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 � Secondary Hopf � Name of bifurcation Hopf Neimark-Sacker
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 x n +1 = f ( x ) mod 1 Sine circle map V. Arnold in the 1960s (Moscow, later also Paris IX, Dauphine, died 2010): � � x n + Ω − K x n +1 = f Ω ,K ( x n ) ≡ 2 π sin(2 πx n ) mod 1 K measures nonlinearity, Ω is basic frequency.
K = 0 = ⇒ x n +1 = [ x n + Ω] mod 1 Ω = p = ⇒ all x are fixed points of f Ω = p/q = ⇒ all x are members of q -cycles of f x + p q + p q + . . . p � � f q ( x ) = mod 1 = [ x + p ]mod 1 = x q � �� � q times 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 � � x n + Ω − K x n +1 = f Ω ,K ( x n ) ≡ 2 π sin(2 πx n ) mod 1 K = 0 K = 1 Ω = 0 . 2 Ω = 0 . 1 f ( x ) = x + Ω saddle-node bifurcation creates pair of fixed points
� � x n + Ω − K x n +1 = f Ω ,K ( x n ) ≡ 2 π sin(2 πx n ) mod 1 Condition for bifurcation Condition for fixed point f ′ ( x ) = 1 f ( x ) = x x + Ω − K 1 − K cos(2 πx ) = 1 2 π sin(2 πx ) = x + n sin(2 πx ) = 2 π cos(2 πx ) = 0 K (Ω − n ) � � x = 1 2 π 1 = 1 = 2 π 4 = ⇒ sin K (Ω − n ) 4 = ⇒ K = 2 π Ω � � x = 3 2 π 3 = − 1 = 2 π 4 = ⇒ sin K (Ω − 1) 4 = ⇒ K = 2 π (1 − Ω)
Saddle-node bifurcations of sine circle map create stable-unstable pairs of fixed points x = 1 x = 3 4 4 K = 0 . 5 Ω = K Ω = 1 − K 2 π = 0 . 08 2 π = 0 . 92
First frequency-locking tongue in (Ω , K ) plane Fixed points (one-cycles) exist inside tongue, for 0 ≤ Ω < K 2 π and 1 − K 2 π < Ω ≤ 1
Two-cycles: fixed points of f 2 ( x ) ≡ f ( f ( x )) (Ω=1 / 2 ,K =0) ( x ) = x ∀ x , set Ω ± ( K ) = 1 Since f 2 2 ± ǫ ( K ) with K, ǫ ≪ 1 K f 2 ( x ) = f ( x ) + Ω ± − 2 π sin(2 πf ( x )) = x + 1 2 ± ǫ − K 1 K 2 π sin(2 πx ) + 2 ± ǫ − 2 π sin(2 πf ( x )) = x + 1 ± 2 ǫ − K K 2 π sin(2 πx ) − 2 π sin(2 πf ( x )) � � �� x + 1 2 ± ǫ − K − sin(2 πf ( x )) = − sin 2 π 2 π sin(2 πx ) = − sin (2 πx + π ± 2 πǫ − K sin(2 πx )) = − sin (2 πx + π ) cos ( ± 2 πǫ − K sin(2 πx )) � �� � � �� � ≈ 1 = sin(2 πx ) − cos (2 πx + π ) sin ( ± 2 πǫ − K sin(2 πx )) � �� � � �� � = cos(2 πx ) ≈± 2 πǫ − K sin(2 πx ) ≈ sin (2 πx ) + cos (2 πx ) ( ± 2 πǫ − K sin(2 πx )) = sin (2 πx ) ± 2 πǫ cos (2 πx ) − K 2 sin(4 πx )
cos(2 πx ) − K 2 f 2 ( x ) ≈ x + 1 ± 2 ǫ ± ǫK 4 π sin(4 πx ) ���� ≪ ǫ Fixed points of f 2 ( x )mod 1 : x = f 2 ( x ) mod 1 ≈ x ± 2 ǫ − K 2 4 π sin(4 πx ) ǫ ≈ ± K 2 8 π sin(4 πx ) At saddle-node bifurcation point of f 2 : 1 = d dxf 2 ( x ) ≈ 1 − K 2 cos(4 πx ) = ⇒ cos(4 πx ) = 0 ⇒ x ≈ 1 8 , 3 8 , 5 8 , 7 = 8 = ⇒ sin(4 πx ) = ± 1 Therefore: ǫ ( K ) ≈ ± K 2 8 π 2 ± K 2 1 2 ± ǫ ( K ) ≈ 1 Ω ± ( K ) = 8 π Note: Period-doubling bifurcation (= pitchfork for f 2 ) requires K ≥ 2 : f ′ ( x ) = 1 − K cos(2 πx ) ≥ 1 − K
Saddle-node bifurcations of f 2 create stable-unstable pairs of 2-cycles K = 0 . 8 2 − K 2 2 + K 2 Ω ≈ 1 Ω ≈ 1 8 π ≈ 0 . 475 8 π ≈ 0 . 525 x ≈ 3 8 , 7 x ≈ 1 8 , 5 8 8
