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Geometry of resonance tongues
∗ † ∗
1 Introduction
1.1 Various contexts
Hopf bifurcation from a fixed point.
e2πpi/q q ≤ 4
e2πpi/q p
q q ≥ 5 |p| < q.
q
q
q
∗
†
q
q
q
Zq
q
Hopf bifurcation and birth of subharmonics in forced oscillators
dX
dt= F(X)
Y(t)
P Y(0) = Y0 Y0 = 0,
P(0) = 0
(dP)0e2πpi/q q < 5
q
p
dX
dt= F(X) + G(t)
2π
G(t) Y0 = 0
F(0) = 0 2π Y(t)
Y(0) = Y0 0
Y0 P X0
X(2π) X(t)
X(0) = X0 P(0) = 0 Y0
q 0 q
q
q Zq
Zq
2π
Coupled cell systems.
1.2 Methodology: generic versus concrete systems.
1.3 Related work
Zq
Chenciner’s degenerate Hopf bifurcation.
ω0
pn/qn ω0,
Zqn
k
The geometric program of Peckam et al.
Z2
Related work by Broer et al.
Z2 D2
2 Bifurcation of periodic points of planar diffeomor-
phisms
2.1 Background and sketch of results
Zq
q
0.20.10
-0.4
-0.3
-0.2
-0.1
00.20.10
0.20.10
-0.3
-0.2
-0.1
0
-0.1
0.20.10
-0.3
-0.2
-0.1
0
-0.1
0 0.1 0.2 0.3
-0.6
-0.4
-0.2
0
0.2
0 0.1 0.2 0.3
e2πip/q
q
q ≥ 7
q
2.2 Reduction to an equivariant bifurcation problem
P q Pq(x) = x
q q
x1, . . . , xq
P(x1) = x2, . . . , P(xq−1) = xq, P(xq) = x1.
P(x1, . . . , xq) = (P(x1) − x2, . . . , P(xq) − x1).
P(0) = 0, P
P(x) = 0
P Zq
σ(x1, . . . , xq) = (x2, . . . , xq, x1).
Pσ = σP.
0 P
J =
A −I 0 0 · · · 0 0
0 A −I 0 · · · 0 0
0 0 0 0 · · · A −I
−I 0 0 0 · · · 0 A
A = (dP)0 J σ
J
Zq Zq
ω qth
Vω
[x]ω =
x
ωx
ωq−1x
.
J[x]ω = [(A − ωI)x]ω.
J A qth
A qth
J
P(x) = 0
R2 → R2 R2
C
g : C → C,
g(0) = 0 (dg)0 = 0
g
σ ω
(dP)0g(ωz) = ωg(z).
p q ω Zq qth
g Zq
Zq
2.3 Zq singularity theory
Zq
g
Zq
The structure of Zq-equivariant maps.
Zq
C∞
Lemma 1 Zq g : C → C
g(z) = K(u, v)z + L(u, v)zq−1,
u = zz v = zq + zq K, L
Zq contact equivalences.
Zq g h Zq
h(z) = S(z)g(Z(z)),
Z(z) Zq S(z) : C → C
z
S(γz)γ = γS(z)
γ ∈ Zq
Normal form theorems.
L(0, 0) $= 0
Theorem 2
h(z) = K(u, v)z + L(u, v)zq−1
K(0, 0) = 0
q ≥ 5 KuL(0, 0) $= 0 h Zq
g(z) = |z|2z + zq−1
G(z,σ) = (σ + |z|2)z + zq−1.
q ≥ 7 Ku(0, 0) = 0 Kuu(0, 0)L(0, 0) $= 0 h Zq
g(z) = |z|4z + zq−1
G(z,σ, τ) = (σ + τ|z|2 + |z|4)z + zq−1,
σ, τ ∈ C
Remark. q = 3 q = 4
2.4 Resonance domains
G(z) = b(u)z + zq−1.
q
p : q
zG = 0
(dG) = 0.
q
G = 0
dG
u = zz v = zq + zq w = i(zq − zq),
b(u)
Theorem 3
|b|2 = uq−2
bb′+ bb ′ = (q − 2)uq−3
q ≥ 5q2 −1
q ≥ 7
q
The nondegenerate singularity when q ≥ 5.
q ≥ 5
b(u) = σ + u
σ = µ + iν (µ,ν)
µ = µ(u),ν = ν(u), u ≥ 0
|b|2 = (µ + u)2 + ν2
bb′+ bb ′ = 2(µ + u).
