Synchronization The Emerging Science of Spontaneous Order

Flashing of Fireflies

SYNC is Ubiquitous Synchronization is common in nature: (1) Synchronously flashing fireflies. (2) Crickets that chirp in unison. (3) Electrically synchronous pacemaker cells. (4) Groups of women whose menstrual cycles become mutually synchronized. (5) The spin of the moon synchronizes with its orbit. (6) Coupled laser arrays. (7) Josephson Junction Arrays (Superconducting Qubit SQ)

Gravity and Tidal Force

In-Phase and Anti-Phase

Pacemaker Cell

Sync Heart Cell of Embryonic Chicks

Pulse Perturbation

Phase resetting

Simulation of Oscillator When x=1, it fires and resets itself to x=0.

Firing Process

Peskin ’ s Conjectures and results (1975) For arbitrary initial conditions, the system approaches a state in which all the oscillators are firing synchronously. This remains true even when the oscillators are not quite identical. Only proved for two Identical oscillators with small Coupling and weak dissipation..

Synchronization of Pulse-Coupled Biological Oscillators Renato E. Mirollo and Steven H. Strogatz 1990 The MS model (1) Oscillators are identical with the same period T. (2) Each oscillator has its phase φ = t – nT, n is an integer. (3) Each oscillator is characterized by a state variable x. (4) Each oscillator has the same state function x = f(φ).

Basic Ingredients: without ODE

Dynamics of Two Oscillators

Return Map Firing Map Return Map Existence of Attractive Fixed Point If

How about N>2 ? By requiring all to all coupling and the process of Absorption (Namely, if two OSCs are synchronous, then they will synchronize forever.). Then N OSCs reduce to N-1. The system for N OSCs becomes synchronized for arbitrary initial Conditions, except for a set of measure zero.

The Weakness of MS-Model Identical Oscillators. It is not realistic to have all to all coupling. Absorption is too strong. (Cheating) Instantaneous response. Absence of Refractory Period. (Time interval in which a second stimulus cannot lead to a subsequent excitation.) …

THRESHOLD EFFECTS ( 1993, PRE 49, 2668) Introducing a threshold The firing Map is Restriction is equivalent to a refractory period. Breaking ALL to ALL condition

Threshold (1)

Threshold (2)

Conclusions The MS- model is a special case, namely, represents the existence of refractory period and also breaks the All to All coupling Scheme. By using Absorption, the system synchronizes for arbitrary initial conditions up to a set of measure zero.

A New Twist on old thinking ( 陳福基 )

Firing map h(φ) = g(ε+f(1-φ)) Return map R(φ) = h(h(φ)) if φ> φ* R(φ) <φ if φ φ Attractive Fixed Point!! So it is commonly believed that this model can not be synchronized!

Some Details Considering A and B has different Firing maps Return Map

Fixed Point Evasion There is no fixed point! A&B will fire synchronously for all initial conditions!

Kuramoto Model (Non-identical) Mean Field solution

SYNC of Kuramoto ’ s model

NETWORKS ( Nature 410, 268, Strogatz) Figure 1 Wiring diagrams for complex networks. a, Food web of Little Rock Lake, Wisconsin, currently the largest food web in the primary literature5. Nodes are functionally distinct ‘ trophic species ’ containing all taxa that share the same set of predators and prey. Height indicates trophic level with mostly phytoplankton at the bottom and fishes at the top. Cannibalism is shown with self-loops, and omnivory (feeding on more than one trophic level) is shown by different coloured links to consumers. (Figure provided by N. D. Martinez). b, New York State electric power grid. Generators and substations are shown as small blue bars. The lines connecting them are transmission lines and transformers. Line thickness and colour indicate the voltage level: red, 765 kV and 500 kV; brown, 345 kV; green, 230 kV; grey, 138 kV and below. Pink dashed lines are transformers. (Figure provided by J. Thorp and H. Wang). c, A portion of the molecular interaction map for the regulatory network that controls the mammalian cell cycle6. Colours indicate different types of interactions: black, binding interactions and stoichiometric conversions; red, covalent modifications and gene expression; green, enzyme actions; blue, stimulations and inhibitions. (Reproduced from Fig. 6a in ref. 6, with permission. Figure provided by K. Kohn.) © 2001 Macmillan Magazines Ltd

Localized Synchronization in Two Coupled Nonidentical Semiconductor Lasers A. Hohl,1 A. Gavrielides,1 T. Erneux,2 and V. Kovanis1 FIG. 1. Schematic of a system of two nonidentical semiconductor lasers mutually coupled at a distance L used to observe localized synchronization. We find that the laser which is pumped at a high level may be forced to entrain to the laser which is pumped at a significantly lower level. VOLUME 78, NUMBER 25 PHY S I CAL REV I EW LETTERS 23 JUNE 1997

Spontaneous synchronization in a network of limit-cycle oscillators with distributed natural frequencies. Theory of phase locking of globally coupled laser arrays PRA 52, 4089 (1995) Kourtchatov et al

Dynamical Evolution Newton ’ s equation Phase Space Trajectory Phase portrait of the pendulum equation.

General dynamical Equations

More Examples on Phase Space Phase portrait of a damped pendulum with a torque. Periodic solutions correspond to closed curves in the phase plane

Deterministic Chaos Nonlinear Equation Dynamical Instability Sensitive Dependence on Initial Condition No Long Term Prediction

Chaotic Signal

The Bunimovich stadium is a chaotic dynamical billiard

Lorenz Attractor (1) dx / dt = a (y - x) a=10,b=28 dy / dt = x (b - z) - y c=8 dz / dt = xy - c z

Lorenz Attractor (2) These figures — made using ρ=28, σ = 10 and β = 8/3 — show three time segments of the 3-D evolution of 2 trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10-5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious. Time t=1Time t=2Time t=3

Chaos Synchronization

Chaotic Laser

SYNC Chaotic Laser Output

Chao and Communication

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