Recent progress on the classical and quantum Kuramoto synchronization

Transcription

1 Recent progress on the classical and quantum Kuramoto synchronization Seung Yeal Ha Department of Mathematical Sciences Seoul National University March 23rd, 2017

2 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

3 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

4 Synchronization "Synchronization (=syn (same, common) + chronous (time))" is an adjustment of rhythms of oscillating objects due to their weak interaction.

5 A phase coupled model for synchronization Consider a weakly coupled limit-cycle oscillators {x i = e p 1 i } rotating along S 1 with natural frequency i which is randomly drawn from some probability distribution g( ). S 1 d i dt = i + K N sin( j i ), i = 1,, N. j=1 This is the most well-known synchronization model like "the Burgers equation" for PDE

6 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

7 From SL oscillators to KM oscillators Coupled Stuart-Landau oscillators: z i = z i (t): state of the i-th LS oscillator at time t. ż i =(1 z i 2 + i i )z i + K N (z j z i ). j=1 For K = 0 and z i = r i e i i, ṙ i = r i (1 r 2 i ), i = i. Unit circle r i = 1 is the limit cycle for a decoupled system.

8 We set The Kuramoto model (1975) z j = e i j. d i dt = i + K N sin( j i ), i = 1,, N. i=1

9 From Lohe oscillators to KM oscillators The Lohe model (2009): i U i U + i = H i + ik 2N j=1 U j U + i U i U + j cf. Lohe, M. A.: Non-abelian Kuramoto model and synchronization. J. Phys. A: Math. Theor. 42, (2009)., U i (t) :d d unitary matrix, H i : d d Hermitian matrix.

10 Why Lohe model? (one-dimensional case d = 1), U i = e i i, H i = i. The Lohe model becomes the Kuramoto model: i = i + K N sin( j i ). j=1 (zero coupling K = 0), i U i U + i = H i, or Ui = ih i U i. This yields U i (t) =e ih i t U i (0), t > 0.

11 From Schrödinger-Lohe to Kuramoto The Schrödinger-Lohe model (2010): Choi-H 14, Cho-Choi-H 16, H-Huh 16, Antonelli-Marcati 17, We set t i = i + V i (x) i + ik 2N ( k h i, k i i ). k=1 V i (x) = i : constant, i (t, x) =e i i, to obtain the Kuramoto model: i = i + K N sin( k i ). k=1

12 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

13 Goal of this talk I will briefly discuss some emergent properties of aforementioned models, e.g., existence, size and structure of relative equilibria The Kuramoto model d i dt The Lohe model = i + K N sin( j i ), i = 1,, N. i=1 i U i U + i = H i ik 2N The Schrödinger-Lohe model j=1 U i U + j U j U + i. i@ t i = i + V i (x) i + ik 2N ( k h i, k i i ). k=1

14 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

15 Preliminary analysis Equilibrium equation: i + K N sin( j i )=0, i = 1,, N. i=1 Simple observations: 1. If there exists an eqilibrium =( 1,, N ), X i + K sin( j i N )=X i i,j=1 i i = 0. Thus, we need to consider "relative equilibrium", i.e., equilibrium in the rotating frame with velocity c = 1 N P i. 2. Note that 3. For K = 0, K N sin( j i ) apple K. i=1 i (t) = i (0)+t i, i = 1,, N completely integrable

16 Hence, to continue the story of equilibria, we need to consider X i = 0, K C i, i = 1,, N. i Definition. Let =( 1,, N ) be a (strong) phase-locked state(pls) if and only if i (t) j (t) = 1 ij, t 0. Clearly, equilibrium solutions are phase-locked states. i + K N sin( j i )=0, i=1 X i = 0.

17 Emergent dynamics of KM osillators The Kuramoto model: Matlab simulations by Jinyeong Park N = 50, i 2 [ 1, 1], K = 0.8

18 N = 50, i 2 [ 1, 1], K = 2.2 Numerical simulations seem to suggest that "phase-locked states" emerge from "all (generic) initial phase configurations" in a large coupling regime.

19 Complete synchronization problem(csp) Problem Statement: For a given generic initial phase configuration 0 and sufficiently large K, show that lim i (t) j (t) = 0, 8 i 6= j. t!1

20 Order parameters: Order parameter approach Re i := 1 N e i j. j=1 Note that R is always bounded, i.e., 0 apple R apple 1 and KM () i = i KR sin( i ). Dynamics of order parameters: H-Slemrod 11 Ṙ = = 1 N 1 RN j=1 j=1 sin( j ) j KR sin( j ), cos( j ) j KR sin( j ).

