Post on 12-Mar-2020
Bifurcation and Chaos in Coupled BVPOscillators
T. Ueta†, T. Kosaka‡, and H. Kawakami††Tokushima University, Japan ‡Fukuyama University Japan
Bifurcation and Chaos in Coupled BVP Oscillators – p.1/42
History of BVP Oscillator• Introduced as a simplified model of
Hodkin-Huxley equation (four-dimensionalautonomous system)
• alias is FitzHugh-Nagumo oscillator• extraction of excitatory activity from HH
equation
x = c(x − x3
3+ y + z)
y = −1c
(x + by − a)
Bifurcation and Chaos in Coupled BVP Oscillators – p.2/42
Analogue of BVP equation
An natural extension of van der Pol equation.• 2nd dimensional autonomous system.• Evaluate a resistance in a coil.• Adding a bias source to destroy symmetric
property.
• A. N. Bautin, “Qualitative investigation of aParticular Nonlinear System,” PPM, vol. 39, No.4, pp. 606–615, 1975. → Topologicalclassification of solutions in BVP equation
• Doi, et. al: Response of BVP with an impulsiveforce
Bifurcation and Chaos in Coupled BVP Oscillators – p.3/42
Coupled BVP equation• O. Papy, H. Kawakami: Analysis on coupled
BVP equations from symmetry point of view.• Kitajima: Chaos generation from symmetry
coupled BVP equations
No chaotic oscillations in symmetrical configuration of
coupled BVP equations
Bifurcation and Chaos in Coupled BVP Oscillators – p.4/42
BVP Oscillator
g(v)C
L
v
E
R
Circuit equation:
Cdvdt= −i − g(v)
Ldidt= v − ri + E
(1)
Bifurcation and Chaos in Coupled BVP Oscillators – p.5/42
Nonlinear conductor2SK30A FET:
47K 47K
100
2SK30AGR5B
g(v) = −a tanh bv
Bifurcation and Chaos in Coupled BVP Oscillators – p.6/42
Fitting
Marquardt-Levenberg method (nonlinear least squaremethod)
a = 6.89099 × 10−3, b = 0.352356
Bifurcation and Chaos in Coupled BVP Oscillators – p.7/42
Approximation error
g(v) = −a tanh bv
with a = 6.89099 × 10−3, b = 0.352356Theoretical value versus measurement value.
-0.001
-0.0005
0
0.0005
0.001
-10 -5 0 5 10
fittin
g er
ror
[A]
v [V]→
A reasonable approximation.Bifurcation and Chaos in Coupled BVP Oscillators – p.8/42
BVP equation
x = −y + tanh γxy = x − ky.
· = d/dτ, x =
√
CL
v, y =ia
τ =1√
LCt, k = r
√
CL, γ = ab
√
LC
Bifurcation and Chaos in Coupled BVP Oscillators – p.9/42
Bifurcation of equilibria for single BVPoscillator
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
Bifurcation and Chaos in Coupled BVP Oscillators – p.10/42
An example phase portrait
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
y →
x →
At region (d):• Two stable sinks.• A saddle.• Two unstable
limit cycles.• A stable limit cy-
cle.
Bifurcation and Chaos in Coupled BVP Oscillators – p.11/42
Circuit parameters setup
L = 10 [mH], C = 0.022 [µF]
⇓
γ = 1.6369909,
√
LC= 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
Bifurcation and Chaos in Coupled BVP Oscillators – p.12/42
Coupled BVP oscillators
A pair of BVP oscillators coupled by a register R.
g(v1)L
v1
C
r1
g(v2)L
v2
C
r2
R
G = 1/R
Bifurcation and Chaos in Coupled BVP Oscillators – p.13/42
Circuit equation
Cdv1
dt= −i1 + a tanh bv1 −G(v1 − v2)
Ldi1dt= v1 − ri1
Cdv2
dt= −i2 + a tanh bv2 −G(v2 − v1)
Ldi2dt= v2 − ri2
(2)
Bifurcation and Chaos in Coupled BVP Oscillators – p.14/42
Scaling
x j =
√
CL
v j, y j =i j
a, k j = r j
√
CL, j = 1, 2.
τ =1√
LCt, γ = ab
√
LC, δ =
√
LC
G.
x1 = −y1 + tanh γx1 − δ(x1 − x2)y1 = x1 − k1y1
x2 = −y2 + tanh γx2 − δ(x2 − x1)y2 = x2 − k2y2
Bifurcation and Chaos in Coupled BVP Oscillators – p.15/42
Symmetry
x = f (x)
where, f : Rn → Rn : C∞ for x ∈ Rn.
P : Rn → Rn
x 7→ Px
P-invariant equation:
f (Px) = P f (x) for all x ∈ Rn
Bifurcation and Chaos in Coupled BVP Oscillators – p.16/42
A matrix P• in case k1 = k2,
P =
0 0 1 00 0 0 11 0 0 00 1 0 0
Γ = P,−P, In,−Informs a group for production.
• in case k1 , k2:
Γ = In,−In
Bifurcation and Chaos in Coupled BVP Oscillators – p.17/42
Bifurcation of Equilibria
0
0.5
1
1.5
2
0 0.5 1 1.5 2
k2
k1
0D
2D
4D
Bifurcation and Chaos in Coupled BVP Oscillators – p.18/42
Poincaré mapping
A solution ϕ(t) :
x(t) = ϕ(t, x0), x(0) = x0 = ϕ(0, x0) .
Poincaré section:
Π = x ∈ Rn | q(x) = 0 ,
T : Π→ Π; x 7→ ϕ(τ(x), x) ,
The fixed point x0 for the limit cyclde ϕ(t):
T (x0) = x0.