µ = −u +q − 2
2uq−3
ν2 = uq−2 −(q − 2)2
4u2(q−3).
(µ,ν) = (0, 0) q−22
ν2 ≈ (−µ)q−2.
q
2 0µ
ν
q
Tongue boundaries in the degenerate case.
g(z) = u2z + zq−1,
q ≥ 7 g G(z) =
b(u)z + zq−1
b(u) = σ + τu + u2.
σ τ
σ = µ + iν
(µ,ν) τ
(σ, τ) q = 7
τ = τ0
τ0
τ
q = 7
q ≥ 7
3 Subharmonics in forced oscillators
q 2qπ
q 2π
3.1 A Normal Form Algorithm
Lie series expansion. m
m Hm
m 0 ∈ C Fm Fm =∏
k≥mHk
Proposition 4 X Y C X
X = X(1) + X(2) + · · · + X(N) FN+1,
X(n) ∈ Hn Y ∈ Hm m ≥ 2 Yt t ∈ R
Y Xt = (Yt)∗(X)
Xt = X +
$ N−1m−1
%∑
k=1
(−1)k
k!tk (Y)k(X) FN+1
= X +
N∑
n=1
$N−nm−1
%∑
k=1
(−1)k
k!tk (Y)k(X(n)) FN+1.
Xt t t = 0
∂
∂tXt = [Xt, Y] = − (Y)(Xt).
∂k
∂tkXt = (−1)k (Y)k(Xt).
t = 0
Xt =∑
k≥0
1
k!tk ∂k
∂tk
∣∣∣∣t=0
Xt
=∑
k≥0
(−1)k
k!tk (Y)k(X).
Y ∈ Hm (Y)k
k(m − 1) X
(Y)k(X) = 0 FN+1,
1 + k(m − 1) > N
Xt =
$ N−1m−1
%∑
k=0
(−1)k
k!tk (Y)k(X) FN+1
= X +
$ N−1m−1
%∑
k=1
(−1)k
k!tk (Y)k(X) FN+1,
Xt =
N∑
n=1
$ N−1m−1
%∑
k=0
(−1)k
k!tk (Y)k(X(n)) FN+1.
(Y)k(X(n)) ∈ Hn+k(m−1)
(Y)k(X(n)) = 0 FN+1,
k > N−nm−1 n
k = &N−nm−1' !
The Normal Form Algorithm. X
S
X N
Lemma 5 (Normal Form Lemma [36])
X
X = S + G(2) + · · · + G(m) Fm+1,
m ≥ 2 G(i) ∈ Hi (S)
X
X = S + G(2) + · · · + G(m−1) + X(m) Fm+1,
X(m) ∈ Hm G(i) ∈ Hi (S) Y ∈ Hm Xt =
(Yt)∗(X)
Xt = X − t (Y)(X(1)) Fm+1.
S
Hm = (S) + (S),
X(m) = G(m) + B(m) G(m) ∈ (S) B(m) ∈ (S)
Y
(S)(Y) = −B(m),
X1 m
X1 = S + G(2) + · · · + G(m−1) + G(m) Fm+1.
!
X N
X
X1 m + 1
X1 m+1
Y Y ∈ (S)
X1 m + 1 m + 1 < N
N + 1
Algorithm (Normal Form Algorithm)Input N S X[2..N]
S
X = S + X[2] + · · · + X[N] FN+1
X[n] ∈ Hn
(∗ X 1 ∗)
for m = 2 to N do
(∗ X m ∗)G ∈ (S) ∩Hm B ∈ (S) ∩Hm
X[m] = G + B
Y Y ∈ (S) ∩Hm
(S)(Y) = −B
(∗ m + 1, . . . ,N ∗)for n = 1 to N do
for k = 1 to &N−nm−1' do
X[n + k(m − 1)] := X[n + k(m − 1)] +(−1)k
k!(Y)k(X[n])
3.2 Applications of the Normal Form Algorithm
z, z
z = iωz
The Lie-subalgebra of real vector fields. R2 C
(x1, x2) R2 x1 + ix2 C X
R2
X = Y1∂
∂x1+ Y2
∂
∂x2,
X = Y∂
∂z+ Y
∂
∂z,
C Y = Y1 + iY2
Example. Y(z, z) = czk+1zk c
X (2) c = a + ib a, b ∈ R
z = x1 + ix2
X = (x21 + x2
2)k(a(x1
∂
∂x1+ x2
∂
∂x2) + b(−x2
∂
∂x1+ x1
∂
∂x2)).
