Bifurcation and Chaos in Coupled BVP Oscillators
Bifurcation and Chaos in Coupled BVP Oscillators
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-dimensional autonomous system) alias is FitzHugh-Nagumo oscillator extraction of excitatory activity from HH equation x3 x˙ = c(x − + y + z) 3 1 y˙ = − (x + by − a) c
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 a Particular Nonlinear System,” PPM, vol. 39, No. 4, pp. 606–615, 1975. → Topological classification of solutions in BVP equation Doi, et. al: Response of BVP with an impulsive force 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 L C
v
g(v)
R E
Circuit equation: dv C = −i − g(v) dt di L = v − ri + E dt
(1)
Bifurcation and Chaos in Coupled BVP Oscillators – p.5/42
Nonlinear conductor 2SK30A 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 square method)
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 = 0.352356 Theoretical value versus measurement value. fitting error [A]
0.001 0.0005 0 -0.0005 -0.001 -10
-5
0
5
10
v [V]→
A reasonable approximation. Bifurcation and Chaos in Coupled BVP Oscillators – p.8/42
BVP equation
x˙ = −y + tanh γx y˙ = x − ky. · = d/dτ, 1 τ = √ t, LC
x= k=r
r r
C v, L
C , L
i y= a r
γ = ab
L C
Bifurcation and Chaos in Coupled BVP Oscillators – p.9/42
Bifurcation of equilibria for single BVP oscillator 2
d
1.5
(c)
k
(e) (a)
1
G
h1
h2
0.5
(d)
(b) Oscillatory
0 0
0.5
1
1.5
2
γ
Bifurcation and Chaos in Coupled BVP Oscillators – p.10/42
An example phase portrait 1.5
1
y→
0.5
0
-0.5
-1
-1.5 -1.5
-1
-0.5
0
x→
0.5
1
At region (d): • Two stable sinks. • A saddle. • Two unstable limit cycles. • A stable limit cy1.5 cle.
Bifurcation and Chaos in Coupled BVP Oscillators – p.11/42
Circuit parameters setup L = 10 [mH],
γ = 1.6369909,
C = 0.022 [µF] ⇓ r
L = 674.19986. C
2
d
1.5
(c)
k
(e) (a)
1
G
h1
h2
0.5
(d)
(b) Oscillatory
0 0
0.5
1
1.5
2
γ
Bifurcation and Chaos in Coupled BVP Oscillators – p.12/42
Coupled BVP oscillators A pair of BVP oscillators coupled by a register R. R v2
v1 L
r1
C
g(v1)
L
C
g(v2)
r2
G = 1/R
Bifurcation and Chaos in Coupled BVP Oscillators – p.13/42
Circuit equation dv1 C dt di1 L dt dv2 C dt di2 L dt
= −i1 + a tanh bv1 − G(v1 − v2 ) = v1 − ri1
(2)
= −i2 + a tanh bv2 − G(v2 − v1 ) = v2 − ri2
Bifurcation and Chaos in Coupled BVP Oscillators – p.14/42
Scaling xj =
r
C v j, L
ij yj = , a
1 τ = √ t, γ = ab LC
kj = rj r
r
L ,δ = C
C , L r
j = 1, 2. L G. C
x˙1 y˙ 1 x˙2 y˙ 2
= = = =
−y1 + tanh γx1 − δ(x1 − x2 ) x1 − k1 y1 −y2 + tanh γx2 − δ(x2 − x1 ) x2 − k2 y2
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 , 0 0 1 0 0 0 P = 1 0 0 0 1 0
0 1 0 0
Γ = {P, −P, In, −In }
•
forms a group for production. in case k1 , k2 :
Γ = {In , −In } Bifurcation and Chaos in Coupled BVP Oscillators – p.17/42
Bifurcation of Equilibria 2
0D 1.5
k2
2D 1
4D
0.5
0 0
0.5
1
k1
1.5
2
Bifurcation and Chaos in Coupled BVP Oscillators – p.18/42
Poincaré mapping A solution ϕ(t) : x(0) = x0 = ϕ(0, x0 ) .
x(t) = ϕ(t, x0 ), Poincaré section:
Π = { x ∈ Rn | q(x) = 0 } , ˆ → Π; T :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 !
∂ϕ χ(µ) = det − µIn . ∂x0 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]) 2 two limit cycles
I1
I2
h G
non-oscillatory Chaotic area
1.5
k2
Pf 1
G
h G
0.5
two limit cycles
single limit cycle
I2
Pf I1
0 0
0.5
1
k1
1.5
2
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 -r1 i1 . (b) v2 -r2 i2 .