Bifurcation and Chaos in Coupled BVP Oscillators – p.19/42
Characteristic equation
χ(µ) = det
(
∂ϕ
∂x0− µIn
)
.
Local bifurcations:• µ = 1: tangent bifurcation G• µ = −1: period-doubling bifurcation I
• µ = e jθ: Neimark-Sacker bifurcation NS• µ = ±1: Pitchfork bifurcation P f
Bifurcation and Chaos in Coupled BVP Oscillators – p.20/42
Bifurcation diagram, δ = 0.337(R =
2000[Ω])
0
0.5
1
1.5
2
0 0.5 1 1.5 2
G
Pf
Pf
Chaotic area
k1
k 2
h
I1
two limit
cycles
two limit
cycles
non-oscillatory
h
I2
I2
I1
G
G
single limit cycle
Bifurcation and Chaos in Coupled BVP Oscillators – p.21/42
Hopf bifurcation k1 = 1.18(r1 ≈ 800)
r2 ≈ 600
r2 ≈ 400
Bifurcation and Chaos in Coupled BVP Oscillators – p.22/42
Period-doubling cascade
r2 ≈ 400( k ≈ 0.72)
r2 ≈ 395[Ω] Rössler-type attractor
Bifurcation and Chaos in Coupled BVP Oscillators – p.23/42
Presence of Double scroll
r2 ≈ 390→ r2 ≈ 380[Ω].
Bifurcation and Chaos in Coupled BVP Oscillators – p.24/42
Chaotic attractor. r1 = 370[Ω]
(a) v1-r1i1. (b) v2-r2i2.
(c) v1-v2, (d) r1v1-r2i2.
Bifurcation and Chaos in Coupled BVP Oscillators – p.25/42
Numerical simulation
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
x 2 →
x1 →
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
x 2 →
x1 →
x1-y1 x2-y2
Bifurcation and Chaos in Coupled BVP Oscillators – p.26/42
Phase portrait between oscillators
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
x 2 →
x1 →
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
x 2 →
x1 →
(c) x1-x2 (d) y1-y2
Bifurcation and Chaos in Coupled BVP Oscillators – p.27/42
Poincaré mapping
x1-x2 plane: Π = x|q(x) = y1 = 0
-1
-0.5
0
0.5
1
-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2
x 2 →
x1 →
0
0.02
0.04
0.06
0.08
0.1
-0.2 -0.19 -0.18 -0.17 -0.16
x 2 →
x1 →
Bifurcation and Chaos in Coupled BVP Oscillators – p.28/42
Time response of Oscillator 1
Chaotic attractor. δ = 0.337, k1 = 1.187, k2 = 0.593,(R = 2000[Ω], r1 = 400[Ω], r2 = 800[Ω]).
-1.5-1
-0.50
0.51
1.5
600 800 1000 1200 1400
x 1 →
t →
-1.5-1
-0.50
0.51
1.5
600 800 1000 1200 1400
y 1 →
t →
Bifurcation and Chaos in Coupled BVP Oscillators – p.29/42
Time response of Oscillator 2
-1.5-1
-0.50
0.51
1.5
600 800 1000 1200 1400x 2
→t →
-1.5-1
-0.50
0.51
1.5
600 800 1000 1200 1400
y 2 →
t →
Bifurcation and Chaos in Coupled BVP Oscillators – p.30/42
Projection into x1-x2-y1—(1)
Bifurcation and Chaos in Coupled BVP Oscillators – p.31/42
Projection into x1-x2-y1—(2)
Bifurcation and Chaos in Coupled BVP Oscillators – p.32/42
Projection into x1-x2-y1—(3)
Bifurcation and Chaos in Coupled BVP Oscillators – p.33/42
Enlargement of bifurcation diagram
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2
NS
G
GPf
Pf
I
I
bi-stable
k1
k 2
Rossler type
Double scrollI
I
Bifurcation and Chaos in Coupled BVP Oscillators – p.34/42
r2 = 390→ 360[Ω]
Bifurcation and Chaos in Coupled BVP Oscillators – p.35/42
The end of scrolling. left: r2 = 360[Ω], right: r2 =
358[Ω], lower: r2 = 346[Ω].
Bifurcation and Chaos in Coupled BVP Oscillators – p.36/42
to be synchronized
upper left: r2 = 342[Ω], upper right: r2 = 340[Ω].
lower left: r2 = 290[Ω], lower right: r2 = 184[Ω].
Bifurcation and Chaos in Coupled BVP Oscillators – p.37/42
Features• There is no period-doubling route for k1 = k2.• (Osc. 1) equilibrium + (Osc. 2) limit cycle =
(Osc. 1+2) chaotic solution.
• Cubic characteristics is essential: g(v) = ax + bx3
with a = −2.27 × 10−3, b = 4.72 × 10−5.
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
x 2 →
x1 →
Bifurcation and Chaos in Coupled BVP Oscillators – p.38/42
An application: Hybrid coupling
g(v1)L
v1
C
r1
L
v2
C
r2
G1
g(v2)
G2
i01 i02
Bifurcation and Chaos in Coupled BVP Oscillators – p.39/42
Period-doubling cascade
Bifurcation and Chaos in Coupled BVP Oscillators – p.40/42
Chaotic response
Bifurcation and Chaos in Coupled BVP Oscillators – p.41/42
RemarksA resistively coupled BVP oscillators• Bifurcation of periodic solutions• Chaos is observed with k1 , k2.• No bifurcation for k1 = k2
Future problems:• synchronization of oscillators• higher dimensional cases
Bifurcation and Chaos in Coupled BVP Oscillators – p.42/42