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Workshop on Network Synchronization: from dynamical systems to neuroscienceLorentz Center, Leiden, 19-30 May 2008 Excitability mediated by dissipative solitons Pere Colet Adrian Jacobo, Damià Gomila, Manuel Matías Claudio J. Tessone, Alessandro Sciré, Raúl Toral
Outline • Introduction • Dissipative solitons in a Kerr cavity • Soliton instabilities • Soliton excitability • Effect of a localized pump • Interaction of oscillating & excitable solitons • Collective firing induced by noise or diversity.
Dissipative solitons Dissipative solitons are localized spatial structures that appear in certain dissipative media: Chemical reactions: J.E. Pearson, Science 261, 189 (1993); K.J. Lee & H.L. Swinney, Science 261, 192 (93). Gas discharges: I. Müller, E. Ammelt & H.G. Purwins, Phys. Rev. Lett. 73, 640, (1994). Fluids: O. Thual & S. Fauve, J. Phys. 49, 1829 (1988). Localized excitations in a vertically vibrated granular layer. P.B. Umbanhowar, F. Melo & H.L. Swinney Nature 382, 793 (1996). Soliton in a Vertical Cavity Surface Emitting Laser S. Barland et al., Nature, 419, 699 (2002). N. Akhmediev & A. Ankiewicz (eds), “Dissipative solitons”, Lecture Notes in Physics 661 (Springer, Berlin, 2005); “Dissipative Solitons: From Optics to Biology and Medicine”, (Springer 2008)
Sodium vapor cell with single mirror feedback T. Ackemann and W. Lange, Appl. Phys. B 72, 21 (2001) Liquid crystal light valve P.L. Ramazza et al., J. Nonlin. Opt. Phys. Mat. 8, 235 (1999) P.L. Ramazza, S. Ducci, S. Boccaletti & F.T. Arecchi, J. Opt. B 2, 399 (2000) Pump field Nonlinear medium Pattern formation in nonlinear optical cavities • Driving • Dissipation • Nonlinearity • Spatial coupling Spontaneous pattern formation F.T. Arecchi, S. Boccaletti & P.L. Ramazza, Phys. Rep. 318, 1 (1999). L.A. Lugiato, M. Brambilla & A. Gatti, Adv. Atom. Mol. Opt. Phys. 40, 229 (1999) N.N. Rosanov, “Spatial Hysteresis and Optical Patterns”, Springer 2002.
Ackemann-Lange Dissipative solitons versus propagation solitons “Dissipative solitons” Dissipative. Unique once the parameters of the system are fixed. Potentially useful for optical storage & information processing. Propagation solitons Conservative Continuous family of solutions depending on energy. Useful for optical communication systems N.N. Rosanov in Progress in Optics, 35 (1996). M. Segev (ed.) Special Issue on Solitons, Opt. Photonics News 13(27), 2002 L.A. Lugiato (ed), Feature section on Cavity Solitons, IEEE J. Quantum Electron. 39(2) (2003); N. Akhmediev & A. Ankiewicz (eds), “Dissipative solitons”, Lecture Notes in Physics 661 (Springer, Berlin, 2005).
Localized structures stabilised by interaction of oscillatory tails. Exist in 1d & 2d systems. P. Coullet, et al PRL 58, 431(1987) G.-L.Oppo et al. J. Opt. B 1, 133 (1999) G.-L.Oppo et al. J. Mod Opt. 47, 2005 (2000) P. Coullet, Int. J. Bif. Chaos 12, 2445 (2002) Subcritical Cellular Pattern Hexagonal Pattern Amplitude Homogeneous Solution Control Parameter Bistability Homogeneous Solution Localized structures as single spot of a cellular pattern. Exist in 1d & 2d systems. W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (1996) Homogeneous Solution Amplitude Homogeneous Solution Control Parameter Scenarios for dissipative solitons Stable droplets: Localized structures stabilised by nonlinear domain wall dynamics due curvature. Exist in 2d systems. D. Gomila et al, PRL 87, 194101 (2001) Excitability mediated by localized structures
Excitability. General ideas • Excitability: has origin in Biology (action potential of nerve cells; also heart), • also found in reaction-diffusion systems. • Simplest minimum ingredients in phase space for excitability: • Stable fixed point • Threshold • Reinjection mechanism in phase space (that leads to refractory period). • Different responses to sub/supra-threshold perturbations. • Three simplest excitability routes (2-D phase space), occur close to bifurcations leading to oscillatory behavior: • a) saddle-node in invariant circle (Andronov-Leontovich) (Adler equation) • b) saddle-loop (homoclinic) bifurcation • c) fast-slow systems with S nullcline (slow manifold): canard (Fitzhugh-Nagumo) Excitable media: spatially extended systems in which the local dynamics is excitable. J.D. Murray, Mathematical Biology, Springer 2002, 3rd ed. E. Meron; Pattern formation in excitable media; Phys. Rep. 218, 1 (1992). B. Lindner, J. García-Ojalvo, A. Neiman & L. Schimansky-Geier; Effects of noise in excitable systems; Phys. Rep. 392, 321 (2004).
