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Coupled Map Lattice (CML) Approach

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Coupled Map Lattice (CML) Approach Different mechanisms of synchronization : Coupled Map Lattice (CML) approach – p.1 Outline Synchronization : an universal phenomena Model (Coupled maps/oscillators on Networks) – p.2 Modelling Spatio-temporal dynamics 2- and 3-dimensional fluid flow, Coupled-oscillator models, Cellular automata, Transport models, Coupled map models Model does not have direct connections with actual physical or biological systems – p.3 Modelling Spatio-temporal dynamics 2- and 3-dimensional fluid flow, Coupled-oscillator models, Cellular automata, Transport models, Coupled map models Model does not have direct connections with actual physical or biological systems General results show some universal features – p.3 We select coupled maps(chaotic), oscillators on Networks Models. – p.4 Coupled Dynamics on Networks ε xi(t +1) = f(xi(t)) + Cijg(xi(t),xj(t)) Cij Pj P Periodic orbit Synchronized chaos Travelling wave Spatio-temporal chaos – p.5 Coupled maps Models Coupled oscillators (- Kuramoto ∼ beginning of 1980’s) Coupled maps (- K. Kaneko) - K. Kaneko, ”Simulating Physics with Coupled Map Lattices —— Pattern Dynamics, Information Flow, and Thermodynamics of Spatiotemporal Chaos”, pp1-52 in Formation, Dynamics, and Statistics of Patterns ed. K. Kawasaki et.al., World. Sci. 1990” – p.6 Studying real systems with coupled maps/oscillators models: - Ott, Kurths, Strogatz, Geisel, Mikhailov, Wolf etc . From Kuramoto to Crawford .... (Physica D, Sep 2000) Chaotic Transients in Complex Networks, (Phys Rev Lett, 2004) Turbulence in oscillatory chemical reactions, (J Chem Phys, 1994) Crowd synchrony on the Millennium Bridge, (Nature, 2005) Species extinction - Amritkar and Rangarajan, Phys. Rev. Lett. (2006) – p.7 SYNCHRONIZATION First observation (Pendulum clock (1665)) - discovered in 1665 by the famous Dutch physicists, Christiaan Huygenes. – p.8 SYNCHRONIZATION First observation (Pendulum clock (1665)) - discovered in 1665 by the famous Dutch physicists, Christiaan Huygenes. And now : Synchronous firing of neurons - Wolf Singer and collaborators, 1989 – p.8 SYNCHRONIZATION First observation (Pendulum clock (1665)) - discovered in 1665 by the famous Dutch physicists, Christiaan Huygenes. And now : Synchronous firing of neurons - Wolf Singer and collaborators, 1989 Synchronization is because of coupling – p.8 Some Examples of synchrony in brain Synchronous firing of neurons - Wolf Singer and collaborators, Nature, (1989) Synchronization among cortical areas - Nature, Vol. 385 (1997). • • Dysfunctional long-range coordination of neural activity during Gestalt perception in schizophrenia. - J Neurosci. (2006) – p.9 Adjustment of heart rate with respiration, cooperative behaviour of insects, animals and even human, synchronously flashing fireflies, cricket chirp in unison, aerobics Bacteria.. Musical instruments, electronic generators, electric power systems .. Synchronization: A Universal Concept in Nonlinear Science, by Pikovsky et. al. (Cambridge University Press, 2001) – p.10 Adjustment of heart rate with respiration, cooperative behaviour of insects, animals and even human, synchronously flashing fireflies, cricket chirp in unison, aerobics