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The work is devoted to the application of various nonlinear dynamics methods to identify interactions in short noisy time series extracted from biological rhythms related to the respiratory, cardiovascular and nervous systems. These interactions are considered as transitions to synchronized states, as the coupling directionality and the delay time in the influence of one system on another. For the analyzed data, the relationship between these rhythms is due to the fact that the neurons of medulla are related to the brain response to changes in blood pressure and respiration during external pain stimulation. Phase synchronization between the variability of blood pressure and respiratory fluctuations in response to these stimuli was established in half of the analyzed time series using methods for assessing phase synchronization based on finding instantaneous phases by the synchrosqueezed wavelet transform and based on calculating the recurrences of phase trajectories. For unsynchronized time series, a predominantly unidirectional influence of respiratory rhythm fluctuations on the blood pressure variability was obtained by the method of phase dynamics modeling and the method of estimating recurrence probabilities. For two third of the data, a unidirectional influence of the blood pressure variability on the neuronal variability was determined, and the remaining data were characterized by a bidirectional relationship between the neuronal and blood pressure variability. Delay time estimation by phase dynamics modeling showed reduced time of influence of the blood pressure variability on the neuronal variability in bidirectional coupling compared to the influence of the neuronal variability on the blood pressure variability. [ABSTRACT FROM AUTHOR]

Microelectromechanical systems (MEMS) based oscillators have been proposed for use in coupled arrays for many applications, including neuromorphic computing, clocks with reduced phase noise, and sensors. In many prior studies, the models for coupled oscillator behavior depended upon the phase dynamics of the individual oscillators, while the amplitude dynamics were assumed to be negligible. However, implementation of practical large-scale MEMS oscillator networks may involve systems with significant amplitude dynamics. This cannot be described by the commonly used phase models or by steady-state models that do not predict transient behavior. We present the design and analysis of a MEMS-Colpitts oscillator topology suitable for monolithic integration (resonators with integrated circuits). Applying multiple scales analysis to the dynamics results in significantly faster computation as compared to direct integration and enables the study of nonlinear behaviors (e.g. bifurcations) and both amplitude and phase dynamics. The strong correlation between the analytical and experimental results demonstrates the effectiveness of our modelling approach. [ABSTRACT FROM AUTHOR]

We propose a general method for optimizing periodic input waveforms for global entrainment of weakly forced limit-cycle oscillators based on phase reduction and nonlinear programming. We derive averaged phase dynamics from the mathematical model of a limit-cycle oscillator driven by a weak periodic input and optimize the Fourier coefficients of the input waveform to maximize prescribed objective functions. In contrast to the optimization methods that rely on the calculus of variations, the proposed method can be applied to a wider class of optimization problems including global entrainment objectives. As an illustration, we consider two optimization problems, one for achieving fast global convergence of the oscillator to the entrained state and the other for realizing prescribed global phase distributions in a population of identical uncoupled noisy oscillators. We show that the proposed method can successfully yield optimal input waveforms to realize the desired states in both cases. [ABSTRACT FROM AUTHOR]

We investigate theoretical aspects of a reduced-order model for transverse galloping, which is derived directly from the full governing equations, i.e., the fluid mass and momentum balances, the fluid far-field boundary condition, the kinematic and dynamic conditions at the interface, and the bluff body equation of motion. We show that the presence of low-intensity turbulence and back-action from the slow transverse oscillations of the bluff body yield a correction to the hydrodynamic force of the quasi-steady theory in the form of additive random excitation. The reduced-order model consists of a pair of nonlinear Langevin equations for the amplitude and phase of the transverse motion of the bluff body. We show that while the phase dynamics is associated with a strongly diffusive random walk motion, the amplitude dynamics is associated with a relatively weak diffusion and can be mapped onto the motion of an overdamped particle trapped in a potential well. This mapping provides a highly useful tool for understanding both the deterministic (no random excitation) and the stochastic (weak random excitation) amplitude dynamics, and hence, for yielding theoretical insights on the overall system dynamics. [ABSTRACT FROM AUTHOR]

Based on the generalized Darboux transformation, the underlying propagation mechanism of localized waves is systematically studied. More specifically, the variable-coefficient coupled nonlinear Schrödinger (NLS) equation is used to accurately describe the dispersion compensation and lumped amplification properties in an inhomogeneous optical fiber. Based on the Lax pair and the seed solutions, the expressions of both the first- and second-order localized wave solutions are calculated. Then, by performing numerical simulations, the evolutionary plots of the interaction of rogue waves with bright-dark solitons and breathers are obtained, and their dynamical characteristics are further analyzed. From the acquired results, it is found that the values of β z and γ z have a direct influence on the propagation shape of the localized waves. Our work provides valuable insights into the dynamical characteristics of the localized waves that the variable-coefficient equations could describe to a certain extent. [ABSTRACT FROM AUTHOR]

This investigation takes an analytical approach to the oscillations of a cantilevered plate immersed in a hypersonic flow with shock impingement. In particular, we derive a mathematical model keyed in on the fact that (1) the shock impingement point moves along the structure as it oscillates and (2) the local curvature of the structure changes the shock reflection angle and thus the compressible flow properties. For cantilever boundary conditions, large motion at the free end my render both effects significant. We show that the movement of the shock impingement point varies approximately with the cotangent of the oblique shock angle and this implicit relationship generates surprisingly strong nonlinear effects in the equation governing the first mode of vibration. A new, geometrically modified third-order Piston Theory is adapted to accurately capture structural curvature induced changes in flow properties as well as possible expansion wave interaction. Our model takes the form of a nonlinearly damped Duffing oscillator with quadratic and cubic nonlinearities stemming from the nonlinear aerodynamic generalized forces as well as geometric and inertial structural nonlinearities. Despite inviscid flow modeling, a perturbation solution to the governing equation shows good agreement with structural oscillations predicted by high-fidelity turbulent computational fluid dynamic simulations. [ABSTRACT FROM AUTHOR]