21 Theorem H-Kim-Ryoo 16 Final Resolution on CSP Suppose that the following conditions hold. R 0 := 1 N j=1 e i j (0) > 0, i0 6= j0, 1 apple i, j apple N, max 1appleiappleN i applel < 1. Then, there exists a large coupling strength K 1 > 0 such that, for any solution =( 1,, N ) to the Kuramoto model with initial data 0 and K K 1, there exists a unique phase-locked state 1 such that lim (t) t!1 1 1 = 0, where the norm 1 is the standard `1-norm in N.

22 Brief outline of the proof The Kuramoto model: d i dt = i + K N sin( j i ), t > 0. j=1 subject to following conditions: i = 0, i=1 i0 = 0, i0 2 [, ), 1 apple i apple N. i=1 Scaled Kuramoto model: d i d = i + 1 N j=1 sin( j i ), = Kt, i := i K. As far as large-time dynamics is concerned, i : fixed K 1 () K = 1, i 1.

23 Key ingredients for the proof The first ingredient van Hemmen-Wreszinski 93, Dong-Xue 13 KM is a gradient flow with analytical potential where = r V ( ), t > 0, V [ ] := k k + K 2N k=1 k,l=1 1 cos( k l ).

24 Theorem Dong-Xue 13, H-Li-Xue 13 Let = (t) be a uniformly bounded global solution to the KM in R N : sup (t) 1 < 1. t>0 Then, there exists a phase locked state 1 such that lim (t) t!1 1 1 = 0, lim (t) 1 = 0. t!1 cf. Identical Kuramoto oscillators (Dong-Xue 13)

25 Formation of a chain of black-holes Lemma Suppose that the initial configuration 0 satisfies i0 2 [, ), 1 apple i apple N, and let n 0,`, and K satisfy N i n \ 2, N, ` 2 0, 2 cos 1 N n 0, max i0 j0 <`, K > 1applei,japplen 0 n 0 D( ) n 0 N sin ` 2(N n 0 ) N sin ` 2 Let be a global solution to KM. Then, we have sup D( (t)) apple 4 + `. t 0.

26 Slightly improved result Theorem H-Ryoo 17 Suppose that 0 and K satisfy R 0 := R( 0 ) > 0 K > 96 p 7 D( ) R D( ) R0 2. (1) Then, Kuramoto flow = (t) issued from 0 tends to the phase-locked state asymptotically.

27 Finiteness result What is the cardinality of the set of all phase-locked states? Example (Identical oscillators) For µ 2 [0, 2 ] and N 4, we define a state µ : 8 2k >< µ N 2 + µ, k = 1,...,N 2, k := 0, k = N 1, >:, k = N. Note that distinct µ s clearly yield non-equivalent phase locked states, and R = 1 2 e i 2k N 2 + e i0 + e i = 0. N k=1 We have a continuum of phase-locked states for the KM for identical oscillators. Thus, for finiteness, we need to restrict R > 0.

28 Theorem 1. For identical Kuramoto oscillators with K > 0, every standard phase-locked state is a permutation of (0,...,0,,..., ), where the number of 0 s is greater than N/2, and the s do not necessarily exist. Thus, we have ( 2 N 1, N odd, P = 2 N 1 1 N 2 N/2, N even, 2. There are at most 2 N phase locked states for the non-identical case, i.e., P apple 2 N.

29 Mathematical Challenge I The Kuramoto model with frustration d i dt = i + K N sin( j i + ij ), i = 1,, N. i=1 cf. No gradient flow structure!!!, Fact: For the case ij = ji, ij < 2, there exists a set of admissible initial data leading to the complete synchronization (H-Kim-Li 14), but numerical simulations suggests

30 Simulatiions by Dongnam Ko. No proof in full generality so far!

31 Theorem (H-Ko-Zhang 17) Identical oscillators Let be a solution with initial data 0 satisfying conditions: 1 < arctan 2 p, K > 0, D( ) = 0, N R 0 cos + sin > cos N, 0 j 6= k 0 for all j 6= k. Then the following dichotomy holds. 1. Either there exists T 0 0 such that for all t T 0 D( (t)) apple D( (T 0 ))e (,T 0,N )t. 2. or there exists T 1 0 and some j 0 2{1,, N} such that for all t T 1 ( j0 (t) )+( j (t) ) apple DĨ( (T 1 ))e (,T 1,Ĩ)t, for all j 6= j 0, j (t) k (t) apple DĨ( (T 1 ))e (,T 1,Ĩ)t, for all j, k 6= j 0.

