Nonlinear and Collective Effects in Mesoscopic Mechanical Oscillators

Dynamics Days Asia-Pacific: Singapore, 2004 1 Nonlinear and Collective Effects in Mesoscopic Mechanical Oscillators Alexander Zumdieck (Max Planck, Dresden), Ron Lifshitz (Tel Aviv), Jeff Rogers (HRL, Malibu) Support: NSF, Nato and EU, BSF, HRL

Dynamics Days Asia-Pacific: Singapore, 2004 2 Message Arrays of micro- and nano-scale oscillators provide an interesting dynamical and pattern forming system Synchronization is an important feature of nonlinear oscillators and may be important technologically in MEMS/NEMS Synchronization in MEMS/NEMS motivates a model of synchronization involving nonlinear frequency pulling and reactive coupling

Dynamics Days Asia-Pacific: Singapore, 2004 3 Outline Motivation: Huyghen s clocks and MEMS Phase and amplitude models Our model Analysis and results onset of synchronization full locking simulations

Dynamics Days Asia-Pacific: Singapore, 2004 4 Huyghen s Clocks (1665) From: Matthew Bennett, Michael Schatz, Heidi Rockwood, and Kurt Wiesenfeld (Proc. Rot. Soc. Lond. 2002)

Dynamics Days Asia-Pacific: Singapore, 2004 5 Experimental Reconstruction

Dynamics Days Asia-Pacific: Singapore, 2004 6 Huyghen s clocks: Model I: Synchronization occurs through increased dissipation acting on the in-phase motion Alternative mechanism: Model II: Synchronization occurs by nonlinear frequency tuning cf. Synchronization by Arkady Pikovsky, Michael Rosenblum, Jürgen Kurths

Dynamics Days Asia-Pacific: Singapore, 2004 7 Array of µm-scale oscillators [From Buks and Roukes (2002)]

Dynamics Days Asia-Pacific: Singapore, 2004 8 MicroElectroMechanicalSystems and NEMS Arrays of tiny mechanical oscillators: driven, dissipative nonequilibrium nonlinear collective noisy (potentially) quantum Technological interest!

Dynamics Days Asia-Pacific: Singapore, 2004 9 [From Buks and Roukes (2001)]

Dynamics Days Asia-Pacific: Singapore, 2004 10 Oscillators are nonlinear... One oscillator

Dynamics Days Asia-Pacific: Singapore, 2004 11 Duffing Oscillator V small driving large driving Amplitude x/g D x Frequency ω D ẍ + x + x 3 = g D cos ω D t

Dynamics Days Asia-Pacific: Singapore, 2004 12 Sweeps of parametric drive frequency

Dynamics Days Asia-Pacific: Singapore, 2004 13 Interesting behavior Suggestion of collective behavior Response beyond maximum linear frequency R. Lifshitz and MCC [Phys. Rev. B 67, 134302 (2003)] Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays

Dynamics Days Asia-Pacific: Singapore, 2004 14 Model for synchronization 0 =ẍ n + (1 + ω n )x n D[x n 1 2 (x n+1 + x n 1 )]

Dynamics Days Asia-Pacific: Singapore, 2004 14 Model for synchronization 0 =ẍ n + (1 + ω n )x n D[x n 1 2 (x n+1 + x n 1 )] ν(1 x 2 n )ẋ n ax 3 n

Dynamics Days Asia-Pacific: Singapore, 2004 14 Model for synchronization 0 =ẍ n + (1 + ω n )x n D[x n 1 2 (x n+1 + x n 1 )] ν(1 x 2 n )ẋ n ax 3 n Assume dispersion, coupling, damping and nonlinear terms are small.

Dynamics Days Asia-Pacific: Singapore, 2004 14 Model for synchronization 0 =ẍ n + (1 + ω n )x n D[x n 2 1 (x n+1 + x n 1 )] ν(1 xn 2 )ẋ n axn 3 Assume dispersion, coupling, damping and nonlinear terms are small. Introduce small parameter ε and write ω n = ε ω n,d= ε D, a = εā, ν = ε ν

Dynamics Days Asia-Pacific: Singapore, 2004 14 Model for synchronization 0 =ẍ n + (1 + ω n )x n D[x n 2 1 (x n+1 + x n 1 )] ν(1 xn 2 )ẋ n axn 3 Assume dispersion, coupling, damping and nonlinear terms are small. Introduce small parameter ε and write ω n = ε ω n,d= ε D, a = εā, ν = ε ν Then x n (t) = [ ] z n (T )e it + c.c. with the slow time scale T = εt. + ɛx n (1) (t) +...

