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At least 19 recordsLinked to original sources
arXiv2010

Optical-Wavelength Paramagnetic Phaser (Lecture Notes). Section 3.1. Nonlinear Balance Equations of Motion

In this work I present a detailed description of the simplest nonlinear model for an optical wavelength paramagnetic phaser, which is an acoustic analog of the class-B lasers. Despite of its simplicity, this model gives a satisfactory explanation of experimental data for optical-wavelength paramagnetic phasers based on high-quality acoustic Fabry-Perot resonators. In particular, this model was successfully used both for qualitative and quantitative interpretation of deterministic chaotic motions observed in spin-phonon system of a nonautonomous ruby phasers at liquid helium temperatures (see arXiv:0704.0123v1 [nlin.CD]).

physics.optics↗
arXiv2012

Classification of Periodic, Chaotic and Random Sequences using NSRPS Complexity Measure

Data compression algorithms are generally perceived as being of interest for data communication and storage purposes only. However, their use in the field of data classification and analysis is also of equal importance. Automatic data classification and analysis finds use in varied fields like bioinformatics, language and sequence recognition and authorship attribution. Different complexity measures proposed in literature like Shannon entropy, Relative entropy, Kolmogrov and Algorithmic complexity have drawbacks that make these methods ineffective in analyzing short sequences that are typical in population dynamics and other fields. In this paper, we study Non-Sequential Recursive Pair Substitution (NSRPS), a lossless compression algorithm first proposed by Ebeling {\it et al.} [Math. Biosc. 52, 1980] and Jiménez-Montaño {\it et al.} [arXiv:cond-mat/0204134, 2002]). Using this algorithm, a new complexity measure was recently proposed (Nagaraj {\it et al.} [arXiv:nlin.CD/1101.4341v1, 2011]). In this work, we use NSRPS complexity measure for analyzing and classifying symbolic sequences generated by 1D chaotic dynamical systems. Even with learning data-sets of length as small as 25 and test data-sets of length as small as 10, NSRPS measure is able to accurately classify the test sequence as periodic, chaotic or random. For such short data lengths, methods which use entropy measure and traditional lossless compression algorithm like LZ77 [A.Lempel and J.Ziv, IEEE Trans. Inform. Theory {\bf 22}, 75 (1976)] (used for instance by {\it Gzip}, {\it Winzip} etc.) fails.

nlin.CD↗
arXiv2005

Breaking a chaos-based secure communication scheme designed by an improved modulation method

Recently Bu and Wang [Chaos, Solitons & Fractals 19 (2004) 919] proposed a simple modulation method aiming to improve the security of chaos-based secure communications against return-map-based attacks. Soon this modulation method was independently cryptanalyzed by Chee et al. [Chaos, Solitons & Fractals 21 (2004) 1129], Wu et al. [Chaos, Solitons & Fractals 22 (2004) 367], and Álvarez et al. [Chaos, Solitons & Fractals, accepted (2004), arXiv:nlin.CD/0406065] via different attacks. As an enhancement to the Bu-Wang method, an improving scheme was suggested by Wu et al. by removing the relationship between the modulating function and the zero-points. The present paper points out that the improved scheme proposed by Wu et al. is still insecure against a new attack. Compared with the existing attacks, the proposed attack is more powerful and can also break the original Bu-Wang scheme. Furthermore, it is pointed out that the security of the modulation-based schemes is not so satisfactory from a pure cryptographical point of view. The synchronization performance of this class of modulation-based schemes is also discussed.

nlin.CD↗
arXiv2002

More ergodic billiards with an infinite cusp

In a previous paper (nlin.CD/0107041) the following class of billiards was studied: For $f: [0, +\infty) \longrightarrow (0, +\infty)$ convex, sufficiently smooth, and vanishing at infinity, let the billiard table be defined by $Q$, the planar domain delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. For a large class of $f$ we proved that the billiard map was hyperbolic. Furthermore we gave an example of a family of $f$ that makes this map ergodic. Here we extend the latter result to a much wider class of functions.

nlin.CD↗
arXiv2002

Comment on ``Analysis of chaotic motion and its shape dependence in a generalized piecewise linear map''

Rajagopalan and Sabir [nlin.CD/0104021 and Phys. Rev. E 63, 057201 (2001)] recently discussed deterministic diffusion in a piecewise linear map using an approach developed by Fujisaka et al. We first show that they rederived the random walk formula for the diffusion coefficient, which is known to be the exact result for maps of Bernoulli type since the work of Fujisaka and Grossmann [Z. Physik B {\bf 48}, 261 (1982)]. However, this correct solution is at variance to the diffusion coefficient curve presented in their paper. Referring to another existing approach based on Markov partitions, we answer the question posed by the authors regarding solutions for more general parameter values by recalling the finding of a fractal diffusion coefficient. We finally argue that their model is not suitable for studying intermittent behavior, in contrast to what was suggested in their paper.