Bacteria.. Musical instruments, electronic generators, electric power systems .. Synchronization: A Universal Concept in Nonlinear Science, by Pikovsky et. al. (Cambridge University Press, 2001) ...And a most recent one: Experimental evidence of synchrony in a chain of three coupled semiconductor lasers - Phys. Rev. Lett. Vol. 97, 123902 (2006) – p.10 Coupled Dynamics f(xi ) i j g(x j) wij Nodes: Neurons, chemicals, species, authors, airports .... Links: Axons, reactions, predator-pray, co-authors, flights .... – p.11 Mechanism of Synchrony ...!?! Meeting with each other or common external threat !?! – p.12 50 40 38 30 20 31 10 0 0 10 20 30 40 50 open circles : nodes in the network close circles : couplings between the nodes – p.13 50 40 30 50 20 40 10 30 0 0 10 20 30 40 50 20 10 0 0 10 20 30 40 50 – p.13 Driven Synchronization N_2 N_1 Model for coupled complete bipartite network can be written as, ǫ N x (t + 1) = (1 − ǫ)f(x (t)) + g(x (t)), for i = 1, ··· , N1 i i N2 X j j=N1+1 1 ǫ N = (1 − ǫ)f(x (t)) + g(x (t)), for i = N1 + 1, ··· , N i N1X j j=1 – p.14 Linear stability analysis for d- synchronized state Jacobian Jt at driven synchronized state (X1,X2) at time t: (1 − ǫ)f ′ 0 ... ǫ g′ ǫ g′ ... 1 N2 2 N2 2 . . . 0 ... (1 − ǫ)f ′ ǫ g′ ǫ g′ ... 1 N2 2 N2 2 ǫ ′ ǫ ′ ′ g1 g1 ... (1 − ǫ)f2 0 ... N1 N1 . . ǫ g′ ǫ g′ ... ... ǫ f ′ N1 1 N1 1 0 (1 − ) 2 ′ ′ 1 ′ ′ f1,g1 are derivatives at X and f2,g2 are derivatives at – p.15 X2 Linear stability analysis for d- synchronized state Jacobian Jt at driven synchronized state (X1,X2) at time t: (1 − ǫ)f ′ 0 ... ǫ g′ ǫ g′ ... 1 N2 2 N2 2 . . . 0 ... (1 − ǫ)f ′ ǫ g′ ǫ g′ ... 1 N2 2 N2 2 ǫ ′ ǫ ′ ′ g1 g1 ... (1 − ǫ)f2 0 ... N1 N1 . . ǫ g′ ǫ g′ ... ... ǫ f ′ N1 1 N1 1 0 (1 − ) 2 ′ ′ 1 ′ ′ f1,g1 are derivatives at X and f2,g2 are derivatives at – p.15 X2 set Eigenvectors Eigenvalue No of eigenvalues condition ′ A (α1 ...αN1, 0 ··· 0) (1 − ǫ)f1 N1 − 1 Σα = 0 ′ B (0 ... 0, β1 ...βN2) (1 − ǫ)f2 N2 − 1 Σβ = 0 C (α,...α,β,...β) - 2 - Eigenvalues corresponding to set C : given from matrix ′ ′ (1 − ǫ)f1 ǫg2 ′ ′ ǫg1 (1 − ǫ)f2 For a complete bi-partite network, three sets of eigenvectors are orthogonal to each other three sets of eigenvectors do not mix with each other under time evolution – p.16 ⇓ Lyapunov exponents corresponding to the transverse eigenvectors 1 τ ′ λi = ln |(1 − ǫ)| + limτ→∞ ln |f (X1)|, τ Pt=1 for i = 1,...,N1 − 1 1 τ ′ λi = ln |(1 − ǫ)| + limτ→∞ ln |f (X2)|, τ Pt=1 for i = N1,...,N − 2 Condition for stable synchronous state : All transverse Lyapunov exponents should be negative. || ⇓ gives boundaries for d- synchronized regions in phase diagram – p.17 For further reading and discussions ”Phase Synchronization in Coupled Map Networks”, SJ and R. E. Amritkar, Phys. Rev. Lett. 90, 014101 (2003); Phys. Rev. E 72, 016211 (2005). ”Stability analysis of driven synchronization”, R.E. Amritkar et. al., Phys. Rev. E 72, 016212 (2005). ”Symbolic Synchronization and the Detection of Global Properties of Coupled Dynamics from Local Information”, SJ, J¨urgen Jost and Fatihcan Atay, Chaos, 16, issue 3 (2006) ”Inferring Global Synchrony from Local Symbolic Dynamics”, SJ, Fatihcan Atay and J¨urgen Jost, (ComplexUS, 2007) ”Complexity, Randomness and Structure”, F. Atay, SJ and J. Jost (Submitted). – p.18.
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