Complex Oscillations Research Articles

Chaotic behavior of quantum cascade lasers at ignition

The ignition of Quantum Cascade Lasers can occur from a state of oscillating field domains. Here, the interplay between lasing and the kinetics of traveling domain boundaries provides complex oscillation scenarios. We analyze our numerical findings in detail for a device operating at terahertz frequencies and manifest chaotic evolution by positive Lyapunov exponents. This shows that these important devices can exhibit chaotic behavior even without periodic driving, which needs to be taken into account in their design.

Classification of Epilepsy Types from Electroencephalogram Time Series Using Continuous Wavelet Transform Scalogram–Based Convolutional Neural Network

Abstract During the supervisory activities of the brain, the electrical activities of nerve cell clusters produce oscillations. These complex biopotential oscillations are called electroencephalogram (EEG) signals. Certain diseases, such as epilepsy, can be detected by measuring these signals. Epilepsy is a disease that manifests itself as seizures. These seizures manifest themselves in different characteristics. These different characteristics divide epilepsy seizure types into two main groups: generalized and partial epilepsy. This study aimed to classify different types of epilepsy from EEG signals. For this purpose, a scalogram-based, deep learning approach has been developed. The utilized classification process had the following main steps: the scalogram images were obtained by using the continuous wavelet transform (CWT) method. So, a one-dimension EEG time series was converted to a two-dimensional time-frequency data set in order to extract more features. Then, the increased dimension data set (CWT scalogram images) was applied to the convolutional neural network (CNN) as input patterns for classifying the images. The EEG signals were taken from Dicle University, Neurology Clinic of Medical School. This data consisted of four classes: healthy brain waves, generalized preseizure, generalized seizure, and partial epilepsy brain waves. With the proposed method, the average accuracy performance of three of the EEG records’ classes (healthy, generalized preseizure, and generalized seizure), and that of all four classes of EEG records were 90.16 % (± 0.20) and 84.66 % (± 0.48). According to these results, regarding the specific accuracy ratings of the recordings, the healthy EEG records scored 91.29 %, generalized epileptic seizure records were at 96.50 %, partial seizure EEG records scored 89.63 %, and the preseizure EEG records had a 90.44 % rating. The results of the proposed method were compared to the results of both similar studies and conventional methods. As a result, the performance of the proposed method was found to be acceptable.

Quantum dynamics of a single fluxon in Josephson-junction parallel arrays with large kinetic inductances

We present a theoretical study of coherent quantum dynamics of a single magnetic fluxon (MF) trapped in Josephson junction parallel arrays (JJPAs) with large kinetic inductances. The MF is the topological excitation carrying one quantum of magnetic flux, $\Phi_0$. The MF is quantitatively described as the $2\pi$-kink in the distribution of Josephson phases, and for JJPAs with high kinetic inductances the characteristic length of such distribution ("the size" of MF) is drastically reduced. Characterizing such MFs by the Josephson phases of three consecutive Josephson junctions we analyse the various coherent macroscopic quantum effects in the MF quantum dynamics. In particular, we obtain the MF energy band originating from the coherent quantum tunnelling of a single MF between adjacent cells of JJPAs. The dependencies of the band width $\Delta$ on the Josephson coupling energy $E_J$, charging energy $E_C$ and the inductive energy of a cell $E_L$, are studied in detail. In long linear JJPAs the coherent quantum dynamics of MF demonstrates decaying quantum oscillations with characteristic frequency $f_{qb}=\Delta/h$. In short annular JJPAs the coherent quantum dynamics of MF displays complex oscillations controlled by the Aharonov-Casher phase $\chi \propto V_g$, where $V_g$ is an externally applied gate voltage. In the presence of externally applied dc bias, $I$, a weakly incoherent dynamics of quantum MF is realized in the form of macroscopic Bloch oscillations leading to a typical "nose" current-voltage characteristics of JJPAs. As ac current with frequency $f$ is applied the current-voltage characteristics displays a set of equidistant current steps at $I_n=2en f$.