Dynamics Days Asia-Pacific: Singapore, 2004 15 Magnitude-phase description ż n = i(ω n α z n 2 )z n + (1 z n 2 )z n + iβ[z n 1 2 (z n+1 + z n 1 )]

Dynamics Days Asia-Pacific: Singapore, 2004 15 Magnitude-phase description ż n = i(ω n α z n 2 )z n + (1 z n 2 )z n + iβ[z n 1 2 (z n+1 + z n 1 )] Matthews, Mirollo, and Strogatz (1991) studied a similar model but with dissipative coupling iβ K, and only saturating nonlinearity α = 0 ż n = iω n z n + (1 z n 2 )z n + K[z n 1 2 (z n+1 + z n 1 )]

Dynamics Days Asia-Pacific: Singapore, 2004 15 Magnitude-phase description ż n = i(ω n α z n 2 )z n + (1 z n 2 )z n + iβ[z n 1 2 (z n+1 + z n 1 )] Matthews, Mirollo, and Strogatz (1991) studied a similar model but with dissipative coupling iβ K, and only saturating nonlinearity α = 0 ż n = iω n z n + (1 z n 2 )z n + K[z n 1 2 (z n+1 + z n 1 )] Phase model (Winfree, Kuramoto): z n 1 e iθ n θ n = ω n K 2 sin(θ n θ n+m ) m=±1 Mean field model θ n = ω n K N sin(θ n θ n+m ) m with ω n taken from distribution g(ω).

Dynamics Days Asia-Pacific: Singapore, 2004 16 Results for the phase model (Kuramoto, 1975) define order parameter = N 1 n r n e iθ n = Re i, r n = 1 synchronization occurs if R = 0 phase evolution equation becomes θ n = ω n F(θ n ) with F(θ) = KR sin θ graphical solution

Dynamics Days Asia-Pacific: Singapore, 2004 17 Locked Free ω = F(θ) R = < cosθ > KRsinθ θ Free

Dynamics Days Asia-Pacific: Singapore, 2004 18 ω = F(θ) R = < cosθ > KRsinθ Locked θ

Dynamics Days Asia-Pacific: Singapore, 2004 19 1.0 Unlocked Partially locked Fully locked R K c K

Dynamics Days Asia-Pacific: Singapore, 2004 20 Our model (mean field version) ż n = i(ω n α z n 2 )z n + (1 z n 2 )z n + iβ N N (z m z n ) m=1 Write as equations for magnitude and phase z = re iθ θ = ω + α(1 r 2 ) + βr r cos θ ṙ = (1 r 2 )r + βr sin θ with θ = θ, ω = ω α β

Dynamics Days Asia-Pacific: Singapore, 2004 21 Easy limit The expectation of synchronization follows from the behavior for narrow frequency distributions and large α. Magnitude relaxes rapidly to the value given by setting ṙ = 0 If r 1 (OK for large α) (1 r 2 )r = βr sin θ 1 r 2 βr sin θ Phase equation becomes (ignoring βr compared with αβr) Kuramoto equation with K = αβ θ ω αβr sin θ

Dynamics Days Asia-Pacific: Singapore, 2004 22 General results θ = ω + α(1 r 2 ) + βr r cos θ ṙ = (1 r 2 )r + βr sin θ For locked oscillators θ =ṙ = 0

Dynamics Days Asia-Pacific: Singapore, 2004 22 General results θ = ω + α(1 r 2 ) + βr r cos θ ṙ = (1 r 2 )r + βr sin θ For locked oscillators θ =ṙ = 0 solve cubic equation for r( θ)

Dynamics Days Asia-Pacific: Singapore, 2004 22 General results θ = ω + α(1 r 2 ) + βr r cos θ ṙ = (1 r 2 )r + βr sin θ For locked oscillators θ =ṙ = 0 solve cubic equation for r( θ) solve phase equation ω = βr r( θ) (α sin θ cos θ) = F( θ)

Dynamics Days Asia-Pacific: Singapore, 2004 22 General results θ = ω + α(1 r 2 ) + βr r cos θ ṙ = (1 r 2 )r + βr sin θ For locked oscillators θ =ṙ = 0 solve cubic equation for r( θ) solve phase equation ω = βr r( θ) (α sin θ cos θ) = F( θ) test stability of solution (r( ω), θ( ω))