ωN(−x2∂
∂x1+x1
∂
∂x2)
S = iωN(z∂
∂z− z
∂
∂z).
∂
∂zX XR
X = XR
∂
∂z+ XR
∂
∂z.
C
Lemma 6 X Y C f : C → C
X(f) = X(f),
[X, Y] = 〈X, Y〉 ∂
∂z+ 〈X, Y〉 ∂
∂z,
〈·, ·〉
〈X, Y〉 = X(YR) − Y(XR).
Derivation of the Hopf Normal Form.
S = iωN(z∂
∂z− z
∂
∂z).
S
(S)(X) = 〈S,X〉 ∂∂z
+ 〈S,X〉 ∂∂z
,
〈S,X〉 = iωN(z∂XR
∂z− z
∂XR
∂z− XR).
Y = YR
∂
∂z+ YR
∂
∂zYR = zkzl
〈S, Y〉 = iωN (k − l − 1) zkzl.
(S) : Hm → Hm m
m = 2k + 1 Y
YR = z|z|2k
Corollary 7 C S = iωN(z∂
∂z− z
∂
∂z)
z = iωz +
m∑
k=1
ckz|z|2k + O(|z|2m+3).
The nondegenerate Hopf bifurcation.c1 X
z = iωz + a0z2 + a1zz + a2z
2 + b0z3 + b1z
2z + b2zz2 + b3z
3 + O(|z|4).
z = iωz +(b1 −
i
3ω(3a0a1 − 3|a1|
2 − 2|a2|2)
)z2z + O(|z|4).
w = iωNw + wb(|w|2, µ) + O(n + 1)
r = r b(r2, µ) + O(n + 1)
ϕ = ωN + b(r2, µ) + O(n + 1)
|hww |2 |hww |2
r = r(µ) b(r2, µ) = 0
ω(µ) = ωN + b(r(µ)2, µ)
c1
b(u, µ) = µ + u µ = a + iδ
wa,δ(t) =√
−aei(ωN+δ)t (a ≤ 0)
a < 0
Hopf-Neımark-Sacker bifurcations in forced oscillators.
2π C
z = F(z, z, µ) + εG(z, z, t, µ),
2π
ε µ ∈ Rk k
q
F z = 0 p : q
µ
2π
C
z = XR(z, z, t, µ),
XR(z, z, t, µ) = iωNz + (α + iδ)z + zP(z, z, µ) + εQ(z, z, t, µ).
µ ∈ Rk ε P
Q z z P(0, 0, µ) = 0
Q(0, 0, t, µ) = 0 Q z
z z = iωNz
q p : q
ωNp
qp q
Theorem 8 (Normal Form to order q)
z = iωNz + (α + iδ)z + zF(|z|2, µ) + d ε zq−1 eipt + O(q + 1),
F(|z|2, µ) q − 1 F(0, µ) = 0
d
2π
C × R/(2πZ)
X = XR
∂
∂z+ XR
∂
∂z+
∂
∂t,
S = iωN(z∂
∂z− z
∂
∂z) +
∂
∂t.
m 2π
m (z, z, µ)∂
∂tHm Fm =
∏k≥mHk
S 2π∂
∂t
(S)(X) = 〈S,X〉R∂
∂z+ 〈S,X〉R
∂
∂z,
〈S,X〉R = iωN(z∂XR
∂z− z
∂XR
∂z− XR) +
∂XR
∂t.
XR = µσzkzleimt |σ| + k + l = n
〈S,X〉R = (iωN(k − l − 1) + im)XR.
k = l + 1 m = 0 ε = 0
ε
|σ| > 0 n ≤ q |σ| > 0
k = 0 l = q − 1 m = p
|σ| = 1
d ε zq−1 eipt,
d !