In this work, stability analysis of plane wave solutions of the cubic-quintic complex Ginzburg–Landau equation for isotropic and anisotropic systems is carried out. In this regard, we perform extensive numerical simulations based on a two-dimensional spatial Fourier discretization and an explicit scheme for temporal differentiation to find the domain of existence of the space-time dynamical behavior of the two-dimensional complex Ginzburg–Landau equation with cubic and quintic nonlinearities. One of the nonlinearity parameters is fixed and the others are varied one by one to determine the regimes in which plane wave solutions exist as stable/instable structures. Moreover, the stability criterion which has been plotted with the state diagram and the different dynamic structures obtained in space parameters has been established. Energy function was also used to characterize spatiotemporal dynamics observed in our system. By performing long simulations for the different parameters of the equation, we found the existence of stable (plane wave, localized defect), intermittent (intermittency state), and unstable (bichaos, phase turbulence, and defect turbulence) structures. [ABSTRACT FROM AUTHOR]

Synchronization is an important phenomenon occurring in many complex systems, such as neural networks, power grids, or social networks. Various methods have been proposed to measure synchronization, each with its advantages and limitations. In this work, we introduce the Poincaré sphere as a generalized measure of synchronization, which provides a useful tool to visualize the similarities between oscillations. We propose the notion of "total synchronization," which considers amplitude, frequency, and phase synchronization, all of which can be measured and visualized simultaneously on the Poincaré sphere. We show that visualizing oscillatory trajectories on the Poincaré sphere has advantages over the more typical two-dimensional representation, particularly in cases where a projection onto a two-dimensional space cannot capture the similarities between different oscillations. Furthermore, we apply this new synchronization measurement to linear and nonlinear oscillators and provide two concrete examples of its application in visualizing phase jitter in noisy oscillations and quantizing solutions of oscillator-based Ising machines. Our work demonstrates the usefulness of the Poincaré sphere as a generalized measure of synchronization and provides a new perspective on studying synchronization in complex systems. [ABSTRACT FROM AUTHOR]

In this paper, we aim at taking a step towards exploring the unstable dynamics of Peregrine soliton as a prototype of rogue waves. Given its spatio-temporal doubly localized structure, the generic Fourier-spectral scheme fails to realize the dimensionality reduction of its nonlinear dynamics. We introduce a modal expansion technique in which physical properties of wave structure are mapped onto proper orthogonal modal basis with space-independent. As a result, the evolution of Peregrine soliton is reduced to an equivalent particle motion in 3-dimension phase space. The govern equation obtained can not only capture correctly underlying behaviors as well as transient natures, but also reveal nicely a complex and subtle structure of the unstable process within the phase-space representation. High consistency with the reconstructed field and the theoretical solution validate the reliability and feasibility of the approach. [ABSTRACT FROM AUTHOR]

Driven by the growing significance of information security, the field of image encryption is attracting increasing interest. This work solves an image encryption issue of experiencing actual chaotic mapping via an optical neural network, extending our knowledge of image encryption based on chaos theory. In this paper, we first construct an optical neural network that is a four-channel signal injection optical reservoir computing (ORC) system relied on an optical dynamics system of the mutually coupled quantum dot spin-polarized vertical cavity surface emitting lasers (MCQD spin-VCSELs) subject to external optical injection, the dynamics of which is analyzed. Then, a secure and general image encryption scheme based on the ORC system is newly developed. The encryption process first requires the image to undergo an ORC-based actual chaotic mapping that achieves a transformation from the original image domain to the chaos domain. Then, one important role of the ORC system outputs is to create secret keys related to plain image combining with chaotic keys derived from the optical chaotic system, and the other one is to act as a transitional medium, being scrambled and diffused by the obtained plaintext-related keys, which further increase the security of our scheme, and finally obtaining the cipher-image. The simulation and analysis results demonstrate that the proposed image encryption scheme is effective and highly secure. Moreover, this study provides an important practical implication that brings the inspiration for future similar research into information security and will help other researchers design new encryption techniques based on chaos theory using neural networks. [ABSTRACT FROM AUTHOR]

This paper aims to present a trilinear neural network method on the basis of the trilinear form of a (3+1)-dimensional p ¯ -GBS equation. This method can be used to construct new exact traveling wave solutions. We set the hidden neurons in three types of tensor functions to some specific functions, and reach a class of periodic bright–dark soliton solutions, two types of breather-like wave solutions and three types of rogue wave solutions for p ¯ -GBS equation successfully. The 3-D and density graphs of those solutions obtained are given to interpret the dynamic characteristics. [ABSTRACT FROM AUTHOR]

Wind tunnel experiments are performed on a pitch-plunge aeroelastic system possessing structural nonlinearity in the stiffness and subjected to dynamic stall conditions. With an increase in the flow speed (U), we observe a well-expected transition from decaying responses to limit cycle oscillations wherein the flutter frequencies of pitch and plunge coalesce perfectly close to the pitch natural frequency, indicating the onset of stall flutter. Interestingly, a further increase in the U yields a secondary frequency peak, emerging approximately at half of the primary frequency peak, accompanied by an intermittent period-1–period-2 (P1–P2) behaviour. Another transition is observed as the flow speed is increased further—the pitch-plunge motion starts exhibiting a beat-like response. We demonstrate that this particular behaviour is '2:1 internal resonance (IR)' and is attributed to a specific combination of structural parameters: the ratio ( ω ¯ ) of natural frequencies of plunge and pitch ( f y and f α , respectively) being 0.44, and the structure possessing quadratic nonlinearity, resulting as the aerodynamic loads tune the flutter frequencies to become commensurate for a 2:1 IR. We show that a typical phase synchronization analysis is inadequate to explain the atypical transitions. To that end, we resort to a frequency-specific synchronization analysis wherein the pitch and plunge time signals are decomposed into two signals of specific frequency range. This reveals that the higher frequency signal (HFS) is always perfectly synchronized. On contrary, the lower frequency signal (LFS) is asynchronous during the intermittent regime and has strong synchrony at the onset of beats. The physical insights presented in this study can be useful for the design of increasingly slender aeroelastic structures that are often subjected to dynamic stall. [ABSTRACT FROM AUTHOR]

