Correlation analysis of dynamical chaos

Classical dissipation and asymptotic equilibrium via interaction with chaotic systems

“…The correlation functions are obtained numerically with a fixed time step symplectic integration algorithm [17] applied to the isolated chaotic system. In our calculations we used time steps of the order of 10 These correlation functions can be well fitted by the expressions tion function is typical of chaotic systems in general [2,16] and also occurs in non-isolated systems subjected to random noise, such as a Brownian particle.…”

Data-driven approach to the estimation of connectivity and time delays in the coupling of interacting neuronal subsystems

“…The lengths of the time series used for the analysis corresponded to 1000 characteristic periods of the chaotic oscillations. The values of the control parameters are chosen in such a way that, in the absence of coupling, each oscillator demonstrates phase-incoherent chaotic oscillations corresponding to the funnel attractor (Anishchenko et al, 2003). Similar to the case considered in (Section 3.1), the chaotic oscillations are characterized by the presence of a broad peak in the power spectrum and might also be considered as a model of oscillations observed in LFPs, whose power spectra frequently contain broad peaks related to the different physiological processes.…”

Statistical properties of dynamical chaos

“…The first important question to be uniquely answered by the experiment is whether a Wiener process approach can be applied to describe the phase of the x(t) process, as assumed in [21,37,39,75]. To define the diffusion coefficient B eff , the instantaneous phase is introduced by using an analytical signal concept and performing the Hilbert transform for experimental realizations x(t) [77].…”

“…In [37,38,39], the correlation and spectral properties of chaotic oscillations are studied for several types of chaotic attractors that can be observed in autonomous differential systems with three-dimensional phase space. Classical models of nonlinear dynamics, such as the Rössler oscillator [40], the Lorenz system [41], and the Anishchenko-Astakhov oscillator that represents a mathematical model of a real radiotechnical device [42,43], were chosen for the studies.…”

“…The aim of this work is to present a brief review of the recent results reported in [21,22,37,38,39]. The presented results concern the method of diagnosing nonhyperbolic chaos, the influence of noise on nonhyperbolic attractors, some probabilistic aspects of chaotic dynamics (such as peculiarities of the relaxation to a stationary probability distiribution), the rate of mixing, and the correlation and spectral analysis of chaotic regimes of different types.…”