301 research outputs found
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
This paper deals with the problem of analytically computing the largest
Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is
succesfully reached within a theoretical framework that makes use of a
geometrization of newtonian dynamics in the language of Riemannian geometry. A
new point of view about the origin of chaos in these systems is obtained
independently of homoclinic intersections. Chaos is here related to curvature
fluctuations of the manifolds whose geodesics are natural motions and is
described by means of Jacobi equation for geodesic spread. Under general
conditions ane effective stability equation is derived; an analytic formula for
the growth-rate of its solutions is worked out and applied to the
Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent
agreement is found the theoretical prediction and the values of the Lyapunov
exponent obtained by numerical simulations for both models.Comment: RevTex, 40 pages, 8 PostScript figures, to be published in Phys. Rev.
E (scheduled for November 1996
Rational approximations, multidimensional continued fractions and lattice reduction
We first survey the current state of the art concerning the dynamical
properties of multidimensional continued fraction algorithms defined
dynamically as piecewise fractional maps and compare them with algorithms based
on lattice reduction. We discuss their convergence properties and the quality
of the rational approximation, and stress the interest for these algorithms to
be obtained by iterating dynamical systems. We then focus on an algorithm based
on the classical Jacobi--Perron algorithm involving the nearest integer part.
We describe its Markov properties and we suggest a possible procedure for
proving the existence of a finite ergodic invariant measure absolutely
continuous with respect to Lebesgue measure.Comment: 30 pages, 4 figure
Chaos and Complexity from Quantum Neural Network: A study with Diffusion Metric in Machine Learning
In this work, our prime objective is to study the phenomena of quantum chaos and complexity in the machine learning dynamics of Quantum Neural Network (QNN). A Parameterized Quantum Circuits (PQCs) in the hybrid quantum-classical framework is introduced as a universal function approximator to perform optimization with Stochastic Gradient Descent (SGD). We employ a statistical and differential geometric approach to study the learning theory of QNN. The evolution of parametrized unitary operators is correlated with the trajectory of parameters in the Diffusion metric. We establish the parametrized version of Quantum Complexity and Quantum Chaos in terms of physically relevant quantities, which are not only essential in determining the stability, but also essential in providing a very significant lower bound to the generalization capability of QNN. We explicitly prove that when the system executes limit cycles or oscillations in the phase space, the generalization capability of QNN is maximized. Moreover, a lower bound on the optimization rate is determined using the well known Maldacena Shenker Stanford (MSS) bound on the Quantum Lyapunov exponent
The Lyapunov Characteristic Exponents and their computation
We present a survey of the theory of the Lyapunov Characteristic Exponents
(LCEs) for dynamical systems, as well as of the numerical techniques developed
for the computation of the maximal, of few and of all of them. After some
historical notes on the first attempts for the numerical evaluation of LCEs, we
discuss in detail the multiplicative ergodic theorem of Oseledec \cite{O_68},
which provides the theoretical basis for the computation of the LCEs. Then, we
analyze the algorithm for the computation of the maximal LCE, whose value has
been extensively used as an indicator of chaos, and the algorithm of the
so--called `standard method', developed by Benettin et al. \cite{BGGS_80b}, for
the computation of many LCEs. We also consider different discrete and
continuous methods for computing the LCEs based on the QR or the singular value
decomposition techniques. Although, we are mainly interested in
finite--dimensional conservative systems, i. e. autonomous Hamiltonian systems
and symplectic maps, we also briefly refer to the evaluation of LCEs of
dissipative systems and time series. The relation of two chaos detection
techniques, namely the fast Lyapunov indicator (FLI) and the generalized
alignment index (GALI), to the computation of the LCEs is also discussed.Comment: 74 pages, 8 figures, accepted for publication in Lecture Notes in
Physic
Information geometric methods for complexity
Research on the use of information geometry (IG) in modern physics has
witnessed significant advances recently. In this review article, we report on
the utilization of IG methods to define measures of complexity in both
classical and, whenever available, quantum physical settings. A paradigmatic
example of a dramatic change in complexity is given by phase transitions (PTs).
Hence we review both global and local aspects of PTs described in terms of the
scalar curvature of the parameter manifold and the components of the metric
tensor, respectively. We also report on the behavior of geodesic paths on the
parameter manifold used to gain insight into the dynamics of PTs. Going
further, we survey measures of complexity arising in the geometric framework.
In particular, we quantify complexity of networks in terms of the Riemannian
volume of the parameter space of a statistical manifold associated with a given
network. We are also concerned with complexity measures that account for the
interactions of a given number of parts of a system that cannot be described in
terms of a smaller number of parts of the system. Finally, we investigate
complexity measures of entropic motion on curved statistical manifolds that
arise from a probabilistic description of physical systems in the presence of
limited information. The Kullback-Leibler divergence, the distance to an
exponential family and volumes of curved parameter manifolds, are examples of
essential IG notions exploited in our discussion of complexity. We conclude by
discussing strengths, limits, and possible future applications of IG methods to
the physics of complexity.Comment: review article, 60 pages, no figure
Extreme phase sensitivity in systems with fractal isochrons
Sensitivity to initial conditions is usually associated with chaotic dynamics
and strange attractors. However, even systems with (quasi)periodic dynamics can
exhibit it. In this context we report on the fractal properties of the
isochrons of some continuous-time asymptotically periodic systems. We define a
global measure of phase sensitivity that we call the phase sensitivity
coefficient and show that it is an invariant of the system related to the
capacity dimension of the isochrons. Similar results are also obtained with
discrete-time systems. As an illustration of the framework, we compute the
phase sensitivity coefficient for popular models of bursting neurons,
suggesting that some elliptic bursting neurons are characterized by isochrons
of high fractal dimensions and exhibit a very sensitive (unreliable) phase
response.Comment: 32 page
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