Many vibrating systems, over some ranges of parameter values, exhibit a single unstable mode. Adding a small resonant secondary system to the unstable system is a well-known stabilization strategy. Here we show that even a nonresonant secondary system, if equipped with a limit cycle of its own, can stabilize the unstable mode of the primary system. The primary system is modeled here as a linear spring block system with negative damping. The secondary system is a van der Pol oscillator. Smallness of the latter's parameters allows use of the method of multiple scales. The resulting slow amplitude equations decouple from the phases and a two-dimensional system is obtained. The secondary system's amplitude evolves faster than that of the primary system, which simplifies analysis. A parameter-dependent transformation casts the system in a canonical form with a single free parameter c 1 > 0 in addition to the small perturbation parameter. The canonical phase portrait involves two key straight lines. When c 1 < 4 these lines intersect and a separatrix passes through that intersection. Solutions on one side of the separatrix show quenching of the primary instability with limit cycle oscillation of the secondary system. Solutions on the other side of the separatrix show significant oscillations of the primary system at its natural frequency, with the secondary limit cycle being quenched. When c 1 > 4 , stabilization fails for all initial conditions. In summary, for the case of a negatively damped oscillator interacting with a small nonresonant secondary limit cycle oscillator, we show stabilization, provide a pair of canonical equations with one free parameter, and present a complete qualitative characterization of the dynamics. [ABSTRACT FROM AUTHOR]

Understanding and controlling the nonlinear coupling in micro/nanomechanical resonators are of great importance to the exploitation of advanced devices. The recently observed electrostatic nonlinear parametric coupling is a very interesting topic. However, the theoretical model of the electrostatic parametric coupling remains unclear. This paper explicitly derives the model and the electrostatically induced dispersive parametric coupling which reveals the ability to tune the bifurcation topology of capacitive resonators is analyzed based on the multiple-time-scale method. A novel displacement-to-frequency transduction scheme based on this electrostatic dispersive parametric coupling effect is proposed. The transduction sensitivity is theoretically given, which indicates that this electrostatic dispersive transduction scheme can provide even more design freedoms than the existing displacement-to-frequency transduction scheme based on tension modulation. In addition, a bifurcation reversal effect is predicted in the strong actuated states of the dispersive parametric coupled system, which reveals the ability to tune the bifurcation topology of capacitive resonators. [ABSTRACT FROM AUTHOR]

This paper focuses on the escape problem of a harmonically forced classical particle from a purely quartic truncated potential well. The latter corresponds to various engineering systems that involve purely cubic restoring force and absence of linear stiffness even under the assumption of small oscillations, such as pre-tensioned metal wires and springs, and compliant structural components made of polymer materials. This is in contrast to previous studies where the equivalent potential well could be treated as linear at first approximation under the assumption of small perturbations. Due to the strong nonlinearity of the current potential well, traditional analytical methods are inapplicable for describing the transient bounded and escape dynamics of the particle. The latter is analyzed in the framework of isolated resonance approximation by canonical transformation to action–angle variables and the corresponding reduced resonance manifold. The escape envelope is formulated analytically. Surprisingly, despite the essential nonlinearity of the well investigated, it exhibits a universal property of a sharp minimum due to the existence of multiple intersecting escape mechanisms. Unlike previous studies, three underlying mechanisms that govern the transient dynamics of the particle were identified: two maximum mechanisms and a saddle mechanism. The first two correspond to a gradual increase in the system's response amplitude for a proportional increase in the excitation intensity, and the latter corresponds to an abrupt increase in the system's response and therefore more potentially hazardous. The response of the particle is described in terms of energy-based response curves. The maximal transient energy is predicted analytically over the space of excitation parameters and described using iso-energy contours. All theoretical predictions are in complete agreement with numerical results. [ABSTRACT FROM AUTHOR]

We show how spectral submanifold theory can be used to construct reduced-order models for harmonically excited mechanical systems with internal resonances. Efficient calculations of periodic and quasi-periodic responses with the reduced-order models are discussed in this paper and its companion, Part II, respectively. The dimension of a reduced-order model is determined by the number of modes involved in the internal resonance, independently of the dimension of the full system. The periodic responses of the full system are obtained as equilibria of the reduced-order model on spectral submanifolds. The forced response curve of periodic orbits then becomes a manifold of equilibria, which can be easily extracted using parameter continuation. To demonstrate the effectiveness and efficiency of the reduction, we compute the forced response curves of several high-dimensional nonlinear mechanical systems, including the finite-element models of a von Kármán beam and a plate. [ABSTRACT FROM AUTHOR]

In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here, we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced models on SSMs to predict the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension. We illustrate the accuracy and efficiency of these computations on finite-element models of beams and plates. [ABSTRACT FROM AUTHOR]

Synchronization phenomena have been studied from a control point of view for many years. However, the vast majority of papers consider complete synchronization, and very few have been devoted to phase synchronization. This paper addresses control and observability aspects of phase synchronization. In order to focus on fundamental aspects, a feedback control framework related to master–slave synchronization is considered. Comparing results using Cartesian and cylindrical coordinates in the context of such a framework and with three known oscillator models, it is argued that (1) observability does not play a significant role in phase synchronization, although it is granted that it might be relevant for complete synchronization, and (2) a practical difficulty is faced when phase synchronization is aimed at, but the control action is not a direct function of the phase error, as it is usually the case in synchronization studies. This difficulty with its fundamental and practical consequences is investigated in some detail. [ABSTRACT FROM AUTHOR]