Chaos and localization in the discrete nonlinear Schrödinger equation

We analyze the chaotic dynamics of a one-dimensional discrete nonlinear Schr\"odinger equation. This nonintegrable model, ubiquitous in several fields of physics, describes the behavior of an array of coupled complex oscillators with a local nonlinear potential. We explore the Lyapunov spectrum for different values of the energy density, finding that the maximal value of the Kolmogorov-Sinai entropy is attained at infinite temperatures. Moreover, we revisit the dynamical freezing of relaxation to equilibrium, occurring when large localized states (discrete breathers) are superposed to a generic finite-temperature background. We show that the localized excitations induce a number of very small, yet not vanishing, Lyapunov exponents, which signal the presence of extremely long characteristic time-scales. We widen our analysis by computing the related Lyapunov covariant vectors, to investigate the interaction of a single breather with the various degrees of freedom.

Modified Morris–Lecar neuron model: effects of very low frequency electric fields and of magnetic fields on the local and network dynamics of an excitable media

A Morris–Lecar neuron model is considered with electric and magnetic field effects where the electric field is a time-varying sinusoid and magnetic field is simulated using an exponential flux memristor. We have shown that the exposure to electric and magnetic fields have significant effects on the neurons and has exhibited complex oscillations. The neurons exhibit a frequency-locked state for the periodic electric field and different ratios of frequency-locked states with respect to the electric field frequency is also presented. To show the impact of the electric and magnetic fields on network of neurons, we have constructed different types of network and have shown the network wave propagation phenomenon. Interestingly, the nodes exposed to both electric and magnetic fields exhibit more stable spiral waves compared to the nodes exhibited only to the magnetic fields. Also, when the number of layers is increased, the range of electric field frequency for which the layers exhibit spiral waves also increases.

Dynamics of a Bistable Current-Controlled Locally-Active Memristor

This paper presents a novel current-controlled locally-active memristor model to reveal the switching and oscillating characteristics of locally-active devices. It is shown that the memristor has two asymptotically stable equilibrium points on its power-off plot and therefore exhibits nonvolatility. Switching from one stable equilibrium point to another is achieved by applying a suitable current pulse. The locally-active characteristic of the memristor is measured by the DC [Formula: see text]–[Formula: see text] plot. A small-signal equivalent circuit on a locally-active operating point with the bias current [Formula: see text] is constructed for describing the characteristic of the locally-active region of the memristor. A periodic oscillator circuit composed of the locally-active memristor, a compensation inductor and a resistor is proposed, whose dynamics is analyzed in detail by using the Hopf bifurcation and the zeros and poles of the impendence function of the circuit. It is found that the locally-active memristor based circuit with different current biases or different initial conditions can exhibit different dynamics such as periodic oscillation and stable equilibrium point. If an energy storage element (capacitor) is added to the periodic oscillation circuit, a chaotic oscillator is obtained, which can exhibit abundant dynamics. The oscillation mechanism of the memristor-based oscillator is analyzed via dynamic route map (DRM), showing that the memristor is an essential device for generating periodic and chaotic oscillations, and its local activity is the cause for complex oscillations.

Overlapping Adding-Doubling Spikes Cascades in a Semiconductor Laser Proxy

A study of the distribution of spikes in periodic oscillations is reported for an optically injected laser diode proxy, as a function of the injected field strength and detuning. A novel and unexpected feature reported here is an abundance of overlapping adding-doubling complexification cascades. Two-parameter spikes complexification cascades of the laser proxy are found to mimic phenomena also found in a state-of-the-art semiconductor laser model. Such cascades should not be difficult to observe experimentally, either in lasers or in other complex oscillators.

Spatiotemporal linear instability analysis of collective neutrino flavor conversion in four-dimensional spacetime

In an environment with high-density neutrinos formed in a core-collapse supernova (CCSN), the neutrinos exhibit nonlinear and complex oscillation behaviors due to their self-interactions. The onset of this nonlinear oscillation can be investigated by linearizing the evolution equation for small perturbations around the flavor eigenstates. While the condition under which the flavor eigenstates are unstable has been investigated in many studies, how the perturbations evolve in spacetime has yet to be elucidated. In this paper, we analytically and correctly derive the asymptotic behaviors of the linear perturbations in 4-dimensional spacetime in the linear regime for a 2-beam neutrino model using the recently proposed Lefschetz thimble formulation. The result suggests that the perturbations grow in the directions between the two neutrino beams. We also briefly discuss the possible effects of neutrino flavor conversion on the explosion mechanism of a CCSN. In particular, the result implies that the flavor instability in the preshock region may propagate into the postshock region, contrary to the previous study focusing on the group velocity in 1-dimensional space. How to treat the case of a more realistic continuous spectrum is also discussed.

