Doctoral course Torino 29.10.2010

Introduction to synchronization: History

1665: Huygens observation of pendula

When pendula are on a common support,they move in synchrony, if not, they slowlydrift apart

α1

ω1

α2

ω2

α1

ω1

α2

ω2

Doctoral course Torino 29.10.2010

Model: two coupled Vanderpol oscillators

11 1

2 2 211 1 1 1 1 1

22 2

2 2 222 2 2

2 1

2 2 2

1 2

dxy

dtdy

x x y ydt

dxy

dtdy

x x y ydt

d x x

d x x

Doctoral course Torino 29.10.2010

Two identical coupled Vanderpol oscillators

Uncoupled: d = 0 Coupled: d = 0.1

x1(t), x2(t)

x2(t) – x1(t)

x1(t), x2(t)

x2(t) – x1(t)

phase difference remains phase difference vanishes

Doctoral course Torino 29.10.2010

Two identical coupled Vanderpol oscillators

1tan ii

i

y

x

2 1 0t

t t

The coupled oscillators synchronize: two different interpretations

for the phases

2 1

2 10

t

x xt t

y y

for the states

phase synchronization state synchronization

generalization: to non identical systems

limitation: to systems where a phase can be defined rhythmic behavior

generalization: to systems with any behavior

limitation: to identical or approximately identical systems

Doctoral course Torino 29.10.2010

Phase synchronization

2 1 for all 0t t C t

1tany

x

Definition:Two systems are phase synchronized, if the difference of their phasesremains bounded:

• Notion depends only on one scalar quantity per system, the phase

• Phase can be defined in different ways:1) If the trajectories circle around a point in a plane:

in higher dimensions: take a 2-dimensional projection

Doctoral course Torino 29.10.2010

Phase synchronization

1tanh

s

2) Take a scalar output signal s(t) from the system, calculate its Hilbert transform h(t) to form the analytical signal z(t) = s(t) + jh(t). Define the phase as in 1) for the complex plane z:

3) For recurrent events suppose that between one event and the next the phase has increased by 2Between events interpolate linearly

0 0.2 0.4 0.6 0.8-0.3

0.05

0.4

0.75

1.1

t [s]

y [mV]

0 1 2 3 4-0.3

0.1

0.5

0.9

1.3

t [s]

y [mV]

2 4 6 8 10

Doctoral course Torino 29.10.2010

Phase synchronization in weakly coupled non-identical oscillators

Weak coupling The trajectory follows approximately the periodic trajectory of each component system. The phases are more or less locked (constant difference)

Example: Two Vanderpol oscillators with different parameters:

1 2 1 20.2, 2, 1, 1.1, 0.1d

Doctoral course Torino 29.10.2010

Phase synchronization in weakly coupled non-identical oscillators

11 1 1 2

22 2 2 1

,

,

ddQ

dtd

dQdt

If asymptotic behavior is periodic, phase synchronization is the same as ina corresponding system of coupled phase oscillators (cf. book by Pikovsky, Rosenblum and Kurths)

1 and 2 are the frequencies of the uncoupled oscillators. Functions Qi

are 2-periodic in both arguments.

phase synchronization common frequency (average derivative of phase)

1 2

Note that phase synchronization may also take place when behavior isnot periodic, e.g. chaotic (but chaos must be rhythmic)

Doctoral course Torino 29.10.2010

State synchronization

State synchronization is not limited to systems with rhythmic behavior

1 1 2 1

2 2 1 2

1

1

x t f x t d f x t f x t

x t f x t d f x t f x t

Example: discrete time system with chaotic behavior:

f:

x1(t)

x2(t)

x1(t) -x2(t)

Doctoral course Torino 29.10.2010

State synchronizationFor chaotic systems, the transition from synchronized to non-synchronized behavior is peculiar: bubbling bifurcation

x1(t)

x2(t)

x1(t) -x2(t)

Doctoral course Torino 29.10.2010

State synchronization in networks of dynamical systems (dynamical networks)

• Arbitrary networks of coupled identical dynamical systems

• Arbitrary dynamics dynamics of individual dynamical system

Multistable:

Oscillatory: Chaos:

Connection graph: n vertices and m edges

Doctoral course Torino 29.10.2010

State synchronization in networks of dynamical systems (dynamical networks)

Synchronization properties depend on

1

, , :n

d dii ij i j

j

dxF x d f x x f

dt

• Individual dynamical systems• Interaction type and strength• Structure of the connection graph

Various notions of synchronization:

• complete vs. partial• global vs. local