1. Introduction to synchronization: History
1665: Huygens observation of pendula α1 α2 α1 α2 ω1 ω2 ω1 ω2
When pendula are on a common support,
they move in synchrony, if not, they slowly
drift apart
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2. Model: two coupled Vanderpol oscillators
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3. Two identical coupled Vanderpol oscillators
w1 = w2,, l1 = l2
Uncoupled: d = 0
Coupled: d = 0.1
x1(t), x2(t)
x1(t), x2(t)
x2(t) – x1(t)
x2(t) – x1(t)
phase difference remains
phase difference vanishes
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4. Two identical coupled Vanderpol oscillators
The coupled oscillators synchronize: two different interpretations
for the phases
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
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5. Phase synchronization
Definition:
Two systems are phase synchronized, if the difference of their phases
remains 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: f
in higher dimensions: take a 2-dimensional projection
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6. Phase synchronization
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 2p. Between events interpolate linearly. 2p 4p 6p 8p
10p
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7. 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:
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8. Phase synchronization in weakly coupled non-identical oscillators
If asymptotic behavior is periodic, phase synchronization is the same as in
a corresponding system of coupled phase oscillators
(cf. book by Pikovsky, Rosenblum and Kurths)
w1 and w2 are the frequencies of the uncoupled oscillators. Functions Qi
are 2p-periodic in both arguments.
phase synchronization  common frequency
(average derivative of phase)
Note that phase synchronization may also take place when behavior is
not periodic, e.g. chaotic (but chaos must be rhythmic)
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9. State synchronization
State synchronization is not limited to systems with rhythmic behavior
Example: discrete time system with chaotic behavior: f:
x1(t)
x2(t)
x1(t) -x2(t)
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10. State synchronization
For chaotic systems, the transition from synchronized to non-synchronized
behavior is peculiar: bubbling bifurcation
x1(t)
x2(t)
x1(t) -x2(t)
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11. Doctoral course Torino 29.10.2010
State synchronization in networks of dynamical systems (dynamical networks)
Arbitrary networks of coupled identical dynamical systems
Connection graph:
n vertices and m edges
Arbitrary dynamics dynamics of individual dynamical system
Oscillatory:
Chaos:
Multistable:
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12. Doctoral course Torino 29.10.2010
State synchronization in networks of dynamical systems (dynamical networks)
Synchronization properties depend on
Individual dynamical systems
Interaction type and strength
Structure of the connection graph
Various notions of synchronization:
complete vs. partial
global vs. local
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