Excitability in optical systems • Some examples of excitability in optical systems (mostly active systems): • Systems with thermal effects (slow variable) that interplay with a hysteresis cycle of a fast variable. Leads to (c), FHN-like excitability. Cavity with T-dependent absorption (Lu et al, PRA 58, 809 (1998)). Semiconductor optical amplifier (Barland et al, PRE 68, 036209 (2003)). • Lasers with saturable absorber (Dubbeldam et al, PRE 60, 6580 (1999)); lasers with optical feedback (Giudicci et al, PRE 55, 6414 (1997); Yacomotti et al, PRL 83, 292 (1999)); lasers with injected signal (Coullet et al, PRE 58, 5347 (1998); Goulding et al, PRL 98, 153903 (2007)) . These lead to (a): saddle-node in an invariant circle. • Lasers with intracavity saturable absorber (Plaza et al, Europhys. Lett. 38, 85 (1997)). Excitability mediated by a saddle-loop bifurcation. • Semiconductor DFB laser (interaction of 2 modes) (Wuensche et al, PRL 88, 023901 (2002)). Homoclinic bifurcation slightly different than (b). • Possible applications: optical switch (responding to sufficiently high optical input signals); optical communications: pulse reshaping.
Homogeneous solution It becomes unstable at Is=1 leading to a subcritical hexagonal pattern Self-focusing Kerr cavity input fieldE0 y x output field z field envelope Lugiato-Lefever model Control parameters E0:pump q :detuning L.A. Lugiato & R. Lefever, PRL 58, 2209 (1988).
Numerical solutions with arbitrary precision: • Discretize r set of nonlinear ordinary eqs. Spatial derivatives computed in Fourier space • Solve using Newton-Raphson • Continuation methods can be used • Linear stability analysis can be performed W.J. Firth & G.K. Harkness, Asian J. Phys 7, 665 (1998); G.-L. Oppo, A.J. Scroggie & W.J. Firth, PRE 63, 066209 (2001); J.M. McSloy, W.J. Firth, G.K. Harkness & G.-L. Oppo, PRE 66, 046606 (2002) Self-focusing Kerr cavity solitons Can be seen as a solution connecting a cell of the pattern with the homogeneous solution Cavity soliton W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (1996); W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta, T67, 12 (96) Radial equation: Soliton profile can be found solving the l.h.s. equated to zero with
Azimuth inst. m=6 m=5 Hopf Is No solitons Saddle-Node W.J. Firth, G.K. Harkness, A. Lord, J. McSloy, D. Gomila & P. Colet, JOSA B 19, 747 (2002) Stability of Kerr cavity solitons Soliton amplitude Stable Hopf Unstable Saddle-Node Hom. solution Is Hopf instability observed in W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta,T67,12 (96)
Azimuth instabilities m=6 t Unstable Eigenmode m=5
Hopf instability Azimuth inst. Hopf solitons No solitons Saddle-node middle branch soliton Cross-section Oscillating soliton still useful for applications since its amplitude is bounded below by middle branch soliton.
Minimum distance of oscillating soliton to middle-branch soliton Saddle-loop bifurcation Is=0.9 oscillating cavity soliton LC q=1.3 Hopf max(|E|) middle-branch cavity soliton SN Saddle-loop q=1.3047 Homogeneous solution q q=1.30478592 q=1.304788 homogeneous solution D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).
middle-branch soliton spectrum l1 numerical simulations 1/1 Saddle-loop bifurcation. Scaling law Close to bifurcation point: T: period of oscillation l1 unstable eigenvalue of saddle (middle-branch soliton) S.H. Strogatz, Nonlinear dynamics and chaos 2004
dA=(E-Esaddle)/Es Oscillatory regime Projection onto ys Projection onto yu Beyond Saddle Loop Projection onto ys Projection onto yu Phase space close to saddle-loop bifurcation middle-branch soliton spectrum yu ys Only two localized modes. Close to saddle: dynamics takes place in the plane(yu, ys) Saddle-node index:n=-ls/lu=2.177/0.177>1 (stable limit cycle) D. Gomila, A. Jacobo, M. Matias and P. Colet, PRA 75, 026217 (2007).
Excitability Beyond saddle-loop bifurcation Small perturbations of homogeneous solution decay. Localized perturbations above middle branch soliton send the system to a long excursion through phase-space. The system is not locally excitable. Excitability emerges from spatial coupling D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).