32 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

33 The Lohe model The Lohe model: Max. Lohe 09 For d = 2, i U i U + i = H i ik 2N j=1 U i U + j U j U + i. 3X U i := e i i xi k k + xi 4 I 2, H i = k=1 3X! i k k + i I 2, where I 2 and i are the identity matrix and Pauli matrices, respectively, defined by i 0 1 I 2 :=, 1 :=, 2 :=, 3 := i k=1.

34 After some algebraic manipulations, we obtain 5N equations for the angles i and the 4-vectors x i : x i 2 i = i + K N x i 2 ẋ i = i x i + K N sin( k i )hx i, x k i, 1 apple i apple N, k=1 cos( k i )( x i 2 x k hx i, x k ix i ), k=1 where i is a real 4 4 antisymmetric matrix given by 0 1 0! i 3! i 2! i 1 i := B! i 3 0! i 1! i 2 i 2! i 1 0! i 3 A, 1 apple i apple N.! i 1! i 2! i 3 0 cf. Choi-H: sufficient conditions for CPS

35 Beauty is a Truth i U i U + i = H i ik 2N Invariance under Lohe flow j=1 U i U + j U j U + i, 1. Invariance of U i U i : d dt (U iu i )=0, t > Invariance under the right-multiplication by a unitary matrix: if L 2 U(d) and V i = U i L, then V i satisfies i V i V i = H i ik 2N j=1 V i V j V j V i, V i (0) =U i0 L.

36 Definition: (Phase-locked states) 1. {U i (t)} is a phase-locked state if and only if U i (t)u j (t) is constant for all i, j and t The Lohe flow {U i (t)} achieves asymptotic phase-locking if and only if the limit of U i U j as t!1exists.

37 In classical Kuramoto oscillators, phase-locked states takes the following form: i (t) = 1 i + c t, c := 1 N i. j=1 Proposition: The phase-locked states(pls) {U i } for the Lohe flow take the form of U i = U 1 i e i t, where Ui 1 2U(d) and is the constant d d Hermitian matrix satisfying the relation U 1 i U 1 i = H i ik 2N k=1 Ui 1 U 1 k Uk 1 U1 i.

38 Main strategy 1. Step A: Introduce an ensemble diameters: D(U) := max ku i U j k, D(H) := max kh i H j k. 1applei,jappleN 1applei,jappleN 2. Step B: Derive a Gronwall type differential inequality for D(U): d dt D(U)2 + 2KD(U) 2 apple 2D(H)D(U)+KD(U) 4 a.e. t 2 (0, 1). 3. Step C: Establish the existence of PLSs. For example, for identical oscillators with D(H) =0, lim D(U(t)) = 0. t!1

39 Theorem: (H-Ryoo 16, J. Stat. Phys.) Suppose that K and U 0 satisfy K > D( ), U0 2 B(0, ) for some >0 Then, we have 1. {U i } achieves asymptotic phase-locking: lim t!1 U iu j converges exponentially fast, with exponential rate bounded above by K (1 3 1 ). 2. There exists a phase-locked state {V i } and L 2U(d) such that converges exponentially fast. lim ku i V i Lk = 0 t!1 3. All phase-locked states {V i } with diameter D(V ) < 2 are right multiplication of this phase-locked state by a unitary matrix. cf. For n = 2, there were some preliminary works by Choi and H.

40 Mathematical Challenges II The Lohe model i U i U + i = H i ik 2N j=1 U i U + j U j U + i. 1. Is the Lohe model gradient flow for suitable potential? 2. Can the Lohe model derivable from more higher-dimensional dynamical systems? 3. Can we introduce a Lohe like system for a set of tensor networks? 4. What is the structure of the emergent phase-locked states?

41 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

42 The Schrödinger-Lohe oscillators The Schrödinger-Lohe model t i = i + V i (x) i + ik 2N a ik ( k h i, k i i ). k=1 Main question Find sufficient conditions leading to the synchronization of wave functions, i.e., 9 d ij := lim t!1 i (, t) j (, t) L 2, 1 apple i, j apple N.

