Methods of dynamical systems, harmonic analysis and wavelets applied to several physical systems (original) (raw)

Time-frequency analysis based on wavelets for Hamiltonian systems

2002

In this work, we present the method of time-frequency analysis based on wavelets for Hamiltonian systems and demonstrate its applications and consequences in the general dynamics of higher dimensional systems. By extracting instantaneous frequencies from the wavelet transform of numerical solutions, we can distinguish regular from chaotic motions, and characterize the global structure of the phase space. The method allows us to determine resonance areas that persists even for high energy levels. We can also show how the existence of resonant motion affects the dynamics of the chaotic motion: we detect when chaotic trajectories are temporarily trapped around resonance areas, or undergo transitions between different resonances. This process is a good indicator of intrinsic transport in the phase space. The method can be applied to a large class of systems, since it is not restricted to nearly integrable systems expressed in action-angle variables, which is the traditional framework fo...

A wavelet-based tool for studying non-periodicity

Computers and Mathematics with Applications, 2010

This paper presents a new numerical approach to the study of nonperiodicity in signals, which can complement the maximal Lyapunov exponent method for determining chaos transitions of a given dynamical system. The proposed technique is based on the continuous wavelet transform and the wavelet multiresolution analysis. A new parameter, the scale index, is introduced and interpreted as a measure of the degree of the signal's non-periodicity. This methodology is successfully applied to three classical dynamical systems: the Bonhoeffer-van der Pol oscillator, the logistic map, and the Henon map.

Quasi-periodic oscillations under wavelet microscope: the application of Matching Pursuit algorithm

We zoom in on the internal structure of the low-frequency quasi-periodic oscillation (LF QPO) often observed in black hole binary systems to investigate the physical nature of the lack of coherence in this feature. We show the limitations of standard Fourier power spectral analysis for following the evolution of the QPO with time and instead use wavelet analysis and a new time-frequency technique -Matching Pursuit algorithm -to maximise the resolution with which we can follow the QPO behaviour. We use the LF QPO seen in a very high state of XTE J1550−564 to illustrate these techniques and show that the best description of the QPO is that it is composed of multiple independent oscillations with a distribution of lifetimes but with constant frequency over this duration. This rules out models where there is continual frequency modulation, such as multiple blobs spiralling inwards. Instead it favours models where the QPO is excited by random turbulence in the flow.

Wavelet analysis of chaotic systems

A nonlinear dynamical system (Van der Pol) is analyzed by investigating the behavior of the corresponding wavelet coefficients. The wavelet coefficients are strictly related to local changes, thus giving information on the differential properties of a function. The main features of the dynamical system, such as periodic motion, asymptotic stability, etc., will be given in terms of the wavelet coefficients, eventually depending on the parameter .

Wavelet Analysis of Umbral Oscillations

The Astrophysical Journal, 2003

We study the temporal behavior of the intensity and velocity chromospheric umbral oscillations, applying wavelet analysis techniques to four sets of observations in the H line and one set of simultaneous observations in the H and the nonmagnetic Fe i (5576.099 Å) line. The wavelet and Fourier power spectra of the intensity and the velocity at chromospheric levels show both 3 and 5 minute oscillations. Oscillations in the 5 minute band are prominent in the intensity power spectra; they are significantly reduced in the velocity power spectra. We observe multiple peaks of closely spaced cospatial frequencies in the 3 minute band (5-8 mHz). Typically, there are three oscillating modes present: (1) a major one near 5.5 mHz, (2) a secondary near 6.3 mHz, and (3) oscillations with time-varying frequencies around 7.5 mHz that are present for limited time intervals. In the frame of current theories, the oscillating mode near 5.5 mHz should be considered as a fingerprint of the photospheric resonator, while the other two modes can be better explained by the chromospheric resonator. The wavelet spectra show a dynamic temporal behavior of the 3 minute oscillations. We observed (1) frequency drifts, (2) modes that are stable over a long time and then fade away or split up into two oscillation modes, and (3) suppression of frequencies for short time intervals. This behavior can be explained by the coupling between modes closely spaced in frequency or/and by long-term variations of the driving source of the resonators.

Decomposition of strong nonlinear oscillations via modified continuous wavelet transform

Physical Review E, 2010

We propose the modification of the complex wavelet transform adapted for an analysis of strong nonlinear oscillations with a shape far from sinusoidal. It is based on the rotation of transform modulus obtained via usage of the Morlet wavelet in such a way that higher harmonics are merged with a main one. The method is illustrated by application to the analysis of regular and chaotic oscillations generated by the Rössler system.

Harmonic Analysis in Discrete Dynamical Systems

International Journal of Modern Nonlinear Theory and Application, 2012

In this paper we review several contributions made in the field of discrete dynamical systems, inspired in the harmonic analysis. Within discrete dynamical systems, we focus exclusively on quadratic maps, both one-dimensional (1D) and two-dimensional (2D), since these maps are the most widely used by experimental scientists. We first review the applications in 1D quadratic maps, in particular the harmonics and antiharmonics introduced by Metropolis, Stein and Stein (MSS). The MSS harmonics of