Direct observation of chaotic resonances in optical microcavities

Optical microcavities play a significant role in the study of classical and quantum chaos. To date, most experimental explorations of their internal wave dynamics have focused on the properties of their inputs and outputs, without directly interrogating the dynamics and the associated mode patterns inside. As a result, this key information is rarely retrieved with certainty, which significantly restricts the verification and understanding of the actual chaotic motion. Here we demonstrate a simple and robust approach to directly and rapidly map the internal mode patterns in chaotic microcavities. By introducing a local index perturbation through a pump laser, we report a spectral response of optical microcavities that is proportional to the internal field distribution. With this technique, chaotic modes with staggered mode spacings can be distinguished. Consequently, a complete chaos assisted tunneling (CAT) and its time-reversed process are experimentally verified in the optical domain with unprecedented certainty.

Digital model of a pseudo-random number generator based on a continuous chaotic system

It is shown that the choice of the time sampling parameter of the digital model of a continuous dynamic system with chaotic modes based on its dynamics makes it possible to control the characteristics of the output sequence, including avoiding short cycles and periodic behavior modes. On the example of the Lorentz system, the analysis of the law of motion of a chaotic system, linearized in the vicinity of points of stable and unstable equilibrium, is carried out, on the basis of which the parameters of the mathematical model of the generator of pseudo-random numbers are selected. The output sequence of numbers generated in proposed way is subjected to statistical and correlation analysis. Based on the results of the tests carried out, we can say that the obtained pseudo-random sequences based on continuous chaotic systems have statistically random properties and can be used in steganographic and cryptographic systems.

On Dynamics of a Fractional-Order Discrete System with Only One Nonlinear Term and without Fixed Points

Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called “self-excited attractors” has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called “hidden attractors”, which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system.

An Experimental Study of Volcanic Tremor Driven by Magma Wagging

<p> </p><p><br>Volcanic tremor is a feature of most explosive eruptions. Pre-eruptive tremors can be characterized by monotonic increases in the maximum frequency, frequency bandwidth and amplitude that are correlated with increases in gas flux from a volcanic vent. An enigmatic feature of this behavior is that is observed at volcanoes with widely ranging conduit geometries and structures. Accordingly, the ``magma wagging'' model introduced by [1] and extended by [2] hypothesizes an underlying mechanism that is only weakly-sensitive to volcano architecture: Within active volcanic conduits, the flow of gas through a permeable foamy annulus of gas bubbles excites and maintains an oscillation of a central magma column through a well-known Bernoulli effect. Furthermore, this oscillation has spectral properties that evolve depending on annulus thickness and permeability and the total flow of gas. </p><p>In this thesis, we carry out a critical experimental test of the underlying mechanism for excitation. We explore the response of columns with prescribed elastic and linear damping properties to forced air annular airflows. From high-speed video measurements of linear and orbital displacements and time series of accelerometer measurements we characterize and understand the excitation, evolution, and steady-state oscillating behaviors of analog magma columns over a broad range of conditions. Where the time scale for damping is much longer than the natural period of free oscillation, column oscillation is continuously excited by relatively short period Bernoulli modes through a reverse energy cascade. We also identify three distinct classes of wagging: i. rotational modes that confirm predictions for whirling modes by [3]; as well as ii. mixed-mode; and iii. chaotic modes that are extensions to previous studies[1,2]. Our results show that rotational modes are favored for symmetric, and high-intensity forcing. Mixed-mode responses are favored for a symmetric and intermediate intensity forcing. Chaotic modes occur in asymmetric or low intensity forcing. To confirm and better understand our laboratory results and also extend them to conditions beyond what is possible in the laboratory we carry out two-dimensional numerical simulations of our analog experiments.</p><p>Taken together, results from our experimental and numerical studies can be extended to a natural system to make qualitative predictions testable in future studies of pre- and syn-eruptive volcano seismicity. Far before an eruption, the total gas flux is low and magma wags in a chaotic mode no matter what is the spatial distribution of the gas flux. At a pre-eruptive state, as gas flux increases, if the distribution of gas flux is approximately symmetric, we expect a transition to mixed and possibly rotational modes. During an eruption, fragmentation and explosions within the foamy annulus can cause spatial heterogeneity in permeability resulting in non-uniform gas flux that favors chaotic wagging behavior. </p><p>[1] A. M. Jellinek and D. Bercovici. Seismic tremors and magma wagging during explosive volcanism. Nature, 470(7335):522-525, 2011</p><p>[2] D. Bercovici, A. M.  Jellinek, C. Michaut, and D. C. Roman. Volcanic tremors and magma wagging: gas flux interactions and forcing mechanism. Geophys. J.Int., 195(2):1001-1022, 2013</p><p>[3] Y. Liao and D. Bercovici. Magma wagging and whirling: excitation by gas flux. Geophys. J.Int., 215(1):713-735, 2018</p>

Influence of Spatial Inhomogeneity on the Formation of Chaotic Modes at the Self-Organization Process

The Lorentz system of equations, in which gradient terms are taken into account, has been solved numerically. Three fundamentally different modes of evolution are considered. In the first mode, the spatial distribution of the order parameter permanently changes in time, and domains of two types with positive and negative order parameter values are formed. In the second mode, the order parameter distribution is close to the stationary one. Finally, in the third mode, the order parameter is identical over the whole space. The dependences of the average area of domains, their number, and their total area on the time are calculated in the first two cases. In the third case, the contribution of gradient terms completely vanishes, and a classical Lorenz attractor is realized.

INFLUENCE OF THE ß SOLAR WIND PARAMETER ON STATISTICAL CHARACTERISTICS OF THE Ap INDEX IN THE SOLAR ACTIVITY CYCLE

We have studied the effect of the β solar wind parameter (equal to the ratio of the plasma pressure to the magnetic pressure) on statistical characteristics of the Ap index reflecting the triggering behavior of the activity of Earth’s magnetosphere. The trigger effect of the dynamics of magnetospheric activity consists in the abrupt transition from the periodic mode to the chaotic mode in the solar activity cycle. It is shown that cumulative amplitude distribution functions and power spectra of the Ap index of both the periodic and chaotic modes are well approximated by power and exponential functions respectively. At the same time, the indices of power functions and the indices characterizing the slope of the Ap index spectrum differ significantly in magnitude for the periodic and chaotic modes. We have found that Ap nonlinearly depends on β for both the modes of magnetospheric dynamics. The maximum of the Ap index amplitude for periodic modes is observed when β>1; and for chaotic ones, when β<1. In almost every cycle of solar activity, the energy of the Ap index fluctuations of chaotic modes is higher than that of periodic ones. The results indicate intermittency and its associated turbulence of magnetospheric activity. The exponential character of the spectral density of the Ap index suggests that the behavior of magnetospheric activity is determined by its internal dynamics, which can be described by a finite number of deterministic equations. The trigger effect of magnetospheric activity is assumed to be due to the angle of inclination of the axis of the solar magnetic dipole to the ecliptic plane, on which the dynamics of the β parameter in the solar activity cycle depends.