Bifurcation and Chaos in Asymmetrically Coupled BVP Oscillators T. Ueta and H. Kawakami Tokushima University, Japan
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