The Refractory Period Matters: Unifying Mechanisms of Macroscopic Brain Waves

The relationship between complex brain oscillations and the dynamics of individual neurons is poorly understood. Here we utilize maximum caliber, a dynamical inference principle, to build a minimal yet general model of the collective (mean field) dynamics of large populations of neurons. In agreement with previous experimental observations, we describe a simple, testable mechanism, involving only a single type of neuron, by which many of these complex oscillatory patterns may emerge. Our model predicts that the refractory period of neurons, which has often been neglected, is essential for these behaviors.

Enhanced multiresolution wavelet analysis of complex dynamics in nonlinear systems.

Multiresolution wavelet analysis (MWA) is a powerful data processing tool that provides a characterization of complex signals over multiple time scales. Typically, the standard deviations of wavelet coefficients are computed depending on the resolution level and such quantities are used as measures for diagnosing different types of system behavior. To enhance the capabilities of this tool, we propose a combination of MWA with detrended fluctuation analysis (DFA) of detail wavelet coefficients. We find that such an MWA&DFA approach is capable of revealing the correlation features of wavelet coefficients in independent ranges of scales, which provide more information about the complex organization of datasets compared to variances or similar statistical measures of the standard MWA. Using this approach, we consider changes in the dynamics of coupled chaotic systems caused by transitions between different types of complex oscillations. We also demonstrate the potential of the MWA&DFA method for characterizing different physiological conditions by analyzing the electrical brain activity in mice.

Особенности сложных колебаний гибких микрополярных сетчатых панелей

In this paper, a mathematical model of complex oscillations of a flexible micropolar cylindrical mesh structure is constructed. Equations are written in displacements. Geometric nonlinearity is taken into account according to the Theodore von Karman model. A non-classical continual model of a panel based on a Cosserat medium with constrained particle rotation (pseudocontinuum) is considered. It is assumed that the fields of displacements and rotations are not independent. An additional independent material parameter of length associated with a symmetric tensor by a rotation gradient is introduced into consideration. The equations of motion of a panel element, the boundary and initial conditions are obtained from the Ostrogradsky – Hamilton variational principle based on the Kirchhoff – Love’s kinematic hypotheses. It is assumed that the cylindrical panel consists of n families of edges of the same material, each of which is characterized by an inclination angle relative to the positive direction of the axis directed along the length of the panel and the distance between adjacent edges. The material is isotropic, elastic and obeys Hooke’s law. To homogenize the rib system over the panel surface, the G. I. Pshenichnov continuous model is used. The dissipative mechanical system is considered. The differential problem in partial derivatives is reduced to an ordinary differential problem with respect to spatial coordinates by the Bubnov – Galerkin method in higher approximations. The Cauchy problem is solved by the Runge – Kutta method of the 4th order of accuracy. Using the establishment method, a study of grid geometry influence and taking account of micropolar theory on the behavior of a grid plate consisting of two families of mutually perpendicular edges was conducted.

Complex oscillations in the Belousov-Zhabotinsky batch reaction with methylmalonic acid and manganese(ii).

The Belousov–Zhabotinsky (BZ) oscillating chemical reaction involves the oxidation of an organic compound by the bromate ion in the presence of a metal ion catalyst such as cerium(iv), manganese(ii), or ferroin. Simple periodic oscillations are generally obtained for the BZ reaction in a batch (closed) system. However, complex oscillations have been observed for the BZ reaction in batch with malonic acid and either cerium or ferroin ions as the catalyst. We report here that fascinating complex oscillations in the potential of a Pt electrode have been found in the batch BZ reaction with methylmalonic acid (MeMA) and manganese(ii). Relatively high initial concentrations of NaBrO3 and MeMA are required, and the [NaBrO3]0/[MeMA]0 ratio is the main factor determining the type of oscillations obtained. Other relevant factors are [NaBr]0, [MnSO4]0, [H2SO4]0 or [NaOH]0, temperature, and stirring rate. Complex phenomena observed include mixed mode oscillations, birhythmicity, quasiperiodicity, bursting, and possible chaos. A mechanism is proposed involving the reversible formation of a manganese(iii) complex with bromomethylmalonic acid followed by two-electron oxidation to methyltartronic acid and Mn2+.

