doctoral course torino 29.10.2010 introduction to synchronization: history 1665: huygens observation...
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![Page 1: Doctoral course Torino 29.10.2010 Introduction to synchronization: History 1665: Huygens observation of pendula When pendula are on a common support, they](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bf701a28abf838c7dc42/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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)
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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
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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