Second frequency-locking tongue in (Ω , K ) plane Two-cycles exist inside tongue 2 − K 2 2 + K 2 1 8 π � Ω � 1 8 π
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 ( K 3 ) 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 I p,q ( K ) surrounding p/q of width O ( K q ) called frequency-locking or Arnold tongues �� � ⇒ � K = 0 = p,q I p,q = rationals measure p,q I p,q = 0 �� � K = 1 = ⇒ measure p,q I p,q = 1
Schematic representation of frequency-locking tongues
Winding number of circle map f f n ( x 0 ) − x 0 f n not truncated to [0 , 1] W ( f ) ≡ lim n n →∞ Poincar´ e: f monotonic & continuous = ⇒ limit exists & independent of x 0 . For sine circle map: K = 0 K = 1 x 0 + n Ω − x 0 W = lim n →∞ = Ω Devil’s staircase n 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 w 0 ≡ 0 1 w n +1 ≡ 1 + w n 1 1 + 0 = 1 w 1 = 1 = 1 1 1 1 + 1 = 1 1 w 2 = = = 1 + w 1 2 1 1 + 1 + 0 1 1 1 = 1 = 2 w 3 = = = 1 + 1 3 1 1 + w 2 3 2 2 1 + 1 1 + 1 + 0 w ∗ ≡ n →∞ w n lim Golden mean:
1 Golden mean: w ∗ ≡ n →∞ w n lim with w n +1 = 1 + w n 1 w ∗ = 1 + w ∗ w ∗ (1 + w ∗ ) = 1 w 2 ∗ + w ∗ − 1 = 0 √ w ∗ = − 1 + √ 1 + 4 5 − 1 = = 0 . 618 . . . 2 2 Parthenon, plants, shells, Greeks, Renaissance, . . . (1 − w ∗ ) : w ∗ = w ∗ : 1
Fibonacci sequence: F 0 = F 1 = 1 , F n +1 = F n + F n − 1 = ⇒ 1 , 1 , 2 , 3 , 5 , 8 , 13 , . . . leads to equivalent definition of w n : F n F n 1 1 w n +1 ≡ = = = F n + F n − 1 F n +1 F n + F n − 1 1 + w n F n Closest rational approximation obtained by truncating continued fraction: 1 a = a 0 + 1 a 1 + a 2 + . . . w ∗ is irrational least well approximated by rational: a 1 = a 2 = . . . = 1 Following path in (Ω , K ) space with W Ω ,K = w ∗ will keep furthest away from frequency-locking tongues
Taylor-Couette flow Laminar Couette Taylor Vortex Wavy Vortex Modulated Wavy Vortex U C ( r ) U T V ( r, z ) U W V ( r, θ, z, t ) U MW 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 x ( t ) evolve according to full nonlinear system: ˙ Let ¯ x = f (¯ ¯ x ( t )) � � Let ǫ ( t ) evolve according to linearized system: ˙ ǫ = Df ¯ ǫ x ( t ) Largest Lyapunov exponent: � � 1 ǫ ( t ) λ (1) ≡ lim � � t ln � � � ǫ (0) � t →∞ 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: ǫ 1 = f ′ (¯ x 0 ) ǫ 0 n − 1 � f ′ (¯ ǫ n = x k ) ǫ 0 k =0 � � n − 1 n − 1 1 � � 1 � � f ′ (¯ ln | f ′ (¯ � � λ = lim n ln x k ) � = lim x k ) | � � n n →∞ n →∞ � k =0 k =0 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: Hopf 1 ( Ω 1 ), Hopf 2 ( Ω 2 ), Hopf 3 ( Ω 3 ), . . . = ⇒ 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 = R n /Z n . If n ≥ 3 , every C 2 neighborhood of v contains a vector field v ′ with a strange Axiom A attractor. If n ≥ 4 , we may take C ∞ instead of C 2 .”
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