Networks of coupled nonlinear oscillators model a broad class of physical, chemical and biological systems. Understanding emergent patterns in such networks is an ongoing effort with profound implications for different fields. In this work, we analytically and numerically study a symmetric ring of N coupled self-oscillators of van der Pol type under external stochastic forcing. The system is proposed as a model of the thermo- and aeroacoustic interactions of sound fields in rigid enclosures with compact source regions in a can-annular combustor. The oscillators are connected via linear resistive coupling with nonlinear saturation. After transforming the system to amplitude-phase coordinates, deterministic and stochastic averaging is performed to eliminate the fast oscillating terms. By projecting the potential of the slow-flow dynamics onto the phase-locked quasi-limit cycle solutions, we obtain a compact, low-order description of the (de-)synchronization transition for an arbitrary number of oscillators. The stationary probability density function of the state variables is derived from the Fokker–Planck equation, studied for varying parameter values and compared to time series simulations. We leverage our analysis to offer explanations for the intermittent energy transfer between Bloch waves observed in acoustic pressure spectrograms observed of real-world gas turbines. [ABSTRACT FROM AUTHOR]

We explore a system of two coupled nonlinear pendula, with the lowest-order coupling term being quadratic, rather than the more typical linear coupling term. We show that synchronization can occur in this system, but with the synchronization frequency lying below either of the original independent frequencies, in contrast to the usual case where the synchronization frequency is between the two frequencies of the independent pendula. Both phase synchronization and amplitude synchronization are found, and one can have situations, for example, where phase synchronization exists, but not amplitude synchronization. We develop analytic formulas to understand the frequency synchronization. [ABSTRACT FROM AUTHOR]

We report on the first numerical implementation of photonic reservoir computing (RC) based on an optically pumped spin vertical-cavity surface-emitting laser (spin VCSEL) with optical feedback and injection. The proposed RC aims at both fast, single task processing and parallel tasks processing, benefiting from feasible tunability and multiplexing of the left and right circularly polarized modes. We evaluate its prediction and classification abilities through two benchmarks, i.e., a Santa Fe time series prediction task and a waveform recognition task. In particular, both the influence of external and internal parameters on the prediction and classification performance is systematically analyzed. The numerical results show that the proposed RC based on a spin VCSEL has remarkable prediction and classification abilities over wider parameter ranges due to the feasible adjustment of the pump intensity and polarization as compared to conventional VCSELs. Most importantly, because of its intrinsic fast response, the spin VCSEL-based RC system is capable of enhancing the information processing rate by significantly reducing the allowable feedback delay time and virtual node interval, reaching 20 Gbps for single task processing and 10 Gbps for parallel tasks processing, respectively. Such a spin VCSEL-based RC system has a potential to achieve high-speed information processing and lower power consumption. [ABSTRACT FROM AUTHOR]

In doubly clamped curved mechanical beams, there are inherent hardening and softening nonlinearities. Thus, their frequency of oscillation is a non-monotonic function of the energy. However, for a sufficiently high energy level there is a zero-dispersion point, where the frequency of oscillation is locally independent of energy even though the beam oscillates deep in its nonlinear regime. This zero-dispersion point is a highly desirable feature in micro-mechanical beams that are used in sensing and time-keeping applications because it effectively eliminates the amplitude-to-frequency noise conversion, and thereby, stabilizes the oscillation frequency. In this paper, we present a detailed analysis of the conservative strongly nonlinear dynamics of curved micro-mechanical beams. Our analysis includes a numerically validated closed-form analytical solution for the strongly nonlinear oscillation of beam, derivation of the condition for the zero-dispersion point in curved beams, and a design scheme for the optimal initial depth of the curved beam that maximizes the frequency of oscillation and the energy level at the zero-dispersion point. We apply our methodology to a physical MEMS device reported in the literature and find the optimal values for this device. Our analysis provides the first step in the development of design tools for exploiting the inherent nonlinearities of curved micro-mechanical beams for frequency stabilization. [ABSTRACT FROM AUTHOR]

In this work, a sufficiently simple model allows one to perform detailed analytic studies to gain insights for the dynamical mechanisms in the general model of gene expression. Through theoretical analysis and numerical simulation, we find that the transcription and translation delays as a bifurcation parameter can drive the system to undergo a Hopf bifurcation, thereby producing oscillation behavior. Meanwhile, the length of these delays can determine the amplitude and period of the oscillations. Moreover, an optimal rate of model parameters is also essential to induce the limit-cycle oscillations. In particular, the transcription factor concentration is called a signal that not only induces twice Hopf bifurcation, but also affects the role of the delay in the system, that is, the varying time delay leads to subcritical Hopf bifurcation and bistability under the small signal strength. On the contrary, supercritical Hopf bifurcation is generated. Furthermore, two-parameter diagram is drawn to further explore how time delay and signal strength coordinate to regulate the oscillation behavior of the system. In addition, the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are determined through the explicit formulae and the global existence of periodic solutions is discussed to verify the authenticity of the simulation. These results may be helpful for us to fully grasp oscillation dynamics and provide clues for further exploring the molecular mechanism of biological rhythms. [ABSTRACT FROM AUTHOR]

A low-dimensional model of a top-tensioned riser under excitations from vortices and time-varying tension is proposed, where the van der Pol wake oscillator is used to simulate the loading caused by the vortex shedding. The governing partial differential equations describing the fluid–structure interactions are formulated and multi-mode approximations are obtained using the Galerkin projection method. The one mode approximation is applied in this study and two different resonances are investigated by employing the method of multiple scales. They are the 1:1 internal resonance between the structure and wake oscillator (also known as 'lock-in' phenomenon) and the combined 1:1 internal and 1:2 parametric resonances. Bifurcations under the varying nondimensional shedding frequency for different mass–damping parameters are investigated and the results of multiple-scale analysis are compared with direct numerical simulations. Analytical responses are calculated using the continuation method and their stability is determined by examining the eigenvalues of the corresponding characteristic equations. Effects of the system parameters including the amplitude of the tension variation, vortex shedding frequency and mass–damping parameter on the system bifurcations have been investigated. The analytical approach has allowed to probe bifurcations occurring in the system and to identify stable and unstable responses. It is shown that the combined resonances can induce large-amplitude vibration of the structure. Counter-intuitively, the amplitude of such responses increases rapidly as the amplitude of the tension variation grows. Comparisons between the analytical and numerical results confirm that the span of the system vibration can be accurately predicted analytically with respect to the obtained response amplitudes of responses. The proposed multi-mode approximation and presented findings of this study can be used to enhance design process of top tension risers. [ABSTRACT FROM AUTHOR]

