Quasiperiodicity Bifurcations undergone by limit cycles before - - PowerPoint PPT Presentation

Universit´ e Pierre et Marie Curie Master Sciences et Technologie (M2) Sp´ ecialit´ e : Concepts fondamentaux de la physique Parcours : Physique des Liquides et Mati` ere Molle Cours : Dynamique Non-Lin´ eaire Laurette TUCKERMAN laurette@pmmh.espci.fr

Quasiperiodicity

Bifurcations undergone by limit cycles

• before

after after after bifurcation pitchfork saddle-node Hopf

Continuous system Poincar´ e map Before bifurcation Limit cycle Fixed point After bifurcation Torus Circle Name of bifurcation Secondary Hopf Neimark-Sacker

• Hopf

Periodic and quasiperiodic behavior in oscillating chemical reaction model

From D. Barkley, J. Ringland & J.S. Turner, J. Chem. Phys. 87, 3812 (1987)

Circle maps

xn+1 = f(x) mod 1

Sine circle map

• V. Arnold in the 1960s (Moscow, later also Paris IX, Dauphine, died 2010):

xn+1 = fΩ,K(xn) ≡

• xn + Ω − K

2π sin(2πxn)

• mod 1

K measures nonlinearity, Ω is basic frequency.

K = 0 = ⇒ xn+1 = [xn + Ω] mod 1

Ω = p = ⇒ all x are fixed points of f Ω = p/q = ⇒ all x are members of q-cycles of f f q(x) =

• x + p

q + p q + . . . p q

• q times
• mod 1 = [x + p]mod 1 = x

Ω = 1/5 Ω = 3/5 Ω 1/5 Ω irrational = ⇒ no fixed points or q-cycles. All x are members of quasiperiodic orbits. Each orbit is dense on the circle.

K > 0: Frequency locking

xn+1 = fΩ,K(xn) ≡

• xn + Ω − K

2π sin(2πxn)

• mod 1

K = 0 K = 1

Ω = 0.2 Ω = 0.1 f(x) = x + Ω saddle-node bifurcation creates pair of fixed points

xn+1 = fΩ,K(xn) ≡

• xn + Ω − K

2π sin(2πxn)

• mod 1

Condition for bifurcation Condition for fixed point f ′(x) = 1 f(x) = x 1 − K cos(2πx) = 1 x + Ω − K 2π sin(2πx) = x + n cos(2πx) = 0 sin(2πx) = 2π K (Ω − n) x = 1 4 = ⇒ sin

• 2π1

4

• = 1 = 2π

K (Ω − n) = ⇒ K = 2πΩ x = 3 4 = ⇒ sin

• 2π3

4

• = −1 = 2π

K (Ω − 1) = ⇒ K = 2π(1 − Ω)

Saddle-node bifurcations of sine circle map create stable-unstable pairs of fixed points x = 1

4

x = 3

4

K = 0.5 Ω = K

2π = 0.08

Ω = 1 − K

2π = 0.92

First frequency-locking tongue in (Ω, K) plane

Fixed points (one-cycles) exist inside tongue, for 0 ≤ Ω < K

2π and 1 − K 2π < Ω ≤ 1

Two-cycles: fixed points of f 2(x) ≡ f(f(x))

Since f 2

(Ω=1/2,K=0)(x) = x ∀x , set Ω±(K) = 1 2 ± ǫ(K) with K, ǫ ≪ 1

f 2(x) = f(x) + Ω± −

K 2π sin(2πf(x))

= x + 1

2 ± ǫ − K 2π sin(2πx) + 1 2 ± ǫ − K 2π sin(2πf(x))

= x + 1 ± 2ǫ − K

2π sin(2πx)

−

K 2π sin(2πf(x))

− sin(2πf(x)) = − sin

• 2π
• x + 1

2 ± ǫ − K 2π sin(2πx)

• = − sin (2πx + π ± 2πǫ − K sin(2πx))

= − sin (2πx + π)

• = sin(2πx)

cos (±2πǫ − K sin(2πx))

• ≈1

− cos (2πx + π)

• = cos(2πx)

sin (±2πǫ − K sin(2πx))

• ≈±2πǫ−K sin(2πx)

