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 Doctoral course Torino 29.10.2010

Model: two coupled Vanderpol oscillators Doctoral course Torino 29.10.2010

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 Doctoral course Torino 29.10.2010

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 Doctoral course Torino 29.10.2010

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 Doctoral course Torino 29.10.2010

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 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: Doctoral course Torino 29.10.2010

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) Doctoral course Torino 29.10.2010

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) Doctoral course Torino 29.10.2010

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) Doctoral course Torino 29.10.2010

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: Doctoral course Torino 29.10.2010

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 Doctoral course Torino 29.10.2010