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We are considering the hyperbolic C-K systems of Anosov–Kolmogorov which are defined on high dimensional tori and are used to generate pseudorandom numbers for Monte-Carlo simulations. All trajectories of the C-K systems are exponentially unstable and pseudorandom numbers are represented in terms of coordinates of very long chaotic trajectories. The C-K systems on a torus have countable set of everywhere dense periodic trajectories and their distribution play a crucial role in coding and implementation of the pseudorandom number generator. The asymptotic distribution of chaotic trajectories of C-K systems with periods less than a given number is well known in mathematical literature, but a deviation from its asymptotic behavior is unknown. Using analytical and computer calculations, we are studying a distribution function of periodic trajectories and their deviation from asymptotic behavior. The corresponding MIXMAX generator has the best combination of speed, size of the state and is currently available generator.

We investigate the connection between local and global dynamics of two N-degree of freedom Hamiltonian systems with different origins describing one-dimensional nonlinear lattices: The Fermi–Pasta–Ulam (FPU) model and a discretized version of the nonlinear Schrödinger equation related to Bose–Einstein Condensation (BEC). We study solutions starting in the vicinity of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase motion (OPM), which are known in closed form and whose linear stability can be analyzed exactly. Our results verify that as the energy E increases for fixed N, beyond the destabilization threshold of these orbits, all positive Lyapunov exponents L_i, i = 1,…, N - 1, exhibit a transition between two power laws, L_i ∝ E^B_k, B_k > 0, k = 1, 2, occurring at the same value of E. The destabilization energy E_c per particle goes to zero as N → ∞ following a simple power-law, E_c/N ∝ N^-α, with α being 1 or 2 for the cases we studied. However, using SALI, a very efficient indicator we have recently introduced for distinguishing order from chaos, we find that the two Hamiltonians have very different dynamics near their stable SPOs: For example, in the case of the FPU system, as the energy increases for fixed N, the islands of stability around the OPM decrease in size, the orbit destabilizes through period-doubling bifurcation and its eigenvalues move steadily away from -1, while for the BEC model the OPM has islands around it which grow in size before it bifurcates through symmetry breaking, while its real eigenvalues return to +1 at very high energies. Furthermore, the IPM orbit of the BEC Hamiltonian never destabilizes, having finite-size islands around it, even for very high N and E. Still, when calculating Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov exponents decrease following an exponential law and yield extensive Kolmogorov–Sinai entropies per particle h_KS/N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

This paper aims to analyze and understand the irregularity and complexity of earthquake ground motions from the perspective of nonlinear dynamics. Chaotic dynamics theory and chaotic time series analysis are suggested to examine the nonlinear dynamical characteristic of strong earthquake ground motions. Based on the power spectral analysis, principal component analysis and modified false nearest neighbors method, it is illustrated qualitatively that the acceleration time series of earthquake ground motions exhibit chaotic property. Next, the chaotic time series analysis is proposed to calculate quantitatively the nonlinear characteristic parameters of acceleration time histories of near-fault ground motions. Numerical results show that the correlation dimension of these ground motions is fractal dimension. Their Kolmogorov entropy is a limited positive value, and their maximal Lyapunov exponent is larger than 0. It is demonstrated that the strong earthquake ground motions present the chaotic property rather than the pure random signals, and the severe irregularity and complexity of ground motions are the reflection of high nonlinearity of earthquake physical process.

A high-flux circulating fluidized bed (CFB) riser (0.076-m I.D. and 10-m high) was operated in a wide range of operating conditions to study its chaotic dynamics, using FCC catalyst particles (d_p = 67 μm, ρ_p = 1500 kg·m^-3). Local solids concentration fluctuations measured using a reflective-type fiber optic probe were processed to determine chaotic invariants (Kolmogorov entropy and correlation dimension). Radial and axial profiles of the chaotic invariants at different operating conditions show that the core region exhibits higher values of the chaotic invariants than the wall region. Both invariants vary strongly with local mean solids concentration. The transition section of the riser exhibits more complex dynamics while the bottom and top sections exhibit a more uniform macroscopic and less-complex microscopic flow structure. Increasing gas velocity leads to more complex and less predictable solids concentration fluctuations, while increasing solids flux generally lowers complexity and increases predictability. Very high solids flux, however, was observed to increase the entropy.

In this chapter, a number of methods for biomedical signal recognition in specific tasks of medical diagnostics are considered. All of them are united by the need to search non-standard approaches for the solution of practical tasks. It is connected with small distinctions of classes in initial multidimensional space, and with specific properties of the recognizable signals which are difficult for revealing by traditional temporary and frequency methods.

The method of recognition of signals with the expressed irregular component by Shannon conditional entropy parameters is considered. A number of parameters of conditional entropy are investigated on model signals. This allowed choosing an assessment of signal irregularity degree and successfully applying it for heart rhythm abnormalities (atrial fibrillation) detection on the sequence of intervals between heart beats. Also interesting task in the field of medical diagnostics devoted to the recognition of signals on the intensity of their nonlinear properties is considered. We used the approach based on the use of approximated entropy which is closely connected with Kolmogorov’s entropy (K-entropy). The algorithm for the calculation of this characteristic and its main properties and parameters is considered in detail. They can be used for the dynamic irregularities detection connected with nonlinear properties of signals. Examples of the detection of cardiac fibrillation problems using the parameters of heart rhythm nonlinearity and also an anesthesia depth assessment according to the electroencephalogram are given.

The method of electrocardiogram signal recognition according to its spectral description on the base of the multiple discriminant analysis is considered in detail. It is shown that the use of Fischer’s criterion with the introduction of weight functions, allows increasing the accuracy of signal classification. Examples of recognition of dangerous arrhythmias, including detection of ventricular fibrillation of the heart at the early stages of its emergence are given. The technique of dangerous arrhythmias recognition according to the electrocardiogram description in the frequency domain is offered. Ways of the spectral description forming and methods of linear decision functions creation by Fischer’s criterion are considered. The experiments carried out showed high efficiency of the offered algorithms.

A novel photonic integrated device has been designed and evaluated as a transmitter for chaos applications in optical communications. The device consists of a semiconductor laser integrated with a semiconductor optical amplifier, a phase section and a waveguide, forming thus an integrated external cavity laser with adjustable optical feedback and phase. Under diverse operating parameters the device behaves in different modes, from stable operation to limit cycles and hyperchaotic broadband behavior. Chaos data analysis from experimental data is performed in order to quantify the complexity of the experimental reconstructed attractors under the influence of noise. Calculation of the correlation dimension and Kolmogorov entropy are compared with numerical predictions from well-known theoretical models.