ПОБУДОВА І АНАЛІЗ КОМП'ЮТЕРНИХ МОДЕЛЕЙ РЕЗОНАТОРІВ ЄМНОСТЕЙ ТЕХНОЛОГІЧНИХ АПАРАТІВ ВІЗУАЛЬНИМИ ЗАСОБАМИ PYTHON

Annotation. The study of nonlinear oscillations and processes in apparatuses that occur under the influence of complex oscillations presents significant mathematical difficulties. The arising strong nonlinear oscillations can significantly intensify technological processes or cause the destruction of structural elements. Therefore, the problem of using the vibrational energy arising in technological devices naturally matured for a long time both for equipment designers and production technologists. A computer model based on a differential equation for determining the frequencies and forms of bending vibrations of a tubular resonator is proposed. The use of the model makes it possible to visualize the modes and frequency of oscillations for a resonator in the form of a cylindrical part of a technological apparatus of any size. This takes into account the thickness of the walls, the outer and inner diameter of the vessel of the apparatus, and its length. The model takes into account the type of tank fastening with variation in the support stiffness. A distinctive feature of the resulting model is that for the first time an approach was used to solve the differential equation of capacity not by obtaining a numerical solution, but an approach was used that includes obtaining an analytical expression for each waveform with subsequent visualization using Python.

Free Response Approach for the Second Order Partial Differential System with Time Periodic Coefficient

In this paper, a new method to approach the free response of the second order partial differential system with time periodic coefficient is presented with a special form of a trigonometric series with mode functions, and it can be expressed in a closed-form solution. As a result that the partial differential equation with time periodic coefficient can be transformed into a linear algebra equations, from which a characteristic equation can be obtained. Then, complex oscillation frequency and all harmonic coefficients can be computed. All arbitrary constants in a general solution can be obtained with initial conditions. Investigations show that the proposed approach not only computes free vibration with multiple modes with higher accuracy, but also it is suitable to analyze vibration instability. So this study is valuable for vibration analysis in an elastic continuum with time periodic coefficient.

Multi-source fidelity sparse representation via convex optimization for gearbox compound fault diagnosis

Industrial automatic control systems have high requirements for manufacturing accuracy, which are often adversely affected by the compound fault of rotating machinery such as gearboxes. Compound fault diagnosis has many challenges because of its many types of faults, complex oscillation characteristics, and mutual interference between various vibration sources. Therefore, it is urgently required for the development of a method which can accurately detect gearbox complex multi-source faults. To address the compound fault problem, a novel multi-source fidelity sparse representation method is proposed, which can accurately realize multiple fault diagnosis of the gearbox without the prior knowledge regarding the number of fault sources. Moreover, to ensure the accuracy of signal reconstruction, the gearbox compound failure mechanism is analyzed, from which the sparse dictionaries are established. The multi-source penalty function is constructed to improve the fidelity of the signal and the convexity condition of the objective function is further discussed for the global minimum. Simulations and engineering signals are used to verify the versatility of the proposed method.

Falling off a limit cycle using phase-agnostic stimuli: Definitions and conceptual framework.

Nearly a half-century of biomedical research has revealed methods and mechanisms by which an oscillator with bistable limit cycle kinetics can be stopped using critical stimuli applied at a specific phase. Is it possible to construct a stimulus that stops oscillation regardless of the phase at which the stimulus is applied? Using a radial isochron clock model, we demonstrate the existence of such stimulus waveforms, which can take on highly complex shapes but with a surprisingly simple mechanism of rhythm suppression. The perturbation, initiated at any phase of the limit cycle, first corrals the oscillator to a narrow range of new phases, then drives the oscillator to its phase singularity. We further constructed a library of waveforms having different durations, each achieving phase-agnostic suppression of rhythm but with varying rates of phase corralling prior to amplitude suppression. The optimal stimulus energy to achieve phase-agnostic suppression of rhythm is dependent on the rate of phase corralling and the configuration of the phaseless set. We speculate that these results are generic and suggest the existence of stimulus waveforms that can stop the rhythm of more complex oscillators irrespective of the applied phase.

