Iram Gleria

Boundedness of solutions and Lyapunov functions in quasi-polynomial systems

Abstract In this Letter we establish sufficient conditions for the existence of a Lyapunov function for a large class of non-linear systems, the Quasi-Polynomial systems [Figueredo et al., J. Math. Phys. 39 (1998) 2929; Figueredo et al., Phys. A 262 (1999) 158; Brenig, Phys. Lett. A 133 (1988) 378] . We also present sufficient conditions such that the solutions are bounded and bounded away from zero componentwise.

Wave-packet dynamics in chains with delayed electronic nonlinear response.

We study the dynamics of one electron wave packet in a chain with a nonadiabatic electron-phonon interaction. The electron-phonon coupling is taken into account in the time-dependent Schrödinger equation by a delayed cubic nonlinearity. In the limit of an adiabatic coupling, the self-trapping phenomenon occurs when the nonlinearity parameter exceeds a critical value of the order of the bandwidth. We show that a weaker nonlinearity is required to produce self-trapping in the regime of short delay times. However, this trend is reversed for slow nonlinear responses, resulting in a reentrant phase diagram. In slowly responding media, self-trapping only takes place for very strong nonlinearities.

Autocorrelation as a source of truncated Lévy flights in foreign exchange rates

We suggest that the ultraslow speed of convergence associated with truncated Levy flights (Phys. Rev. Lett. 73 (1994) 2946) may well be explained by autocorrelations in data. We show how a particular type of autocorrelation generates power laws consistent with a truncated Levy flight. Stock exchanges have been suggested to be modeled by a truncated Levy flight (Nature 376 (1995) 46; Physica A 297 (2001) 509; Econom. Bull. 7 (2002) 1). Here foreign exchange rate data are taken instead. Scaling power laws in the “probability of return to the origin” are shown to emerge for most currencies. A novel approach to measure how distant a process is from a Gaussian regime is presented.

Algorithmic complexity theory and the relative efficiency of financial markets

Financial economists usually assess market efficiency in absolute terms. This is to be viewed as a shortcoming. One way of dealing with the relative efficiency of markets is to resort to the efficiency interpretation provided by algorithmic complexity theory. We employ such an approach in order to rank 36 stock exchanges and 20 US dollar exchange rates in terms of their relative efficiency.

Sistemas complexos, criticalidade e leis de potência

Neste texto fazemos um apanhado inicial e geral das principais id¥eias relacionadas a teoria dos sistemas complexos.

Interplay of XPM and nonlinear response time in the modulational instability of copropagating optical pulses

We investigate the modulational instability induced by cross-phase modulation (XPM) of two incoherently coupled optical pulses copropagating in a lossless fiber with a finite nonlinear response time. The non-instantaneous character of the nonlinear response is introduced through a Debye relaxation process. We analytically obtain the exact dispersion relation for weak harmonic perturbations over the stationary solution. We show that the instability spectrum, present in both normal and anomalous dispersive regimes in instantaneously responding Kerr media, develops a double peak structure whose relative strength and typical frequency range depend on the response time. Further, we reveal that there are two unstable modes in the entire frequency spectrum. We report the dependence of the maximum gain and central frequency within each unstable mode as a function of the group velocity mismatch and response time, showing the crossover between the regimes of fast and slow nonlinear responses.

On the Origins of Truncated Levy Flights

Abstract We show that truncated Levy flights appear due to the presence of particular features of autocorrelation in data. We present and analyze ‘physical’ reasons sufficient to ensure the scaling power laws and sluggish convergence associated with truncated Levy flights. Our approach is exemplified with currency data for the British pound and Chinese yuan against the US dollar. We further compare these examples with a simulated Lorentzian distribution.

Multifractality and heteroscedastic dynamics: An application to time series analysis

An increasingly important problem in physics concerns scale invariance symmetry in diverse complex systems, often characterized by heteroscedastic dynamics. We investigate the nature of the relationship between the heteroscedastic and fractal aspects of the dynamics of complex systems, by analyzing the sensitivity to heteroscedasticity of the scaling properties of weakly nonstationary time series. By using multifractal detrended fluctuation analysis, we study the singularity spectra of currency exchange rate fluctuations, after partially or completely eliminating n-point correlations via data shuffling techniques. We conclude that heteroscedasticity can significantly increase multifractality and interpret these findings in the context of self-organizing and adaptive complex systems.

A novel approach for the stability problem in non-linear dynamical systems

We present a methodology for the determination of sufficient conditions for the existence of a Lyapunov function in a general class of non-linear dynamical systems. The algorithm can be applied in the cases where the system parameters are numerically specified or not. The numerical algorithm involves the resolution of a linear programming problem. The algebraic version is implemented using the MAPLE programming system in the package Lyapunov.

Fat Tails, Long-Range Correlations and Multifractality as Emergent Properties in Nonstationary Time Series

An important open problem concerns the physical origin of long-range correlations, multifractality and fat-tailed distributions observed in heteroscedastic time series associated with complex systems. Financial stylized facts provides one useful example usually not explained by traditional economic models. We investigate the behavior of an agent-based model consisting of N agents which interact with each other via fixed rules. We show that fat-tailed distributions, long-range correlations, heteroscedasticity and multifractality arise as N becomes large. Our findings suggest that such stylized facts can in principle arise as emergent properties.