nlin.CD↗
arXiv2002

Particle transport in a random velocity field with Lagrangian statistics

The transport properties of a random velocity field with Kolmogorov spectrum and time correlations defined along Lagrangian trajectories are analyzed. The analysis is carried on in the limit of short correlation times, as a perturbation theory in the ratio, scale by scale, of the eddy decay and turn-over time. Various quantities such as the Batchelor constant and the dimensionless constants entering the expression for particle relative and self-diffusion are given in terms of this ratio and of the Kolmogorov constant. Particular attention is paid to particles with finite inertia. The self-diffusion properties of a particle with Stokes time longer than the Kolmogorov time are determined, verifying on an analytical example the dimensional results of [nlin.CD/0103018]. Expressions for the fluid velocity Lagrangian correlations and correlation times along a solid particle trajectory, are provided in several parameter regimes, including the infinite Stokes time limit corresponding to Eulerian correlations. The concentration fluctuation spectrum and the non-ergodic properties of a suspension of heavy particles in a turbulent flow, in the same regime, are analyzed. The concentration spectrum is predicted to obey, above the scale of eddies with lifetime equal to the Stokes time, a power law with universal -4/3 exponent, and to be otherwise independent of the nature of the turbulent flow. A preference of the solid particle to lie in less energetic regions of the flow is observed.

nlin.CD↗
arXiv2001

Percolation model for nodal domains of chaotic wave functions

Nodal domains are regions where a function has definite sign. In recent paper [nlin.CD/0109029] it is conjectured that the distribution of nodal domains for quantum eigenfunctions of chaotic systems is universal. We propose a percolation-like model for description of these nodal domains which permits to calculate all interesting quantities analytically, agrees well with numerical simulations, and due to the relation to percolation theory opens the way of deeper understanding of the structure of chaotic wave functions.

nlin.CD↗
arXiv2004

Spectral statistics of a quantum interval-exchange map

Curious spectral properties of an ensemble of random unitary matrices appearing in the quantization of a map p -> p+alpha, q -> q+f(p+alpha) in [Giraud et al. nlin.CD/0403033] are investigated. When alpha=m/n with integer co-prime m,n and matrix dimension N -> infinity is such that mN = 1 or -1 mod n, local spectral statistics of this ensemble tends to the semi-Poisson distribution [Bogomolny et al. Eur. Phys. J. B 19, 121 (2001)] with arbitrary integer or half-integer level repulsion at small distances: R(s)-> s^{beta} when s -> 0 and beta=n-1 or n/2-1 depending on time-reversal symmetry of the map.

nlin.CD↗
arXiv2012

Sampled-Data and Harmonic Balance Analyses of Average Current-Mode Controlled Buck Converter

Dynamics and stability of average current-mode control of buck converters are analyzed by sampled-data and harmonic balance analyses. An exact sampled-data model is derived. A new continuous-time model "lifted" from the sampled-data model is also derived, and has frequency response matched with experimental data reported previously. Orbital stability is studied and it is found unrelated to the ripple size of the current-loop compensator output. An unstable window of the current-loop compensator pole is found by simulations, and it can be accurately predicted by sampled-data and harmonic balance analyses. A new S plot accurately predicting the subharmonic oscillation is proposed. The S plot assists pole assignment and shows the required ramp slope to avoid instability.

eess.SY↗
arXiv2005

Approximating multi-dimensional Hamiltonian flows by billiards

Consider a family of smooth potentials $V_ε$, which, in the limit $ε\to0$, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as $ε\to0$ to the original billiard, and provide asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the $C^{r}$ norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials.

nlin.CD↗
arXiv2012

Boundary Conditions of Subharmonic Oscillations in Fixed-Switching-Frequency DC-DC Converters

Design-oriented boundary conditions for subharmonic oscillations are of great interest recently. Based on a subharmonic oscillation boundary condition reported in a PhD thesis more than a decade ago, extended new boundary conditions are derived in closed forms for general switching DC-DC converters. Sampled-data and harmonic balance analyses are applied and generate equivalent results. It is shown that equivalent series resistance causes the boundary conditions for voltage/current mode control to have similar forms. Some recently reported boundary conditions become special cases in view of the general boundary conditions derived. New Nyquist-like design-oriented plots are proposed to predict or prevent the occurrence of the subharmonic oscillation. The relation between the crossover frequency and the subharmonic oscillation is also analyzed.