A symmetric model based on optically pumped quantum dot (QD) spin-polarized vertical cavity surface emitting lasers (spin-VCSELs) subject to mutual coupling is proposed. Compared with the conventional solitary model, more plentiful dynamical regimes associated with different dynamical regions are displayed visually by numerous bifurcation maps, and the detailed direct numerical simulations accurately reveal the dependence of dynamical behavior on external and internal parameters through introducing mutual coupling scheme. We then find that the roles played by the mutual coupling strength and delay time are remarkably significant in intensifying the regime of complex dynamical oscillation. Beyond that, the crucial effects of capture rate, gain parameter, linewidth enhancement factor and frequency detuning on determining the evolvement of dynamical behavior in the mutually coupled QD spin-VCSELs model are shown evidently in the plane of the optical pump intensity and polarization. Through a comprehensive investigation of the dynamical behavior dependent on key parameters, the dynamical mechanism expressed in the current model can be adjusted and even controlled well. Therefore, the involved efforts are enlightening and foresighted to promote a deeply research in the field of QD lasers-based expansion. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of a MEMS resonator dynamics under the action of phase-locked and automatic gain control loops. Particular attention is directed to the study of the nonlinearity factor of the resonator elastic restoring force. It was found that the determination of control system parameters based on the stability analysis of the operating resonant mode, in the general case, does not provide the required phase adjustment and stabilization of the oscillation amplitude. Stable multifrequency modes of oscillations are found, and an analytical study of the mechanisms of their appearance and evolution is carried out under variation of the key parameters of the system. The real regions of the control system stable operation are determined (which do not coincide, as was found, with the regions of stability of the operating resonant mode, due to the presence of hidden attractors in the phase space of the system). A methodology has been developed for identifying such areas of stable operation. A significant complication of the structure of possible motions in the system with an increase in the Q-factor of the resonator is revealed. [ABSTRACT FROM AUTHOR]

In this work, we study the complex dynamics and synchronization of two coupled Duffing-type circuits within the framework of recurrence quantification analysis (RQA). For the case of a Duffing oscillator driven by a sinusoidal voltage source, the behavior of various RQA parameters has been shown to reveal complex chaotic transitions taking place in its oscillatory behavior as the amplitude of forcing term is varied. The problem of synchronization of two identical Duffing oscillators coupled together resistively is further investigated using RQA. The simulated recurrence plot (RP) and plot for the τ -recurrence/recurrence probability, p (τ) , have been used extensively over different control parameter regime to understand the dynamical complexity including chaos synchronization for both one-way and two-way coupled Duffing-type oscillators. The critical parameter beyond which chaos synchronization occurs in such systems is determined. [ABSTRACT FROM AUTHOR]

The collective behavior of the ensembles of coupled nonlinear oscillator is one of the most interesting and important problems in modern nonlinear dynamics. In this paper, we study rotational dynamics, in particular space-time structures, in locally coupled identical pendulum-type elements chains that describe the behavior of phase-locked-loop systems, distributed Josephson junctions, coupled electrical machines, etc. The control parameters in the considered chains are: dissipation, coupling strength, and number of elements. In the system under consideration, the realized modes are synchronous in frequency and synchronous (in-phase) or asynchronous (out-of-phase) in phase. In the low dissipation case, the in-phase synchronous rotational regime instability region boundaries are theoretically found and the bifurcations leading to the loss of its stability are determined. The analysis was carried out for chains of arbitrary length. The existence of various out-of-phase synchronous rotational modes types is revealed: completely asynchronous in phases and the cluster in-phase synchronization regime. Regularities of transitions from one type of out-of-phase synchronous mode to another are established. It was found that at certain coupling parameter values, the coexistence of stable in-phase and out-of-phase synchronous modes is possible. It was found that for arbitrary chain length, the number of possible stable out-of-phase modes is always one less than the chain elements number. Analytical results are confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper considers nonlinear interactions between vibration modes with a focus on recent studies relevant to micro- and nanoscale mechanical resonators. Due to their inherently small damping and high susceptibility to nonlinearity, these devices have brought to light new phenomena and offer the potential for novel applications. Nonlinear interactions between vibration modes are well known to have the potential for generating a "zoo" of complicated bifurcation patterns and a wide variety of dynamic behaviors, including chaos. Here, we focus on more regular, robust, and predictable aspects of their dynamics, since these are most relevant to applications. The investigation is based on relatively simple two-mode models that are able to capture and predict a wide range of transient and sustained dynamical behaviors. The paper emphasizes modeling and analysis that has been done in support of recent experimental investigations and describes in full detail the analysis and attendant insights obtained from the models that are briefly described in the experimental papers. Standard analytical tools are employed, but the questions posed and the conclusions drawn are novel, as motivated by observations from experiments. The paper considers transient dynamics, response to harmonic forcing, and self-excited systems and describes phenomena such as extended coherence time during transient decay, zero dispersion response, and nonlinear frequency veering. The paper closes with some suggested directions for future studies in this area. [ABSTRACT FROM AUTHOR]

Information propagation in the cortical neural network has attracted a large amount of attention. In this paper, a feedforward network is constructed with each subnetwork being a recurrent neuronal population, consisting of neurons coupled by excitatory and inhibitory synapses, in which neurons have both types of outgoing synapses. Namely, neurons in the constructed feedforward neuronal networks have mixed excitatory–inhibitory synapses. Here, propagation of population firing rate and pulse packets are investigated. For the propagation of population firing rate, it is found that proper feedforward connection probability and strength could promote the propagation of population firing rate. Properly larger feedforward synaptic time constant could considerably enhance the fidelity of the propagating population firing rate. For the propagation of pulse packet, we find suitably adjusting the relative strength, recurrent probability, and feedforward synaptic connection can promote propagation of pulse packet. Notably, different from each other, larger feedforward synaptic time constant can considerably promote the propagation of population firing rate while hindering the propagation of pulse packets. [ABSTRACT FROM AUTHOR]