3.3 Via covering spaces to the Takens Normal Form
Existence of 2πq-periodic orbits. The Van der Pol transformation.
q 2π q
2π P : C → C
q C × R/(2πZ)
q
Π : C × R/(2πqZ) → C × R/(2πZ),
(z, t) .→ (z itp/q, t 2πZ))
q
(z, t) .→ (z 2πip/q, t − 2π).
ζ = ze−iωNt
ζ = (α + iδ)ζ + ζP(ζeiωNt, ζe−iωNt, µ) + εQ(ζeiωNt, ζe−iωNt, t, µ)
C × R/(2πqZ) Zq
Theorem 9 Equivariant Normal Form of order q
Zq
ζ = (α + iδ)ζ + ζF(|ζ|2, µ) + d ε ζq−1
+ O(q + 1),
O(q + 1) 2πq
Resonance tongues for families of forced oscillators. q
P
P q
NN
Π∗N = N .
P
2πq
P = N 2πq + O(q + 1),
N 2πq 2πq N .
P X
P = R2πωN◦N 2π + O(q + 1),
R2πωN2πωN = 2πp/q,
P (z, µ) = (0, 0),
q Pµ,
Pµ.
R3 = {a, δ, ε}
(a + iδ)ζ + ζF(|ζ|2, µ) + εdζq−1
,
d $= 0 $= Fu(0, 0)
Fu(0, 0)
F(u, µ) u
Theorem 10 (Bifurcation equations modulo contact equivalence)d $= 0 Fu(0, 0) $= 0 Zq
G(ζ, µ) = (a + iδ + |ζ|2)ζ + εζq−1
.
G(ζ, µ)
δ = ±ε(−a)(q−2)/2 + O(ε2).
|ζ|2ζ + εζq−1
Zq
p : q
G(ζ, µ) = 0,
(dG)(ζ, µ) = 0.
u = |z|2 b(u, µ) = a + iδ+ u G(ζ, µ) =
b(u, µ)ζ + εζq−1
|b|2 = ε2uq−2,
bb′+ bb ′ = (q − 2)ε2uq−3,
b ′ =∂b
∂u(u, µ)
(a + u)2 + δ2 = ε2uq−2,
a + u =1
2(q − 2)ε2uq−3.
u
!
a
εε
δ
aa
εε
δ
C×R/(2πZ)2πip/q
q
4 Generic Hopf-Neımark-Sacker bifurcations in feed
forward systems?
Coupled Cell Systems.
1 2 3
x1 = f(x1, x1)
x2 = f(x2, x1)
x3 = f(x3, x2)
xj ∈ R2
16
12
f
S1
f(eiθz2, eiθz1) = eiθf(z2, z1),
θ
C zj = xj1+ ixj2
λ, µ
Dynamics of the first and second cell. S1
fλ,µ(0, 0) = 0 f
fλ,µ(z1, z1)
z1 = 0
z1 = 0
z2 = fλ,µ(z2, z1) = fλ,µ(z2, 0),
fλ,µ
fλ,µ
fλ,µ(z2, 0) = (λ + i − |z2|2) z2,
λ = 0
λ > 0
z2(t) =√λeit.
Dynamics of the third cell.
z3 = eity S1
ieity + eity = = fλ,µ(eity,√λ eit)
= eit fλ,µ(y,√λ).
y = −iy + fλ,µ(y,√λ).
f = fλ,µ(z2, z1)
Fλ,µ,ε(z2, z1) := fλ,µ(z2, z1) + εP(z2, z1).
P(z2, z1)
z1 = Fλ,µ,ε(z1, z1) = fλ,µ(z1, z1) + εP(z1, z1),
z2 = Fλ,µ,ε(z2, 0),
z1 = 0
z2 =√λeit λ > 0)
P(z2, 0) ≡ 0.
y = e−itz3
y = −iy + fλ,µ(y,√λ) + εe−itP(y eit,
√λ eit).
1.
Hε (λ, µ)
5 Conclusion and future work
Zq
Zq
g(z)
P
References
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