(c) v1 -v2 , (d) r1 v1 -r2 i2 . Bifurcation and Chaos in Coupled BVP Oscillators – p.25/42
1.5
1.5
1
1
0.5
0.5
x2 →
x2 →
Numerical simulation
0
0
-0.5
-0.5
-1
-1
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1.5
-1
-0.5
x1 →
x1 -y1
0
0.5
1
1.5
x1 →
x2 -y2
Bifurcation and Chaos in Coupled BVP Oscillators – p.26/42
1.5
1.5
1
1
0.5
0.5
x2 →
x2 →
Phase portrait between oscillators
0
0
-0.5
-0.5
-1
-1
-1.5 -1.5
-1
-0.5
0
0.5
x1 →
(c) x1 -x2
1
1.5
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
x1 →
(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
x2 →
0.5
0
-0.5
-1 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
x1 →
-0.1
0
0.1
0.2
0.1
x2 →
0.08 0.06 0.04 0.02 0
Bifurcation and Chaos in Coupled BVP Oscillators – p.28/42
Time response of Oscillator 1
y1 →
x1 →
Chaotic attractor. δ = 0.337, k1 = 1.187, k2 = 0.593, (R = 2000[Ω], r1 = 400[Ω], r2 = 800[Ω]). 1.5 1 0.5 0 -0.5 -1 -1.5 600
800
1000
1200
1400
600
800
1000
1200
1400
t→
1.5 1 0.5 0 -0.5 -1 -1.5
t→
Bifurcation and Chaos in Coupled BVP Oscillators – p.29/42
y2 →
x2 →
Time response of Oscillator 2 1.5 1 0.5 0 -0.5 -1 -1.5 600
800
1000
1200
1400
600
800
1000
1200
1400
t→
1.5 1 0.5 0 -0.5 -1 -1.5
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 0.75
0.7
I
Rossler type
0.65
I
k2
0.6
Pf
I
Double scroll
0.55
0.5
I
NS 0.45
bi-stable
G
Pf
G 0.4
0.35 1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
k1 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
x2 →
•
0
-0.5
-1 -1
-0.5
0
0.5
1
x1 → Bifurcation and Chaos in Coupled BVP Oscillators – p.38/42
An application: Hybrid coupling
i01 v1 g(v1)
L
G1
G2
i02 v2 L
C
C r1
g(v2)
r2
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
Remarks A 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
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-dimensional autonomous system) alias is FitzHugh-Nagumo oscillator extraction of excitatory activity from HH equation x3 x˙ = c(x − + y + z) 3 1 y˙ = − (x + by − a) c
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 a Particular Nonlinear System,” PPM, vol. 39, No. 4, pp. 606–615, 1975. → Topological classification of solutions in BVP equation Doi, et. al: Response of BVP with an impulsive force 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 L C
v
g(v)
R E
Circuit equation: dv C = −i − g(v) dt di L = v − ri + E dt
(1)
Bifurcation and Chaos in Coupled BVP Oscillators – p.5/42
Nonlinear conductor 2SK30A 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 square method)
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 = 0.352356 Theoretical value versus measurement value. fitting error [A]
0.001 0.0005 0 -0.0005 -0.001 -10
-5
0
5
10
v [V]→
A reasonable approximation. Bifurcation and Chaos in Coupled BVP Oscillators – p.8/42
BVP equation
x˙ = −y + tanh γx y˙ = x − ky. · = d/dτ, 1 τ = √ t, LC
x= k=r
r r
C v, L
C , L
i y= a r
γ = ab
L C
Bifurcation and Chaos in Coupled BVP Oscillators – p.9/42
Bifurcation of equilibria for single BVP oscillator 2
d
1.5
(c)
k
(e) (a)
1
G
h1
h2
0.5
(d)
(b) Oscillatory
0 0
0.5
1
1.5
2
γ
Bifurcation and Chaos in Coupled BVP Oscillators – p.10/42
An example phase portrait 1.5
1
y→
0.5
0
-0.5
-1
-1.5 -1.5
-1
-0.5
0
x→
0.5
1
At region (d): • Two stable sinks. • A saddle. • Two unstable limit cycles. • A stable limit cy1.5 cle.
