doctoral course torino 29.10.2010 introduction to synchronization: history 1665: huygens observation...
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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