≈ sin (2πx) + cos (2πx) (±2πǫ − K sin(2πx)) = sin (2πx) ± 2πǫ cos (2πx) − K

2 sin(4πx)

f 2(x) ≈ x + 1 ± 2ǫ ± ǫK

• ≪ǫ

cos(2πx) − K2

4π sin(4πx)

Fixed points of f 2(x)mod 1: x = f 2(x) mod 1 ≈ x ± 2ǫ − K2

4π sin(4πx)

ǫ ≈ ±K2

8π sin(4πx)

At saddle-node bifurcation point of f 2: 1 = d dxf 2(x) ≈ 1 − K2 cos(4πx) = ⇒ cos(4πx) = 0 = ⇒ x ≈ 1

8, 3 8, 5 8, 7 8 =

⇒ sin(4πx) = ±1 Therefore: ǫ(K) ≈ ±K2

8π

Ω±(K) =

1 2 ± ǫ(K) ≈ 1 2 ± K2 8π

Note: Period-doubling bifurcation (= pitchfork for f 2) requires K ≥ 2: f ′(x) = 1 − K cos(2πx) ≥ 1 − K

Saddle-node bifurcations of f 2 create stable-unstable pairs of 2-cycles

K = 0.8 Ω ≈ 1

2 − K2 8π ≈ 0.475

Ω ≈ 1

2 + K2 8π ≈ 0.525

x ≈ 3

8, 7 8

x ≈ 1

8, 5 8

Second frequency-locking tongue in (Ω, K) plane

Two-cycles exist inside tongue

1 2 − K2 8π Ω 1 2 + K2 8π

Three-cycles: fixed points of f 3(x)

Emerge from K = 0, Ω = 1/3 and Ω = 2/3 via saddle-nodes of f 3 Exist within Ω-intervals of width O(K3)

For any (p, q), seek Ω, K, x such that f q

Ω,K(x) = x + p

q-cycles are produced via saddle-node bifurcations of f q Exist within Ω-intervals Ip,q(K) surrounding p/q of width O(Kq) called

frequency-locking or Arnold tongues

K = 0 = ⇒

p,q Ip,q = rationals

measure

• p,q Ip,q
• = 0

K = 1 = ⇒ measure

• p,q Ip,q
• = 1

Schematic representation of frequency-locking tongues

Winding number of circle map f

W (f) ≡ lim

n→∞

f n(x0) − x0 n f n not truncated to [0, 1] Poincar´ e: f monotonic & continuous = ⇒ limit exists & independent of x0. For sine circle map: K = 0 K = 1 W = limn→∞

x0+nΩ−x0 n

= Ω Devil’s staircase continuous, diagonal line constant on set of measure one, jumps at each irrational number

The golden mean: “most irrational” number

Stays furthest away from frequency-locking tongues w0 ≡ 0 wn+1 ≡ 1 1 + wn w1 = 1 1 + 0 = 1 1 = 1 w2 = 1 1 + 1 1 + 0 = 1 1 + w1 = 1 1 + 1 = 1 2 w3 = 1 1 + 1 1 + 1 1 + 0 = 1 1 + w2 = 1 1 + 1

2

= 1

3 2

= 2 3 Golden mean: w∗ ≡ lim

n→∞ wn

Golden mean: w∗ ≡ lim

n→∞ wn

with wn+1 = 1 1 + wn w∗ = 1 1 + w∗ w∗(1 + w∗) = 1 w2

∗ + w∗ − 1 = 0

w∗ = −1 + √1 + 4 2 = √ 5 − 1 2 = 0.618 . . . (1 − w∗) : w∗ = w∗ : 1 Parthenon, plants, shells, Greeks, Renaissance, . . .

Fibonacci sequence: F0 = F1 = 1, Fn+1 = Fn + Fn−1 = ⇒ 1, 1, 2, 3, 5, 8, 13, . . . leads to equivalent definition of wn: wn+1 ≡ Fn Fn+1 = Fn Fn + Fn−1 = 1

Fn+Fn−1 Fn

= 1 1 + wn Closest rational approximation obtained by truncating continued fraction: a = a0 + 1 a1 + 1 a2 + . . . w∗ is irrational least well approximated by rational: a1 = a2 = . . . = 1 Following path in (Ω, K) space with WΩ,K = w∗ will keep furthest away from frequency-locking tongues

Taylor-Couette flow

Laminar Couette Taylor Vortex Wavy Vortex Modulated Wavy Vortex UC(r) UT V (r, z) UW V (r, θ, z, t) UMW V (r, θ, z, t)

• No frequency-locking in modulated wavy vortex flow! Why not?