eess.SY↗
arXiv2017

Copy the dynamics using a learning machine

Is it possible to generally construct a dynamical system to simulate a black system without recovering the equations of motion of the latter? Here we show that this goal can be approached by a learning machine. Trained by a set of input-output responses or a segment of time series of a black system, a learning machine can be served as a copy system to mimic the dynamics of various black systems. It can not only behave as the black system at the parameter set that the training data are made, but also recur the evolution history of the black system. As a result, the learning machine provides an effective way for prediction, and enables one to probe the global dynamics of a black system. These findings have significance for practical systems whose equations of motion cannot be approached accurately. Examples of copying the dynamics of an artificial neural network, the Lorenz system, and a variable star are given. Our idea paves a possible way towards copy a living brain.

stat.ML↗
arXiv2025

Geometry of Almost-Conserved Quantities in Symplectic Maps. Part I: Perturbation Theory

Noether's theorem, which connects continuous symmetries to exact conservation laws, remains one of the most fundamental principles in physics and dynamical systems. In this work, we draw a conceptual parallel between two paradigms: the emergence of exact invariants from continuous symmetries, and the appearance of approximate invariants from discrete symmetries associated with reversibility in symplectic maps. We demonstrate that by constructing approximating functions that preserve these discrete symmetries order by order, one can systematically uncover hidden structures, closely echoing Noether's framework. The resulting functions serve not only as diagnostic tools but also as compact representations of near-integrable behavior. The first article establishes the formal foundations of the method. Using the symmetric form of the map as a flexible test case, we benchmark the perturbative construction against established techniques, including the Lie algebra method for twist coefficients. To resolve the inherent ambiguity in the perturbation series, we introduce an averaging procedure that naturally leads to a resonant theory -- capable of treating rational rotation numbers and small-denominator divergences. This enables an accurate and structured description of low-order resonances, including singular and non-singular features in the quadratic and cubic Hénon maps. The approach is systematic, requiring only linear algebra and integrals of elementary functions, yet it yields results in striking agreement with both theory and numerical experiment. We conclude by outlining extensions to more general maps and discussing implications for stability estimates in practical systems such as particle accelerators.

nlin.CD↗
arXiv2013

Chaotic motion of three-body problem : an origin of macroscopic randomness of the universe

The famous three-body problem is investigated by means of a numerical approach with negligible numerical noises in a long enough time interval, namely the Clean Numerical Simulation (CNS). From physical viewpoints, position of any bodies contains inherent micro-level uncertainty. The evaluations of such kind of inherent micro-level uncertainty are accurately simulated by means of the CNS. Our reliable, very accurate CNS results indicate that the inherent micro-level uncertainty of position of a star/planet might transfer into macroscopic randomness. Thus, the inherent micro-level uncertainty of a body might be an origin of macroscopic randomness of the universe. In addition, from physical viewpoints, orbits of some three-body systems at large time are inherently random, and thus it has no physical meanings to talk about the accurate long-term prediction of the chaotic orbits. Note that such kind of uncertainty and randomness has nothing to do with the ability of human being. All of these might enrich our knowledge and deepen our understandings about not only the three-body problem but also chaos.

nlin.CD↗
arXiv2013

Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps

Invariant circles play an important role as barriers to transport in the dynamics of area-preserving maps. KAM theory guarantees the persistence of some circles for near-integrable maps, but far from the integrable case all circles can be destroyed. A standard method for determining the existence or nonexistence of a circle, Greene's residue criterion, requires the computation of long-period orbits, which can be difficult if the map has no reversing symmetry. We use de la Llave's quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a Diophantine circle conjugate to rigid rotation, and the singularity of a norm of a derivative of the conjugacy to predict criticality. We study near-critical conjugacies for families of rotational invariant circles in generalizations of Chirikov's standard map. A first goal is to obtain evidence to support the long-standing conjecture that when circles breakup they form cantori, as is known for twist maps by Aubry-Mather theory. The location of the largest gaps is compared to the maxima of the potential when anti-integrable theory applies. A second goal is to support the conjecture that locally most robust circles have noble rotation numbers, even when the map is not reversible. We show that relative robustness varies inversely with the discriminant for rotation numbers in quadratic algebraic fields. Finally, we observe that the rotation number of the globally most robust circle generically appears to be a piecewise-constant function in two-parameter families of maps.

nlin.CD↗