It is well known that the oscillation frequency relates approximately quadratically with amplitude in a Duffing nonlinear oscillator while the frequency is independent of amplitude in a linear oscillator. In this article, the dynamics of a micromechanical oscillator during synchronization is studied and anomalous amplitude-frequency (a-f) dependence in a micromechanical resonator is observed. We theoretically and experimentally observed that in a linear resonator the amplitude is tuned quadratically with frequency while tuned linearly in a hardening as well as a softening nonlinear resonator, when the self-sustained resonator is synchronized to an external weak perturbation. Further investigation shows that the tunable range of the oscillation amplitude of a certain oscillator directly relies on the synchronization bandwidth, perturbation amplitude and frequency difference. The slope of the dependence can be tuned by phase delay in the feedback loop, while the feedback force dominantly determines the properties of the dependence from nonlinear relation to linear relation. This anomalous a-f effect provides a convenient technique for precise amplitude control. [ABSTRACT FROM AUTHOR]

The phase reduction approach has manifested its power in analyzing the rhythmic behaviors for limit cycle oscillators. For coherent oscillation purely induced by noise, e.g., the coherence resonance oscillator, the stochastic dynamics exhibit almost deterministic limit cycle phenomenon which inspires the application of the phase reduction approach to this kind of systems. In this paper, the FitzHugh–Nagumo system in coherence resonance is modeled as a jump process. The phase sensitivity is obtained by applying the phase reduction approach for the hybrid system. A modified direct method is proposed to compare the theoretical results with those of Monte Carlo simulation for the stochastic system, which shows a relatively good agreement for the perturbation not being too small. The phase reduction results of the coherent oscillators are applied to two coupled FitzHugh–Nagumo neurons. It is interesting that the phase difference of the coupled coherent oscillators does not converge as the deterministic oscillators, but forms a distribution where the peaks and valleys correspond to the stable and unstable synchronizations predicted by the phase coupling functions. The idea in this paper could be applied to other coherent oscillators where the stochastic dynamics follow almost deterministic paths. [ABSTRACT FROM AUTHOR]

We formulate and experimentally validate a theoretical reduced-order model for the transverse galloping of nonlinear structures, namely a pair of identical, parallel-oriented cantilever beams whose free ends are attached to square prisms. We derive the structural nonlinearities from (a) a single-mode approximation of the nonlinear (truncated at cubic order) equation of motion, calculated for conservative cantilever beams augmented by a non-conservative aerodynamic force acting on a prism; and (b) phenomenological linear, quadratic, and cubic damping forces. We estimate the coefficients of the damping forces from the ring-down responses of the structures in still air. We analyze the deterministic dynamics of transverse galloping that stem from the aerodynamic force of the quasi-steady theory, and the stochastic effect of spectral line broadening that stem from turbulence-induced random fluctuations. Our findings clearly show that standard nonlinear macroscopic structures exhibit considerably different steady-state response curves than the universal curve of Parkinson obtained for linear mass–spring–damper structures. Importantly, the amplitudes of the oscillations are attenuated at high upstream velocities due to nonlinear damping, while the spectral line broadens due to turbulence-induced random fluctuations and an amplitude-to-phase noise conversion, which lowers the quality of the self-sustained oscillations. These two phenomena should be considered in the design of efficient transverse galloping-based energy harvesters—a rapidly growing field of research. [ABSTRACT FROM AUTHOR]

The stability of synchronous states is analyzed in the context of two populations of inhibitory and excitatory neurons, characterized by two different pulse-widths. The problem is reduced to that of determining the eigenvalues of a suitable class of sparse random matrices, randomness being a consequence of the network structure. A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the overall stability depends crucially on the relative pulse-width. This has implications for the overall property of the asynchronous (balanced) regime. [ABSTRACT FROM AUTHOR]

The realization of soliton control and soliton quality balance is one of the key research contents of optical communications. In order to better reflect the nonlinear interactions during soliton transmission and investigate their change rule, it is necessary to use the transmission model based on nonlinear interaction of solitons to study their characteristics. In this paper, we consider the fifth-order nonlinear Schrödinger equation with variable dispersion and nonlinear effects to realize soliton control and improve soliton quality. Analytic solutions of this equation are presented by the bilinear method, and the transmission and interaction characteristics of solitons in real fiber systems are studied. Influences of dispersion and nonlinear effects on them are analyzed. The nonlinear control based on the linear optical fiber is realized by studying the interactions between two adjacent solitons. [ABSTRACT FROM AUTHOR]

Neural coding is a key problem in neuroscience aimed to understand the information processing mechanism in brain. Among the classical theories of neural coding, population rate coding has been studied widely in many works. In computational studies, neurons are usually classified into excitatory or inhibitory ones. Excitatory neurons have excitatory output synapses, and inhibitory neurons have inhibitory output synapses. However, according to physiological observations, neurons potentially have both types of output synapses. Thus, in this paper, neuronal networks consisting of neurons with mixed excitatory–inhibitory synapses are constructed to investigate the population rate coding fidelity of neuronal systems. It is revealed that, under intermediate values of recurrent probability, inhibitory–excitatory strength ratio, and noise intensity, the performance of population rate coding could be improved by both excitatory synaptic strength and synaptic time constant. It is indicated that external stimuli can be encoded in the form of population firing rate by the studied neuronal networks very well. What is more exciting is that we find the neuronal networks considered in our work have higher coding efficiency than the traditional ones. Therefore, neurons with mixed excitatory–inhibitory synapses may be much more rational. [ABSTRACT FROM AUTHOR]