Hopf saddle-node solitons oscillating solitons q=1.7 q=1.6 saddle-loop saddle-node q=1.5 Hopf No solitons q =1.5 Distance between saddle-node and Hopf The Hopf frequency when it meets the saddle-node is zero.Takens-Bogdanov point. Unfolding of TB yields a Saddle-Loop TB d → 0 for →∞ andIs→ 0 NLSE Takens-Bogdanov point Saddle-loop bifurcation is not generic. Why it is present here?
Excitability Hom. pump Ish=0.7, q=1.34 max(|E|2) Pattern Hopf Saddle Node Is 1 Excitability Oscillations max(|E|2) SNIC Pattern Is 1 Pump: Plane wave + Localized Gaussian Beam Excitability arising from a saddle-loop bifurcation have a large threshold. To reduce the threshold we consider for the pump:
Projection onto ys Projection onto yu Saddle-node in the circle (SNIC) bifurcation Close to bifurcation point: From the new oscillatory regime to the excitable regime. Ish=0.3, q=1.45 unstable upper branch soliton Is=0.927 Is=0.907 Is=0.8871 middle-branch cavity soliton Is=0.8 fundamental solution
Full scenario • Excitability can appear as a result of: • Saddle loop (oscillating and middle branch solitons collide) • Saddle node on the invariant circle (fundamental solution and middle branch soliton collide). • Controllable excitability threshold. Ish=0.3 I Only fundamental solution II Stationary DS, fundamental solution stable III Oscillating DS, fundamental solution stable IV Excitable DS, fundamental solution stable V Oscillating DS, no fundamental solution
Noise effects, coherence resonance In excitable systems a moderate level of noise induces a more regular firing (coherence resonance) A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 (1997). Introducing white spatiotemporal noise excitable solitons show coherence resonance.
In-phase oscillation. T=8.93 Out-phase oscillation. T=8.94 Strong interaction. In & out-phase oscillation depending on initial condition. Tin=8.59 < Tout=10.45 Interaction of two oscillating solitons q =1.27, Is=0.9, homogeneous pump Single structure period T=8.66 Oscillating solitons move until they reach equilibrium positions given by tails interaction. Three equilibrium distances are found:
Pulse on Pulse on Pulse off Firing Firing 1 1 1 0 Firing induced by interaction Firing induced by interaction 1 1 Firingbit 1 OR logical gate Interaction of excitable solitons
AND 1 1 1 0 Pulse on Pulse on Pulse off 1 0
Kuramoto order parameter Global variables: Approximate equation • Global phase dynamics similar to individual units but with scaled frequency. • A degradation in entrainment r lowers excitablity threshold allowing for synchronous firing. • The precise origin of the degradation of r is irrelevant. C.J. Tessone, A. Scirè, R. Toral and P. Colet, Phys. Rev. E 75, 016203 (2007). Collective firing induced by noise or diversity Globally coupled active rotators wj<1excitable. wj>1rotates. noise Diversity: natural frequencies
Numerical simulations noise diversity s=0 D=0.4 No firing s=1.6 D=1.0 Synchronized firing s=3.0 D=5.0 Desynchronized firing Diversity and noise play a similar role and induce coherent firing.
N=50 xN=100 N=1000 N=10000 Self-consistent approximation Shinomoto-Kuramoto order parameter Self-consistent approx. • No firing • I Collective firing • II Desynchronized firing C.J. Tessone, A. Scirè, R. Toral and P. Colet, Phys. Rev. E 75, 016203 (2007).
Summary • Dissipative solitons in a nonlinear Kerr medium: subcritical cellular patterns • Oscillating solitons: Still useful for applications envisioned for static solitons. New ones? • Excitable regime associated with the existence of cavity solitons. • Extended systems, in order to exhibit excitability, do not require local excitable behavior. Excitability in a whole new class of systems. • For homogeneous pump excitability appears as a result of a saddle-loop bifurcation: oscillating and middle-branch soliton collide. • Scenario organized by a Takens-Bogdanov codimension 2 point (at→∞ &Is→0) • For pump composed of a Gaussian localized beam on top of homogeneous background excitability also mediated by a SNIC:fundamental solution and middle branch soliton collide. • Lower (controllable) excitability threshold. • A suitable amount of white noise induces coherence resonance. • Coupled oscillatory solitons lock to distances given by tail interaction. • Depending on the locking distance solitons oscillate in or out-of-phase. • For strong coupling in-phase and out-of phase oscillations coexists. • Interaction of excitable solitons may be used for logical gates. • In coupled excitable systemsdisorder can induce collective firing. • Any source of disorder plays a similar role.