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Francesco Ginelli University of Aberdeen - ICSMB Characterizing dynamics with covariant Lyapunov Vectors Joint work with: A. Politi (Aberdeen), H. Chaté (CEA/Saclay), R. Livi (Florence), K.A. Takeuchi (Tokyo) BIRS 2015: RDSMET

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Lyapunov exponents, especially in the physics/applied math community, have long been the main tool of choice to characterize dynamical systems. Introduction For much of this talk I will discuss autonomous dynamical systems in a rather practical setup ( M, L with an evolution operator L over the Riemaniann manifold M equipped with a preserved measure . We also suppose that at each point x in M we can identify the tangent space T x M Indeed, to fix ideas I will simplify further, identifying M with and the evolution operator with ordinary differential equations or discrete maps With tangent space evolution operator Satisfying the cocycle properties

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Lyapunov exponents, at least in the physics/applied math community, have long been the main tool of choice to characterize dynamical systems. They characterize the exponential sensitivity to initial conditions, i.e. divergence of nearby orbits They are easily computed numerically, even in systems with many degrees of freedom via the Benettin et al Gram Schmidt algorithm. How to characterize the dynamics? I am particularly interested in large dynamical systems, where I have a large number of degrees of freedom (i.e. a sort of thermodynamics limit) and my approach is numerical

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LEs quantify the growth of volumes in tangent space Entropy production (Kolmogorov-Sinai entropy): Attractor dimension (Kaplan Yorke Formula) There exist a thermodynamic limit for Lyapunov spectra in spatially ext. systems:

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The existence of a complete set of N LEs is granted by the Oseledets theorem: Oseledets theorem There exist a mesurable filtration of the dynamics (or of its time reversal) composed of nested subspaces (spanned by the eigenvectors of the Oseledets matrix) such that: Oseledets matrix (with multiplicity g j )

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Tangent space structure It is somehow natural, together with Lyapunov exponents (i.e. eigenvalues), to also consider the associated tangent space directions in order to quantify stable and unstable directions in tangent space. Hierarchical decomposition of spatiotemporal chaos Optimal forecast in nonlinear models (e.g. in geophysics) Study of “hydrodynamical modes” in near-zero exponents and vectors (access to transport properties ?) Characterized the degree of non-hyperbolicity in large dynamical systems

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Covariant Lyapunov vectors (Oseledets splitting) Ruelle (1979) – Oseledets splitting Oseledets splitting determines a measurable decomposition of the tangent space which is independent of the chosen norm (at least for a large class of norms), covariant with the dynamics and (obviously) invariant under time reversal. Oseledets decomposition into such covariant subspaces exists for any map (11 ), which is continuous and measurable together with its inverse and whose Jacobian matrix exists and is finite in each element.

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Covariant Lyapunov vectors (CLVs) Covariant Lyapunov vectors (Oseledets splitting) CLVs are norm independent and invariant under time reversal. CLVs span the stable and unstable manifolds CLVsare covariant with dynamics and do yield correct growth factors (LEs):

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Politi et. al. (1998) – Covariant vectors satisfy a node theorem for periodic orbits Wolfe & Samelson (2007) – Intersection algorithm, more efficient for j << N Lack of a practical algorithm to compute them No studies of ensemble properties in large systems Legras & Vautard; Trevisan & Pancotti (1996) – Covariant vectors in Lorenz 63 After Ruelle, little attention in numerical applications… Brown, Bryant & Abarbanel (1991) – Covariant vectors in time series data analysis

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Gram Schmidt vectors ?

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A by-product of Benettin et al. -- Gram Schmidt vectors Gram Schmidt vectors are obtained by GS orthogonalization (QR decomposition) (Benettin et al. 1980) Upper triangular orthonormalization

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It can be shown that any orthonormal set of vectors eventually converge to a well defined basis (Ershov and Potapov, 1998) For time-invertible systems they coincide with the eigenvectors of the backward Oseledets matrix: They both probe the past part of the trajectory (or viceversa if time is reversed)

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Dynamical properties are “washed away” by orthonormalization, which is norm dependent, while LEs are not (for a wide class of norms). But… They are not invariant under time reversal, while LEs are (sign-wise): They are not covariant with dynamics and do not yield correct growth factors: They are orthogonal, while stable and unstable manifolds are generally not.