43 Lyapunov functional approach a ik = 1, D( ) = max i,j Theorem: Choi-H 14, Cho-Choi-H 16 i j L 2, D(V) :=max V i V j L 1. i,j 1. Suppose that the coupling strength and initial data satisfy K > 0, V j = V, 0 j 2 = 1, 1 apple j apple N, D( 0 ) < 1 2. Then, the diameter D( ) satisfies D( (t)) apple D( 0 ) D( 0 )+(1 2D( 0, t 0. ))ekt 2. Suppose that the coupling strength and initial data satisfy K > 54D(V), i0 = 1, j = 1,, N, D( 0) 1. Then, we can achieve practical synchronization: lim K!1 lim sup D( (t)) = 0. t!1

44 Dynamical systems approach Finite-dimensional reduction from the SL model Define correlation functions Z h ij (t) :=h i (t), j (t)i = i(x) j (x)dx, 1 apple i, j apple N. Then, for congruent one-body potentials V i = V + i, 1 apple i apple N, i 2 R h ij satisfies ḣ ij = i ij h ij + apple 2N k=1 h i a ik (h kj h ik h ij )+a jk (h ik h kj h ij ). cf. a ik = 1: H-Huh 16, Antonelli-Marcati 17

45 We set Then h satisfies dh dt A two-oscillator system = K h := h 12 and := 12. apple h + i 2! 2 + 1, t > 0. 2K 4K 2 We need to consider the following three cases: K > 2, K = 2, K < 2. Case 1 (K > ): Two equilibria 2 h 1, := 1 r 2 4 i 2 K 2K and h1,+ := 1 r 4 2 h(t) = h1,+(h0 h 1, )+h 1, (h 0 h 1,+)e p h 0 h 1, (h 0 h 1,+)e 4K 2 Thus, it is easy to see that for any initial data h 0 6= h 1, h(t)! h 1,+ as t!1. K p 4K 2 2 t 2 2 t, we have. i 2K.

46 Case 2 (K = ): the unique equilibrium: 2 and explicit solution: Thus, we have Case 3 (K < ): the equilibria: 2 h 1, = i 2K 1 2 Explicit periodic solution: h 0 cos h(t) = cos h 1 := i, h(t) = h0 i(h 0 + i)kt. 1 +(h 0 + i)kt h(t)! h 1 as t!±1. r 2 4, h 1,+ = i K t p 2 4K 2 2 t p 2 4K 2 2 p K i h 0 2 sin 2 4K 2 K K + p 2h 0 + i sin 2 4K 2 K 2K + 1 r K t p 2 4K 2 2 t p 2 4K 2 2. Bifurcation at K = 2.

47 Mathematical Challenges III Establish the bifurcation (phase-transition phenomenon) for many-body system

48 Define a functional: Identical potentials V i = V H(t) := max 1appleiappleN k=1 1 h ik (t). Lemma: Assume that a ik > 0. Then, the functional H satisfies where d dt H(t) apple 2K (A m D(A)) H(t)+ 2Ka 1 N H(t)2, 1 A m : min i N a ik, k=1 D(A) :=max i,j max k a ik a jk, a 1 := max a ik > 0. i,k

49 Theorem: H-Huh-Kim 17 Suppose that initial data h 0 ij satisfies max 1appleiappleN 1 N j=1 1 h 0 ij applea m D(A)) a 1. Then, we have 1 h ij (t) applee Ct.

50 Outline PROLOGUE What is synchronization? Three main characters Goal of this talk INTERLOGUE Kuramoto oscillators Lohe oscillators Schrödinger-Lohe oscillators EPILOGUE Take-Home message

51 Take-Home message So far, I have discussed some emergent properties of The Kuramoto model d i dt The Lohe model = i + K N sin( j i ), i = 1,, N. i=1 i U i U + i = H i ik 2N The Schrodiger-Lohe model j=1 U i U + j U j U + i. i@ t i = i + V i (x) i + ik 2N ( k h i, k i i ), (x, t) 2 R d R +. k=1

52 Good News "Never ending story" There are still lots of "Beauty and Mystery" to be explored in the synchronization models Ha, S.-Y. et al "Collective synchronization of classical and quantum oscillators ". EMS Surveys in Mathematical Sciences.3, 2016, pp

53 Thank you for your attention!!!