Spike-frequency adaptation, which reduces the firing rate of a neuron during a constant stimulus, is a prominent property observed in many neurons. In this work, we studied the effects of the I AHP spiking adaptation on the information transmission and efficiency of the Morris–Lecar (ML) neuron model in the spike-timing coding scheme. We showed that this kind of adaptation caused non-trivial spiking dynamics when the input rate was high. Under the stimulation of high-rate inputs, although I AHP adaptation neurons could not outperform non-adaptation neurons in terms of information rate, I AHP adaptation neurons yielded higher coding efficiency than non-adaptation neurons. We also found that the noise could enlarge the range of input rates that the adaptation takes effect to enhance coding efficiency. Increasing the calcium-activated K + current could also extend the range of input rates in which adaptation takes effect. Therefore, we argue that the I AHP adaptation mechanism may play a role of adaptive noise cancelation mechanism in neuronal information processing, i.e., it maintains information transmission when neurons receive low-rate inputs but substantially enhancing coding efficiency for high-rate inputs and highly noise environments by suppressing the spike train variability caused by the noise. [ABSTRACT FROM AUTHOR]

This paper develops a design method for the interconnections of a network of Andronov–Hopf oscillators such that the system exhibits a desired strange attractor. Because of the structure of the oscillators, the desired behavior can be achieved via weak linear coupling, which destabilizes the oscillators' phase difference. First, a set of sufficient conditions are established that result in phase destabilization, and thus instability, of a desired periodic solution. Then, an additional condition is determined to ensure that all harmonic periodic orbits will be unstable. Finally, additional numerical properties are assessed, where tuning of a small parameter can result in chaos. [ABSTRACT FROM AUTHOR]

We discuss the role played by the time correlation properties of stochastic sources and by model nonlinearities in single-degree of freedom energy-harvesting systems. After transforming the state equations into energy-angle coordinates, we apply a stochastic projection operator technique to obtain the system power balance equation. The latter allows to evaluate both the magnitude of the power injected by noise into the system and the harvested power, thus providing a tool instrumental for designers to optimize the harvester. We show that for systems with modulated (multiplicative) noise, nonlinear energy harvesters can outperform their linear counterparts, highlighting the physical mechanism that explains their better performance. [ABSTRACT FROM AUTHOR]

In this paper, we design a memristive system involving magnetic coupling with time-delayed feedback. In a way of autaptic connection, the memristive magnetic coupling feedback is mapped into a neuronal model with ion-channel effect. Compared with the resistive coupling feedback in line with proposed autaptic connection, the memristive magnetic coupling feedback can enhance firing rates more effectively. Furthermore, we analyze collective behaviors of a neural lattice composed of the individual neuronal model connected electrically by bidirectional gap junction. Under the different intensities of gap junction, we find the memristive magnetic coupling feedback can induce a target wave or make a transition from a regular spiral wave to an irregular pattern mixed by both target wave and spiral wave and then to an inverted U-type steady traveling wave. Indeed, the memristive magnetic coupling feedback is able to trigger the transition between different wave patterns. Thereby, we expect that designing the neural lattice with memristive magnetic coupling feedback will be helpful for more advanced applications like neuromorphic circuits. [ABSTRACT FROM AUTHOR]

The feedforward neural network is widely applied in various machine learning architectures, in which the synaptic weight between layers plays an important role in the weak signal propagation. In this paper, the five-layer Izhikevich neural networks with excitatory or excitatory–inhibition neurons are employed to study the effect of Gaussian white noise and synaptic weight between layers on the weak signal transmission characteristics of the subthreshold excitatory postsynaptic currents signal imposed on the input layer. It can be found that there is an optimum value of noise intensity (a medium noise intensity) at which the weak signal can be stably propagated in the five-layer Izhikevich neural networks. The spike timing precision (STP) under the optimal noise intensity will become maximum by increasing the synaptic weight. The noise intensity and synaptic weight corresponding to the maximum value of the STP in the excitatory–inhibition network are smaller than those in excitatory–inhibition network. For the smaller or the bigger noise intensity, however, the STP will become very small, and the weak signal cannot be transmitted from the input layer to the output layer. Furthermore, the weak signal is propagated from the input layer to the output layer and enhanced under a larger synaptic weight in the feedforward neural networks. [ABSTRACT FROM AUTHOR]

We conside the dynamics of a nonlinear resonator that is nonlinearly coupled to a linear resonator that has a relatively short decay time. In this case, the secondary (linear) resonator adiabatically tracks the primary (nonlinear) resonator. This model, which is motivated by ongoing experimental work in nano-resonators, is analyzed analytically and numerically to show that the linear and nonlinear characteristics of the primary resonator can be altered in a significant manner by the coupling to the secondary resonator. Such an arrangement may provide a practical means of tuning resonator characteristics in applications. [ABSTRACT FROM AUTHOR]

Chaotic dynamics is ubiquitous in nature. Traditionally, a good model representation of a certain system can help in predicting this system's future behavior. However, for a complex system, a physics-based model may not be easy to construct given the complexity of a system, in particular, those that exhibit chaotic behavior. Furthermore, due to the aperiodic nature of the motion and finite precision, a model-based prediction may only have relatively high accuracy over a short-time horizon, before significant growth of error occurs in the prediction. In this article, the authors explore an alternate modeling approach, which is based on data-driven modeling, to explore forecasting viability for systems that display chaotic dynamics. Specifically, a deep recurrent neural network architecture, a neural machine, is constructed for forecasting temporal evolution of different chaotic systems. Data obtained from simulations with well-known nonlinear dynamical system prototypes serve as training data for the chosen neural network. In practice, this simulation data may be replaced with field data. The trained system is studied to examine the forecasting ability. Two ordinary differential dynamical systems, namely the Lorenz'63 system and the Lorenz'96 system, and a partial differential system, the Kuramoto–Sivashinsky equation, are studied, and the numerical experiments conducted are presented here to demonstrate the forecasting viability of the constructed neural network. [ABSTRACT FROM AUTHOR]