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An efficient dynamical algorithm for covariant Lyapunov vectors Upper triangular Express j-th covariant vectors as linear Combination of the first j Gram-Schmidt vectors g

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Covariant evolution means: Covariant vector j is a linear combination of the first j GS vectors 2. Moving backwards insures convergence to the “right” covariant vectors 1. The matrix R evolves the covariant vectors coefficients Diagonal matrix of local growth factors

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1. The matrix R evolves the covariant vectors coefficients (Expand CLV on GS basis) Covariant evolution means: one gets the evolution rule (use QR decomposition)

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2. Moving backwards insures convergence to the “right” covariant vectors (consider two different initial conditions for C, a random one and a true one ) By simple manipulations

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(by matrix components) Which for large times implies All random initial conditions converge exponentially to the same ones, apart a prefactor Thus this reversed dynamics converges to covariant vectors for almost any initial condition C

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The dynamical algorithm for computing covariant Lyapunov Vectors

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Is a generic nonsingular upper triangular matrix

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Further comments on the dynamical algorithm CLVs exist and can be computed for non time reversible systems too by following backward a stored forward trajectory It is not need to store GS vectors if is only interested in angles between CLVs. Some further tricks to ease memory storage in RAM are possible The convergence to the j-th (j > 1) covariant vector is exponential in the LEs difference, at least in 0-norm. Higher norms have smaller exponential decay rates due to FTLE fluctuations For spatially extended or globally coupled systems, computational time scales as otherwise Matrices C, R are upper triangular. is diagonal. This makes inversion simple

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Angles between CLVs or linear combinations of CLVs: measure (lack of) hyperbolicity. Tangent space decomposition of spatially extended chaotic systems (de)-localization properties and collective modes in large chaotic systems Some applications Hydrodynamic Lyapunov modes Identify spurious LEs in timeseries analysis Data assimilation algorithms ?

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1. Density of homoclinic tangencies Hénon Map Lozi Map Detect tangencies between the stable and unstable manifolds

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In more then 2 dimensions, linear combinations between vectors should be considered (Kuptsov & Kuznetsov ArXiv: (2009)) Minimum angle between stable and unstable manifold

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FPU chain e = 10 chain of Henon maps Many degrees of freedom….

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High energy density, above SST Low energy density, nelow SST

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Crossover in L at high energy densities

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2. Tangent space decomposition in spatially extended systems Kuramoto Sivashinsky Eq. L = 96 Lyapunov spectrum for different frequency cut-offs

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2. Tangent space decomposition in spatially extended systems Kuramoto Sivashinsky Eq. L = 96 Lyapunov spectrum for different frequency cut-offs

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2. Tangent space decomposition in spatially extended systems Typical trajectory and CLVs

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2. Tangent space decomposition in spatially extended systems Typical angles between consecutive vectors Separiation between physical CLVs and spurious CLVs

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2. Tangent space decomposition in spatially extended systems We conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor The number of physical modes scales linearly with system size. The inertial manifold is extensive?

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Localized: nonvanishing Y 2 Delocalized: vanishing Y 2 3. Localization properties in large chaotic systems: a tool to characterize collective modes Localization properties of vector j can be characterized by the inverse participation ratio Where the amplitude per oscillator

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Localized, extensive covariant Lyapunov vectors corresponding to microscopic dynamics Delocalized, nonextensive covariant Lyapunov vectors corresponding to collective modes

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Localization large systems – GSVs vs CLVs CLVs - localize GSVs – spurious delocalization 1D Chains: a) CML of Tent maps b) Simplectic maps c) Rotors d) FPU

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: individual oscillators : collective dynamics A model system: Globally coupled limit cycle oscillators Ginzburg Landau oscillators Kuramoto & Nakagawa (1994, 1995)

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: individual oscillators : collective dynamics A model system: Globally coupled limit cycle oscillators Ginzburg Landau oscillators Kuramoto & Nakagawa (1994, 1995)

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individual oscillators: chaotic collective dynamics:quasi-periodic like (weakly chaotic) Intermediate couplung: nontrivial collective behavior Density plot

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Parametric plot of vs Delocalized – collective, macroscopic dynamics Localized –microscopic dynamics

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Parametric plot of vs

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Bibliography FG, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi, Phys Rev Lett 99, (2007). FG, H. Chaté, R. Livi, and A. Politi, J Phys A 46, (2013). K. Takeuchi, FG, H. Chaté, Phys. Rev. Lett. 103, (2009). H-l.Yang, K. A. Takeuchi, FG, H. Chaté, and G. Radons Phys. Rev. Lett (2009). K. A. Takeuchi, H-l.Yang, FG, G. Radons and H. Chaté, Phys Rev E (2011) K. A. Takeuchi, and H. Chaté, J Phys A 46, (2013). Thank you

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More then 2 dimensions, linear combinations between vectors should be considered (Kuptsov & Kuznetsov ArXiv: (2009)) Minimum angle between stable and unstable manifold

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1. R evolves the coefficients C according to tangent dynamics (Expand CLV on GS basis) Covariant evolution means: one gets the evolution rule (use QR decomposition)

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2. Moving backwards insures convergence to the “right” covariant vectors (consider two different random initial conditions) A. If C are upper triangular with non-zero diagonal, one can verify that B. By simple manipulations

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(by matrix components) If we follow the reversed dynamics (diagonal matrix) All random initial conditions converge to the same ones, apart a prefactor Thus this reversed dynamics converges to covariant vectors for almost any initial condition

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On Wolfe & Samelson (2007): vector n-th out of N where since n - 1 forward and n backward GSV are needed to compute the kernel

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Fourier analysis of “last positive” vector

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Localization length

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