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Effective power-law dependence of Lyapunov exponents on the central mass in galaxies  [PDF]
N. Delis,C. Efthymiopoulos,C. Kalapotharakos
Physics , 2015, DOI: 10.1093/mnras/stv064
Abstract: Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation $L\propto m^p$ between the mean Lyapunov exponent $L$ of stellar orbits chaotically scattered by a supermassive black hole in the center of a galaxy and the mass parameter $m$, i.e. ratio of the mass of the black hole over the mass of the galaxy. The exponent $p$ is found numerically to obtain values in the range $p \approx 0.3$--$0.5$. We propose a theoretical interpretation of these exponents, based on estimates of local `stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the black hole's sphere of influence. We thus predict $p=2/3-q$ with $q\approx 0.1$--$0.2$. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power law scaling of $L$ with $m$ holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the black hole affects mainly populations of thick tube-like orbits surrounding some low-order branches of the $x_1$ family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the Inner Lindbland resonance and the 4/1 resonance. Implications of the correlations between $L$ and $m$ to determining the rate of secular evolution of galaxies are discussed.
Large deviations of Lyapunov exponents  [PDF]
Tanguy Laffargue,Khanh-Dang Nguyen Thu Lam,Jorge Kurchan,Julien Tailleur
Physics , 2013, DOI: 10.1088/1751-8113/46/25/254002
Abstract: Generic dynamical systems have `typical' Lyapunov exponents, measuring the sensitivity to small perturbations of almost all trajectories. A generic system has also trajectories with exceptional values of the exponents, corresponding to unusually stable or chaotic situations. From a more mathematical point of view, large deviations of Lyapunov exponents characterize phase-space topological structures such as separatrices, homoclinic trajectories and degenerate tori. Numerically sampling such large deviations using the Lyapunov Weighted Dynamics allows one to pinpoint, for example, stable configurations in celestial mechanics or collections of interacting chaotic `breathers' in nonlinear media. Furthermore, we show that this numerical method allows one to compute the topological pressure of extended dynamical systems, a central quantity in the Thermodynamic of Trajectories of Ruelle.
Genericity of zero Lyapunov exponents  [PDF]
Jairo Bochi
Mathematics , 2002, DOI: 10.1017/S0143385702001165
Abstract: We show that, for any compact surface, there is a residual (dense $G_\delta$) set of $C^1$ area preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was announced by R. Mane, but no proof was available. We also show that for any fixed ergodic dynamical system over a compact space, there is a residual set of continuous $SL(2,R)$-cocycles which either are uniformly hyperbolic or have zero exponents a.e.
Entropy, Lyapunov Exponents and Escape Rates in Open Systems  [PDF]
Mark Demers,Paul Wright,Lai-Sang Young
Mathematics , 2011,
Abstract: We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions including Lebesgue and SRB measures. Lower bounds do not hold in such generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon with holes.
On the abundance of non-zero central Lyapunov exponents, physical measures and stable ergodicity for partially hyperbolic dynamics  [PDF]
Vitor Araujo,Carlos H. Vasquez
Mathematics , 2010,
Abstract: We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical measure with full basin, which is $C^r$ stably ergodic. Our method is perturbative and does not rely on preservation of a smooth measure.
On quantum Lyapunov exponents  [PDF]
Wladyslaw A. Majewski,Marcin Marciniak
Physics , 2005,
Abstract: It was shown that quantum analysis constitutes the proper analytic basis for quantization of Lyapunov exponents in the Heisenberg picture. Differences among various quantizations of Lyapunov exponents are clarified.
Positive Lyapunov exponents for symplectic cocycles  [PDF]
Mario Bessa,Paulo Varandas
Mathematics , 2014,
Abstract: In the present paper we give a positive answer to a question posed by Viana on the existence of positive Lyapunov exponents for symplectic cocycles. Actually, we prove that for an open and dense set of Holder symplectic cocycles over a non-uniformly hyperbolic diffeomorphism there are non-zero Lyapunov exponents with respect to any invariant ergodic measure with the local product structure.
Lyapunov Exponents without Rescaling and Reorthogonalization  [PDF]
Govindan Rangarajan,Salman Habib,Robert D. Ryne
Physics , 1998, DOI: 10.1103/PhysRevLett.80.3747
Abstract: We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or MM_trans where M is the tangent map. This method uses a minimal set of variables, does not require renormalization or reorthogonalization, can be used to efficiently compute partial Lyapunov spectra, and does not break down when the Lyapunov spectrum is degenerate.
Global vector-field reconstruction of nonlinear dynamical systems from a time series with SVD method and validation with Lyapunov exponents
Liu Wei-Dong,K F Ren,S Meunier-Guttin-Cluzel,G Gouesbet,
,K. F. Ren,S. Meunier-Guttin-Cluzel,G. Gouesbet

中国物理 B , 2003,
Abstract: A method for the global vector-field reconstruction of nonlinear dynamical systems from a time series is studied in this paper. It employs a complete set of polynomials and singular value decomposition (SVD) to estimate a standard function which is central to the algorithm. Lyapunov exponents and dimension, calculated from the differential equations of a standard system, are used for the validation of the reconstruction. The algorithm is proven to be practical by applying it to a R?ssler system.
Recurrence and lyapunov exponents  [PDF]
B. Saussol,S. Troubetzkoy,S. Vaienti
Mathematics , 2002,
Abstract: We prove two inequalities between the Lyapunov exponents of a diffeomorphism and its local recurrence properties. We give examples showing that each of the inequalities is optimal.
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