Scale-invariant Lyapunov exponents for classical hamiltonian systems (original) (raw)
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Lyapunov Exponents without Rescaling and Reorthogonalization
Physical Review Letters, 1998
We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or MM 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.
Meccanica, 1980
Da diversi anni gli esponenti caratteristici di Lyapunov sono divenuti di notevole interesse nello studio dei sistemi dinamici al fine di caratterizzare quantitativamente le proprietà di stocasticità, legate essenzialmente alla divergenza esponenziale di orbite vicine. Si presenta dunque il problema del calcolo esplicito di tali esponenti, già risolto solo per il massimo di essi. Nel presente lavoro si dà un metodo per il calcolo di tutti tali esponenti, basato sul calcolo degli esponenti di ordine maggiore di uno, legati alla crescita di volumi. A tal fine si dà un teorema che mette in relazione gli esponenti di ordine uno con quelli di ordine superiore. Il metodo numerico e alcune applicazioni saranno date in un sucessivo articolo. Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.
Semi-analytic estimates of Lyapunov exponents in lower-dimensional systems
Physics Letters A, 2003
Statistical arguments, seemingly well-justified in higher dimensions, can also be used to derive reasonable estimates of Lyapunov exponents χ in lower-dimensional Hamiltonian systems. This Letter explores the assumptions incorporated into these arguments. The predicted χ's are insensitive to most details, but do depend sensitively on the nongeneric form of the auto-correlation function characterising the time-dependence of an orbit. This dependence on dynamics implies a fundamental limitation to the application of thermodynamic arguments to lower-dimensional systems.
Journal of Physics A: Mathematical and General, 2006
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.
Theoretical Computation of Lyapunov Exponents for Almost Periodic Hamiltonian Systems
2011
Lyapunov exponents are an important concept to describe qualitative properties of dynamical systems. For instance, chaotic systems can be caracterized with the positivity of the largest Lyapunov exponent. In this paper, we use the Iwasawa decomposition of the semisimple Lie group Sp(n,R) and the enlargement of the phase space to give a theoretical computation of Lyapunov exponents of almost periodic Hamiltonian systems. In particular, we obtain the existence of Lyapunov exponents everywhere in the surface of constant energy of the Hamiltonian H . It turns out that, in this context, the Oseledec’s assumption is not necessary to guarantee the existence and the finitness of Lyapunov exponents.
On the notion of quantum Lyapunov exponent
arXiv (Cornell University), 2005
Classical chaos refers to the property of trajectories to diverge exponentially as time t → ∞. It is characterized by a positive Lyapunov exponent. There are many different descriptions of quantum chaos. The one related to the notion of generalized (quantum) Lyapunov exponent is based either on qualitative physical considerations or on the so-called symplectic tomography map [1, 2]. The purpose of this note is to show how the definition of quantum Lyapunov exponent naturally arises in the framework of the Moyal phase space formulation of quantum mechanics [3], and is based on the notions of quantum trajectories and the family of quantizers [4]. The role of the Heisenberg uncertainty principle in the statement of the criteria for quantum chaos is made explicit.
A semi-classical approach for Lyapunov exponents of a quantum mechanical system
We attempt to construct Lyapunov exponents for quantum mechanical systems. Lyapunov exponents have always been a key concept of classical chaotic dynamics, but difficult to use in a quantum context. By constructing a semi-classical potential from the ground state wavefunction of a system and analyzing the classical dynamics produced by this potential, one can use concepts from classical dynamics, such as Lyapunov exponents, to probe the dynamics of the quantum system. In this manner, we find non trivial corrections to the classical dynamics arising from quantum information: the chaotic behavior is suppressed although not entirely eliminated.
Lyapunov exponents for continuous transformations and dimension theory
Discrete and Continuous Dynamical Systems, 2005
We generalize the concept of Lyapunov exponent to transformations that are not necessarily differentiable. For fairly large classes of repellers and of hyperbolic sets of differentiable maps, the new exponents are shown to coincide with the classical ones. We also discuss the relation of the new Lyapunov exponents with the dimension theory of dynamical systems for invariant sets of continuous transformations.
Scaling relations in the Lyapunov exponents of one dimensional maps
Pramana-journal of Physics, 1994
We establish numerically the validity of Huberman-Rudnick scaling relation for Lyapunov exponents during the period doubling route to chaos in one dimensional maps. We extend our studies to the context of a combination map. where the scaling index is found to be different.