The experimental characterization of a network of three resistively coupled microelectromechanical systems (MEMS)-based Colpitts oscillators is presented. The oscillators were fabricated on a printed circuit board and comprised of a 16 MHz quartz resonator in a positive feedback loop, which generated a sustained limit cycle oscillation when powered by a DC voltage supply. It was demonstrated that by varying the coupling strength and uncoupled frequency distribution between the oscillators, frequency synchronization and phase relationships could be controlled. This was examined in isolated pairs of oscillators and in cases where the entire network was interacting. Coupling was achieved with discrete resistors that could be autonomously interchanged with digitally controlled switches. These results build upon a small body of experimental studies by examining a coupling architecture wherein impedance interactions between the oscillators made amplitude dynamics relevant. This is a fundamental step toward fully understanding patterns and behavior in practical networks of coupled MEMS oscillators. The potential for the network to be used in neuromorphic computing and pattern recognition applications is also discussed. [ABSTRACT FROM AUTHOR]

Coupled oscillators can exhibit complex spatiotemporal dynamics. Here, we study the propagation of nonlinear waves into an unstable state in dissipative coupled oscillators. To this, we consider the dissipative Frenkel–Kontorova model, which accounts for a chain of coupled pendulums or Josephson junctions and coupling superconducting quantum interference devices. As a function of the dissipation parameter, the front that links the stable and unstable state is characterized by having a transition from monotonous to non-monotonous profile. In the conservative limit, these traveling nonlinear waves are unstable as a consequence of the energy released in the propagation. Traveling waves into unstable states are peculiar of dissipative coupling systems. When the coupling and the dissipation parameter are increased, the average front speed decreases. Based on an averaging method, we analytically determine the front speed. Numerical simulations show a quite fair agreement with the theoretical predictions. To show that our results are generic, we analyze a chain of coupled logistic equations. This model presents the predicted dynamics, opening the door to investigate a wider class of systems. [ABSTRACT FROM AUTHOR]

We explore the dynamical transitions facilitated by the symmetry breaking dynamical states in a directly coupled limit-cycle oscillators, when the oscillators are interacting through attractive and repulsive couplings. Initially, we show that the transition from out-of-phase synchronized (OPS) state to in-phase synchronized (IPS) state is mediated by the onset of symmetry breaking oscillatory (SBO) state using two coupled systems. Similarly, we find the emergence of symmetry breaking oscillation death (AOD) state when the system transits from oscillation death (OD) state to nontrivial amplitude death (NAD) state. Further, we deduce the analytical stability conditions for all the observed dynamical states, namely the IPS, OPS, NAD, OD, including SBO and AOD states. Additionally, the investigation on symmetry breaking dynamics in a ring of unidirectionally coupled oscillators reveals that the breaking of symmetry leads to the traveling chimera (TC) and symmetry breaking incoherent oscillation death states. We also report that the transient nature of TC state obeys a power-law relation near the stable TC region as a function of the coupling strength. From the results, it is evident that the trade-off between the attractive and the repulsive couplings induces the symmetry breaking dynamics in the entire range of the system frequency. [ABSTRACT FROM AUTHOR]

This paper presents necessary and sufficient conditions for the existence of bright/dark solitary solutions in the Hodgkin–Huxley model. The second-order analytic solitary solutions are derived using the generalized differential operator technique. It is shown that the heteroclinic bifurcation in the Hodgkin–Huxley model yields a symmetry breaking effect. Trajectories of solitary solutions before the bifurcation lie on manifolds of one of the saddle points and the separatrix between periodic and non-periodic solutions. A new separatrix emerges after the heteroclinic bifurcation—but solitary solutions do not lie on this trajectory. This symmetry breaking effect is demonstrated using analytic and computational experiments. [ABSTRACT FROM AUTHOR]

Amplitude death, which occurs in a system when one or more macroscopic wavefunctions collapse to zero, has been observed in mutually coupled solid-state lasers, analog circuits, and thermoacoustic oscillators, to name a few applications. While studies have considered amplitude death on oscillator systems and in externally forced complex Ginzburg–Landau systems, a route to amplitude death has not been studied in autonomous continuum systems. We derive simple analytic conditions for the onset of amplitude death of one macroscopic wavefunction in a system of two coupled complex Ginzburg–Landau equations with general nonlinear self- and cross-interaction terms. Our results give a more general theoretical underpinning for recent amplitude death results reported in the literature, and suggest an approach for tuning parameters in such systems so that they either permit or prohibit amplitude death of a wavefunction (depending on the application). Numerical simulation of the coupled complex Ginzburg–Landau equations, for example, including cubic, cubic–quintic, and saturable nonlinearities, is used to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

Multivariate singular spectrum analysis (M-SSA) is a useful tool to detect phase synchronization (PS) without any a priori need for phase estimation. The discriminatory power of M-SSA is often enhanced by using only the time series of the variable that provides the best observability of the dynamics. In the case of a network, however, diverse factors could prevent access to this variable at some nodes. Hence, other variables should be used instead, resulting in a mixed set of variables. The aim of the present work is to investigate, in a systematic way, the impact of using a mixed/incomplete measurement set in the M-SSA of chains of Rössler systems and cord oscillators. Results show that (i) the measurement of some variable from all oscillators does not guarantee detection of PS; (ii) typically one good observable per cluster should be recorded in order to detect PS among such clusters and that (iii) dropping poor variables does not reveal new PS transitions but improves on the resolution of what was already seen with such variables. The procedure is robust to noise. [ABSTRACT FROM AUTHOR]

We study the effects of different coupling strengths and network topologies on signal detection in small-world neuronal networks. Research has previously revealed that the ability of detecting subthreshold signals could be significantly enhanced by appropriately fine-tuning the noise intensity. Here we show that the coupling strength and the structure of the underlying network can also lead toward enhanced signal detection. In particular, we show that there are two levels of the coupling strength at which the subthreshold signal can be detected at an appropriate noise intensity and network structure. We also show that the network structure has little impact on signal detection if the coupling is weak. On the other hand, for intermediate coupling strengths, we show that the shorter the average path length, the better the signal detection. Finally, if the coupling is strong, we show that there exists an intermediate average path length at which signal detection becomes optimal. [ABSTRACT FROM AUTHOR]