√√√√√L

C, δ=G

√√√√√L

C.

η =1

1 + δk1

15

Normalized equation (cont.)

dx1

dτ= −y1 + tanh γx1

dy1

dτ= x1 − k1y1 + δk1η(k1y1 − x2)

dx2

dτ= −y2 + tanh γx2 + δη(k1y1 − x2)

dy2

dτ= x2 − k2y2

16

Symmetry

x = f(x)

where, f : Rn → Rn : C∞ for x ∈ Rn.

P : Rn → Rn

x 7→ Px

P -invariant equation:

f(Px) = Pf(x) for all x ∈ Rn

17

Matrix P

• If k1 = k2, then

P =

0 0 1 00 0 0 11 0 0 00 1 0 0

group for product:Γ = {P,−P, In,−In}

• If k1 6= k2 then Γ = {In,−In}18

Poincare 写像解 ϕ(t) :

x(t) = ϕ(t, x0), x(0) = x0 = ϕ(0, x0) .

Poincare 切断面:

Π = {x ∈ Rn | q(x) = 0 } ,

T : Π → Π; x 7→ ϕ(τ(x), x) ,

周期解 ϕ(t) について，固定点が対応する :

T (x0) = x0.

19

特性方程式と局所分岐

χ(µ) = det

∂ϕ

∂x0− µIn

.

分岐の種類：

• µ = 1: 接線分岐

• µ = −1: 周期倍分岐

• µ = ejθ: Neimark-Sacker 分岐

• Pitchfork 分岐20

0.7

0.8

0.9

1

1.1

0.7 0.8 0.9 1 1.1

k2

k1

h

G

G

non-oscillatory

oscillatoryoscillatory

Bifurcation diagram

21

周期解の分岐図

0.7

0.8

0.9

1

1.1

0.7 0.8 0.9 1 1.1

k2

k1

h

G

G

non-oscillatory

oscillatoryoscillatory

Pf

G

I

22

周期解の分岐図(拡大図)

0.88

0.9

0.92

0.94

0.96

0.66 0.68 0.7 0.72 0.74

k2

k1

G

G

G

I

Pf

h

period-doubling

cascade

23

位相平面図(x1-x2)

(a) -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 → (b) -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

(c) -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 → (d) -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

24

平衡点・周期解の分岐図

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0.8 1.2 1.6 2 2.4 2.8

k2

δ

G

G

h

25

平衡点・周期解の分岐図

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

k2

δ

period-doubling

cascade

h

G

I

Pf

26

平衡点・周期解の分岐図(拡大図)

0.95

1

1.05

1.1

1.15

1.2

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

k2

δ

I1

I2I4

I1

G

Pf

chaotic

h

27

周期解の分岐 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 2 →

x1 →

28

リアプノフ指数

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15

ν →

δ →

ν1

ν2

ν3

ν4

29