Rand (1981): Symmetry! In rotating frame, wavy vortex flow is steady and modulated wavy vortex flow is periodic. Points on circle (phases in θ) dynamically equivalent = ⇒ no saddle-nodes.

Lyapunov exponents

Steady state ¯ x: eigenvalues of Jacobian matrix Limit cycle ¯ x(t mod T ): Floquet exponents Any attractor: Lyapunov exponents Let ¯ x(t) evolve according to full nonlinear system: ˙ ¯ x = f(¯ x(t)) Let ǫ(t) evolve according to linearized system: ˙ ǫ =

• Df¯

x(t)

• ǫ

Largest Lyapunov exponent: λ(1) ≡ lim

t→∞

1 t ln

• ǫ(t)

ǫ(0)

• Independent of initial condition if within same attractor

Integrate perturbed non-linear system: Initial slope is largest Lyapunov exponent Stop when trajectory reaches attractor boundary.

Winding number: average rotation per iteration Lyapunov exponent: average growth or decay per iteration Rate of growth of area: λ(1) + λ(2) Rate of growth of volume: λ(1) + λ(2) + λ(3), etc. Map: ǫ1 = f ′(¯ x0)ǫ0 ǫn =

n−1

• k=0

f ′(¯ xk)ǫ0 λ = lim

n→∞

1 n ln

• n−1
• k=0

f ′(¯ xk)

• = lim

n→∞

1 n

n−1

• k=0

ln |f ′(¯ xk)| Chaotic attractors: nearby initial conditions eventually diverge = ⇒ at least one Lyapunov exponent is positive One of the definitions of chaos

Wrinkling of a torus

When K > 1, sine circle map becomes non-invertible = ⇒ it cannot be the Poincar´ e mapping of a flow = ⇒ it can become chaotic (an invertible map cannot become chaotic) Attractor can no longer be mapped onto a circle and may become wrinkled

(a) Torus (quasiperiodic) (b) Frequency locking (1:49) (c) Bands on wrinkled torus (d) Wrinkled torus (e) Frequency locking (1:48) (c) Wrinkled torus From D. Barkley, J. Ringland & J.S. Turner, J. Chem. Phys. 87, 3812 (1987).

Route to chaos from a torus

< 1970s Landau: Hopf1 (Ω1), Hopf2 (Ω2), Hopf3 (Ω3), . . . = ⇒ Turbulence ≈ 1980s Lorenz, May, Feigenbaum, etc.: Small number (3) of ODEs can display chaos Ruelle & Takens (1971); Newhouse, Ruelle & Takens (1978): Theorem concerning quasiperiodic motion (motion on torus) of dimension n ≥ 3. Perturbations can lead to chaos: “Let v be a constant vector field on the torus T n = Rn/Zn. If n ≥ 3, every C2 neighborhood of v contains a vector field v′ with a strange Axiom A attractor. If n ≥ 4, we may take C∞ instead of C2.”

Curry & Yorke (1978); Grebogi, Ott & Yorke (1985): Numerical investiga- tion of probability of random perturbations leading to chaos Flow Poincar´ e map two-torus circle map three-torus pair of coupled circle maps θn+1 = θn + ω1 + KP1(θn, φn) mod 1 φn+1 = φn + ω2 + KP2(θn, φn) mod 1 Solutions: quasiperiodic with three frequencies, quasiperiodic with two frequencies, periodic, or chaotic. Map is non-invertible for K ≥ Kc. Attractor Lyapunov exp

K Kc = 3 8 K Kc = 3 4 K Kc = 9 8

Three-frequency quasiperiodic 0, 0 82% 44% 0% Two-frequency quasiperiodic 0, − 16% 38% 33% Periodic −, − 2% 11% 31% Chaotic +, ? 0% 7% 36%