Francesco Ginelli

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

2. Introduction • For much of this talk I will discuss autonomous dynamical systems in a rather • practical setup • (M,m, L) • with an evolution operator L over the Riemaniann manifold M equipped with a • preserved measure m . We also suppose that at each point x in M we can identify • the tangent space Tx M • Lyapunov exponents, especially in the physics/applied math community, have • long been the main tool of choice to characterize dynamical systems. • 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

3. 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 • 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.

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

5. Oseledets theorem • The existence of a complete set of N LEs is granted by the Oseledets theorem: Oseledets matrix (with multiplicity gj) • 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:

6. 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

7. 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.

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

9. After Ruelle, little attention in numerical applications… • Brown, Bryant & Abarbanel (1991) – Covariant vectors in time series data analysis • Legras & Vautard; Trevisan & Pancotti (1996) – Covariant vectors in Lorenz 63 • 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

10. Gram Schmidt vectors ?

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

12. 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)

13. But… • They are orthogonal, while stable and unstable manifolds are generally not. • Dynamical properties are “washed away” by orthonormalization, which is norm • dependent, while LEs are not (for a wide class of norms). • 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:

14. 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

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

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

17. 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

18. (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

19. The dynamical algorithm for computing covariant Lyapunov Vectors

20. The dynamical algorithm for computing covariant Lyapunov Vectors Is a generic nonsingular upper triangular matrix

21. Further comments on the dynamical algorithm • Matrices C, R are upper triangular. D is diagonal. This makes inversion simple • For spatially extended or globally coupled systems, computational time scales as • otherwise • 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 • 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

22. Some applications • 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 • Hydrodynamic Lyapunov modes • Identify spurious LEs in timeseries analysis • Data assimilation algorithms ?

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

24. In more then 2 dimensions, linear combinations between vectors should be considered (Kuptsov & Kuznetsov ArXiv:0812.4823 (2009)) Minimum angle between stable and unstable manifold

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

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

27. Crossover in L at high energy densities

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

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

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

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

32. 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?

33. Localized: nonvanishing Y2 • Delocalized: vanishing Y2 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

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

35. Localization large systems – GSVs vs CLVs CLVs - localize GSVs – spurious delocalization 1D Chains: CML of Tent maps Simplectic maps Rotors FPU

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

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

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

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

40. Parametric plot of vs

41. Parametric plot of vs

42. Thank you Bibliography FG, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi, Phys Rev Lett99, 130601 (2007). FG, H. Chaté, R. Livi, and A. Politi, J Phys A 46, 254005 (2013). K. Takeuchi, FG, H. Chaté, Phys. Rev. Lett. 103, 154103 (2009). H-l.Yang, K. A. Takeuchi, FG, H. Chaté, and G. Radons Phys. Rev. Lett. 102074102 (2009). K. A. Takeuchi, H-l.Yang, FG, G. Radons and H. Chaté, Phys Rev E84 046214 (2011) K. A. Takeuchi, and H. Chaté, J Phys A 46, 254007 (2013).

43. More then 2 dimensions, linear combinations between vectors should be considered (Kuptsov & Kuznetsov ArXiv:0812.4823 (2009)) Minimum angle between stable and unstable manifold

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

47. 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

48. (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

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

50. Fourier analysis of “last positive” vector