Dynamics Days Asia-Pacific: Singapore, 2004 23 Results of analysis (α, β > 0) There is always a stable solution at θ = 0, r = 1 For βr < 2/ 27 a real, positive solution for r exists for all θ (but the solution is not necessarily stable). For βr > 2/ 27 a real positive solution for r only exists for θ B < θ <π+ θ B with θ B = sin 1 (2/βR 27). Solutions with r<1/ 2orF ( θ) < 0 are unstable Moving away from θ = 0 instability may occur through a stationary bifurcation (λ = 0) where F ( θ) = 0 or by a Hopf bifurcation (Re λ = 0, Im λ = 0) when r = 1/ 2.

Dynamics Days Asia-Pacific: Singapore, 2004 24 α=1.0, Β=1.2 ω = F λ θ θ

Dynamics Days Asia-Pacific: Singapore, 2004 25 Partial locking Locked Free ω = F θ Free

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Dynamics Days Asia-Pacific: Singapore, 2004 27 Full locking ω = F Locked θ

Dynamics Days Asia-Pacific: Singapore, 2004 28 Full locking Self-consistency transverse R = r n e i θ n g( ω)r( ω) sin[ θ( ω)] d ω = 0 determines order parameter frequency which is not in general the mean of the oscillator frequencies longitudinal g( ω)r( ω) cos[ θ( ω)] d ω = R

Dynamics Days Asia-Pacific: Singapore, 2004 29 Results for a triangular distribution g(ω) 2/w -w/2 w/2 ω Show results for w = 2

Dynamics Days Asia-Pacific: Singapore, 2004 30 1.0 Unsynchronized state unstable α 0.5 Unsynchronized state stable 0 0 1 2 3 4 β

Dynamics Days Asia-Pacific: Singapore, 2004 31 1.0 Synchronized Fully locked stable α 0.5 Fully locked unstable Unsynchronized state stable 0 0 1 2 3 4 β

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Dynamics Days Asia-Pacific: Singapore, 2004 33 1.0 Synchronized Fully locked α 0.5 Partially locked Unsynchronized Unsynchronized + Synchronized 0 0 1 2 3 4 β

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Dynamics Days Asia-Pacific: Singapore, 2004 35 1.0 Synchronized Fully locked α 0.5 2 Synchronized States Partially locked Unsynchronized Unsynchronized + Synchronized 0 0 1 2 3 4 β

Dynamics Days Asia-Pacific: Singapore, 2004 36

Dynamics Days Asia-Pacific: Singapore, 2004 37 1.0 Synchronized Fully locked α 0.5 2 Synchronized States Partially locked Unsynchronized Unsynchronized + Synchronized 0 0 1 2 3 4 β

Dynamics Days Asia-Pacific: Singapore, 2004 38 1.0 Synchronized Fully locked α 0.5 2 Synchronized States Partially locked Unsynchronized Unsynchronized + Synchronized 0 0 1 2 3 4 β

Dynamics Days Asia-Pacific: Singapore, 2004 39 Conclusions Micromechanical devices suggests a model for synchronization due to reactive coupling and nonlinear frequency pulling Linear stability results for unsynchronized state (triangular, tophat, and Lorentzian distributions) and fully locked state (triangular, tophat) Together with simulations yields a rich phase diagram of synchronized behavior Novel state that is synchronized R>0 but not frequency locked (no oscillator with θ = )

Dynamics Days Asia-Pacific: Singapore, 2004 40 Derivation of equation for amplitude A n (T ) ( ) ẋ n = [ia n + ɛa n ]eit + c.c. + ɛẋ n (1) ( ẍ n = [ A n + 2iɛA n + ɛ2 A n ]eit + c.c. At O(ε) ẍ n (1) + x n (1) +ā = ω n A n e it + ν ( ) 2iA n eit + c.c. ( ia n e it + c.c.) [ 1 (t) +... ) + ɛẍ n (1) (t) +... ( ) ] 2 A n e it + c.c. ( A n e it + c.c.) 3 D[ ( A n 1 2 (A n + A n 1 ) ) e it + c.c.] To eliminate secular terms in e it 2A n = ( ν +i ω n)a n ( ν +3iā) A n 2 A n i D[A n 1 2 (A n+1 +A n 1 )] = 0