Bifurcation and Chaos in Coupled BVP Oscillators – p.11/42
Circuit parameters setup L = 10 [mH],
γ = 1.6369909,
C = 0.022 [µF] ⇓ r
L = 674.19986. C
2
d
1.5
(c)
k
(e) (a)
1
G
h1
h2
0.5
(d)
(b) Oscillatory
0 0
0.5
1
1.5
2
γ
Bifurcation and Chaos in Coupled BVP Oscillators – p.12/42
Coupled BVP oscillators A pair of BVP oscillators coupled by a register R. R v2
v1 L
r1
C
g(v1)
L
C
g(v2)
r2
G = 1/R
Bifurcation and Chaos in Coupled BVP Oscillators – p.13/42
Circuit equation dv1 C dt di1 L dt dv2 C dt di2 L dt
= −i1 + a tanh bv1 − G(v1 − v2 ) = v1 − ri1
(2)
= −i2 + a tanh bv2 − G(v2 − v1 ) = v2 − ri2
Bifurcation and Chaos in Coupled BVP Oscillators – p.14/42
Scaling xj =
r
C v j, L
ij yj = , a
1 τ = √ t, γ = ab LC
kj = rj r
r
L ,δ = C
C , L r
j = 1, 2. L G. C
x˙1 y˙ 1 x˙2 y˙ 2
= = = =
−y1 + tanh γx1 − δ(x1 − x2 ) x1 − k1 y1 −y2 + tanh γx2 − δ(x2 − x1 ) x2 − k2 y2
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 , 0 0 1 0 0 0 P = 1 0 0 0 1 0
0 1 0 0
Γ = {P, −P, In, −In }
•
forms a group for production. in case k1 , k2 :
Γ = {In , −In } Bifurcation and Chaos in Coupled BVP Oscillators – p.17/42
Bifurcation of Equilibria 2
0D 1.5
k2
2D 1
4D
0.5
0 0
0.5
1
k1
1.5
2
Bifurcation and Chaos in Coupled BVP Oscillators – p.18/42
Poincaré mapping A solution ϕ(t) : x(0) = x0 = ϕ(0, x0 ) .
x(t) = ϕ(t, x0 ), Poincaré section:
Π = { x ∈ Rn | q(x) = 0 } , ˆ → Π; T :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 !
∂ϕ χ(µ) = det − µIn . ∂x0 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]) 2 two limit cycles
I1
I2
h G
non-oscillatory Chaotic area
1.5
k2
Pf 1
G
h G
0.5
two limit cycles
single limit cycle
I2
Pf I1
0 0
0.5
1
k1
1.5
2
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 -r1 i1 . (b) v2 -r2 i2 .
(c) v1 -v2 , (d) r1 v1 -r2 i2 . Bifurcation and Chaos in Coupled BVP Oscillators – p.25/42
1.5
1.5
1
1
0.5
0.5
x2 →
x2 →
Numerical simulation
0
0
-0.5
-0.5
-1
-1
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1.5
-1
-0.5
x1 →
x1 -y1
0
0.5
1
1.5
x1 →
x2 -y2
Bifurcation and Chaos in Coupled BVP Oscillators – p.26/42
1.5
1.5
1
1
0.5
0.5
x2 →
x2 →
Phase portrait between oscillators
0
0
-0.5
-0.5
-1
-1
-1.5 -1.5
-1
-0.5
0
0.5
x1 →
(c) x1 -x2
1
1.5
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
x1 →
(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
x2 →
0.5
0
-0.5
-1 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
x1 →
-0.1
0
0.1
0.2
0.1
x2 →
0.08 0.06 0.04 0.02 0
Bifurcation and Chaos in Coupled BVP Oscillators – p.28/42
Time response of Oscillator 1
y1 →
x1 →
Chaotic attractor. δ = 0.337, k1 = 1.187, k2 = 0.593, (R = 2000[Ω], r1 = 400[Ω], r2 = 800[Ω]). 1.5 1 0.5 0 -0.5 -1 -1.5 600
800
1000
1200
1400
600
800
1000
1200
1400
t→
1.5 1 0.5 0 -0.5 -1 -1.5
t→
Bifurcation and Chaos in Coupled BVP Oscillators – p.29/42
y2 →
x2 →
Time response of Oscillator 2 1.5 1 0.5 0 -0.5 -1 -1.5 600
800
1000
1200
1400
600
800
1000
1200
1400
t→
1.5 1 0.5 0 -0.5 -1 -1.5
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 0.75
0.7
I
Rossler type
0.65
I
k2
0.6
Pf
I
Double scroll
0.55
0.5
I
NS 0.45
bi-stable
G
Pf
G 0.4
0.35 1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
k1 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
x2 →
•
0
-0.5
-1 -1
-0.5
0
0.5
1
x1 → Bifurcation and Chaos in Coupled BVP Oscillators – p.38/42
An application: Hybrid coupling
i01 v1 g(v1)
L
G1
G2
i02 v2 L
C
C r1
g(v2)
r2
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
Remarks A 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