Neural mass modeling of slow-fast dynamics of seizure initiation and abortion.

Epilepsy is a dynamic and complex neurological disease affecting about 1% of the worldwide population, among which 30% of the patients are drug-resistant. Epilepsy is characterized by recurrent episodes of paroxysmal neural discharges (the so-called seizures), which manifest themselves through a large-amplitude rhythmic activity observed in depth-EEG recordings, in particular in local field potentials (LFPs). The signature characterizing the transition to seizures involves complex oscillatory patterns, which could serve as a marker to prevent seizure initiation by triggering appropriate therapeutic neurostimulation methods. To investigate such protocols, neurophysiological lumped-parameter models at the mesoscopic scale, namely neural mass models, are powerful tools that not only mimic the LFP signals but also give insights on the neural mechanisms related to different stages of seizures. Here, we analyze the multiple time-scale dynamics of a neural mass model and explain the underlying structure of the complex oscillations observed before seizure initiation. We investigate population-specific effects of the stimulation and the dependence of stimulation parameters on synaptic timescales. In particular, we show that intermediate stimulation frequencies (>20 Hz) can abort seizures if the timescale difference is pronounced. Those results have the potential in the design of therapeutic brain stimulation protocols based on the neurophysiological properties of tissue.

On the Complex Oscillation of Meromorphic Solutions of Nonhomogeneous Linear Differential Equations with Meromorphic Coefficients

In this paper, we study the higher order differential equation f k + B f = H , where B is a rational function, having a pole at ∞ of order n > 0 , and H ≡ 0 is a meromorphic function with finite order, and obtain some properties related to the order and zeros of its meromorphic solutions.

Self-organizations with fast-slow time scale in a memristor-based Shinriki’s circuit

The structure of complex oscillations with two time scales generated by a novel memristor-based Shinriki’s circuit is studied in detail. Employing the stability phase diagrams, we identify the parameter regions of mixed-mode oscillations (MMOs) produced by each state variable, and classify the fast and slow properties of state variables according to the characteristics of these MMOs. Additionally, we exhibit the self-organizations of phases and describe how they are interconnected and how their complexity unfolds with control parameters varying. For several parameter combinations, in addition to recently reported shrimp-shaped cascades and “eye” of chaos, two kinds of interesting self-organizations, the shrimp cascade covered by four distinct numbers of peaks and “eye” of anti-chaos, are also discovered. A significant feature of these structures indicates that the fast-slow dynamics of the system depend not only on the state variables but also strongly on the parameter interval. The results of theoretical analyses obtained are verified by the Multisim simulation experiments. This paper will contribute to understanding the profound dynamical behaviors of the memristor-based circuit and open the possibility of studying hitherto unobserved phenomena.

A Practical Guide to Resonance Frequency Assessment for Heart Rate Variability Biofeedback.

Heart rate variability (HRV) represents fluctuations in the time intervals between successive heartbeats, which are termed interbeat intervals. HRV is an emergent property of complex cardiac-brain interactions and non-linear autonomic nervous system (ANS) processes. A healthy heart is not a metronome because it exhibits complex non-linear oscillations characterized by mathematical chaos. HRV biofeedback displays both heart rate and frequently, respiration, to individuals who can then adjust their physiology to improve affective, cognitive, and cardiovascular functioning. The central premise of the HRV biofeedback resonance frequency model is that the adult cardiorespiratory system has a fixed resonance frequency. Stimulation at rates near the resonance frequency produces large-amplitude blood pressure oscillations that can increase baroreflex sensitivity over time. The authors explain the rationale for the resonance frequency model and provide detailed instructions on how to monitor and assess the resonance frequency. They caution that patterns of physiological change must be compared across several breathing rates to evaluate candidate resonance frequencies. They describe how to fine-tune the resonance frequency following an initial assessment. Furthermore, the authors critically assess the minimum epochs required to measure key HRV indices, resonance frequency test-retest reliability, and whether rhythmic skeletal muscle tension can replace slow paced breathing in resonance frequency assessment.