bifurcation and chaos in asymmetrically coupled …tetsushi/talks/iscas...† asymmetrically coupled...
TRANSCRIPT
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