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

k