Nonlinear Oscillators Research Articles

Non-Hermitian Global Synchronization.

Synchronization of coupled nonlinear oscillators is a prevalent phenomenon in natural systems and can play important roles in various fields of modern science, such as laser arrays and electric networks. However, achieving robust global synchronization has always been a significant challenge due to its extreme susceptibility to initial conditions and structural perturbations. Here, a novel approach is presented to achieve robust global synchronization by manipulating the interplay between non-Hermitian physics and nonlinear dynamics. Remarkably, the initial-state-independent non-Hermitian skin and topological global synchronization are proposed, exhibiting diverse anomalous effects such as the enlarged-size triggered non-Hermitian global synchronization and nonlinear skin states-dominated global synchronization. To validate the findings, nonlinear topoelectrical circuits for experimental observation of non-Hermitian global synchronization are designed and fabricated. The work opens up a promising avenue for establishing resilient global synchronization with potential applications in constructing high-radiance laser arrays and topologically synchronized networks.

On the Problem of Nonlinear Oscillations of a Conservative System in the Absence of Resonance

On the Problem of Nonlinear Oscillations of a Conservative System in the Absence of Resonance

Vibration suppressing study of a simplified floating raft system by mixing using a nonlinear connecting intercalary plate and connecting nonlinear oscillators

Vibration suppressing study of a simplified floating raft system by mixing using a nonlinear connecting intercalary plate and connecting nonlinear oscillators

Standing Solitons Tuned by Impurities in Damped Nonlinear Lattices under External Excitation.

Periodic chains of nonlinear oscillators are known to support solitonic solutions within a specific range of physical parameters when damping effect is considered. This Letter investigates the dynamics of stationary solitons in damped nonlinear lattices under external excitation, focusing on the influence of impurities related to the natural frequency of the oscillators. We demonstrate experimentally and numerically that incorporating impurities into externally driven periodic lattices can expand the solitonic stability diagram under high-damping areas and near the Hopf bifurcation of periodic structures. A mathematical description that closely aligns with experimental realities is presented through the disordered damped nonlinear Schrödinger equation. Specifically, we prove how impurities along the chain can spontaneously nucleate the lattice solitons. The obtained results open the way toward the functionalization of disorder to control nonlinear energy localization in damped nonperiodic structures.

Onset of global instability in a premixed annular V-flame

We investigate self-excited axisymmetric oscillations of a lean premixed methane–air V-flame in a laminar annular jet. The flame is anchored near the rim of the centrebody, forming an inverted cone, while the strongest vorticity is concentrated along the outer shear layer of the annular jet. Consequently, the reaction and vorticity dynamics are largely separated, except where they coalesce near the flame tip. The global eigenmodes corresponding to the linearised reacting flow equations around the steady base state are computed in an axisymmetric setting. We identify an arc branch of eigenmodes exhibiting strong oscillations at the flame tip. The associated eigenvalues are robust with respect to domain truncation and numerical discretisation, and they become destabilised as the Reynolds number increases. The frequency of the leading eigenmode is found to correspond to the Lagrangian disturbance advection time from the nozzle outlet to the flame tip. The essential role of this convective mechanism is also supported by resolvent analysis, which finds that the same flame-tip disturbance structure and frequency are optimally amplified when the flame is subjected to external white noise forcing. Strong non-modal effects in the form of pseudo-resonance are not found. Nonlinear time-resolved simulation further reveals notable hysteresis phenomena in the subcritical regime prior to instability. Hence, even when the flame is linearly stable, perturbations of sufficient amplitude can trigger limit-cycle oscillations and higher-dimensional dynamics sustained by nonlinear feedback. A Monte Carlo simulation of passive tracers in the unsteady flame suggests a nonlinear non-local instability mechanism. Notably, linear analysis of the subcritical time-averaged limit-cycle state yields eigenvalues that do not match the nonlinear periodic oscillation frequencies. This mismatch is attributed to the fundamentally nonlinear dynamics of the subcritical V-flame instability, where the dichromatic, non-local interaction between the heat release rate along the flame surface and the vortex dynamics in the jet shear layer cannot be approximated as a simple distortion of the mean flow.

Vibration Suppression and Energy Harvesting of Nonlinear Energy Sink-Based Piezoelectric System with Combined Damping and Bistability Properties for Nonlinear Oscillator

Nonlinear energy sink (NES) as a new type of dynamic absorber has received widespread attention due to its broadband vibration suppression capability, but it has a certain requirement of energy triggering threshold. In this work, a bistable NES-based piezoelectric system with combined damping (BCPNES) and low threshold requirement is proposed. Electromechanical-coupled equations for the nonlinear oscillator coupled with BCPNES (NO-BCPNES) are established. The vibration suppression and vibration energy harvesting performance for the coupled system are investigated numerically and analytically. The frequency–energy plots of the conservative system are investigated using the complex-variable averaging method and the electromechanical decoupling method. The influence of circuit parameters on the target energy transfer threshold and harvested energy is discussed through the equivalent stiffness and damping of the circuit. The vibration suppression performance and the target energy transfer characteristics are analyzed in terms of the time–history response, the time–frequency response and the slowly varying dynamic flow, respectively. The results show that the electromechanical coupling coefficient affects the skeleton curve of frequency–energy plots in the form of equivalent stiffness. The NO-BCPNES system exhibits higher energy dissipation performance and better targeted energy transfer form under various excitations. From the phase perspective, NES-based piezoelectric system achieves vibration suppression of the main structure through the phase locking phenomenon. The mechanism of the influence of piezoelectric device parameters on the energy harvesting performance of the systems is clarified.

Exploring Non-linear Dynamics: Constructing the Bifurcation Diagram of a Damped Driven Pendulum using Python

In this paper, we present a detailed analysis and construction of the bifurcation diagram for the damped-driven pendulum system. The bifurcation diagram in general presents the qualitative changes of the steady-state behavior for the pendulum. For this purpose, we implement the use of the Python programming language with the inclusion of scientific libraries. This nonlinear dynamical system is an example of a system that exhibits a chaotic regime, which is the sensitivity of its behavior to the initial conditions and the parameters of the system. We investigate the response of the system to a range of drive strengths γ applied. By changing the driving strength, the system reveals patterns of periodicity, quasi-periodic, and chaotic regimes. The critical values where it goes from regular motion to chaotic one are highlighted, offering further understanding of the mechanisms of the transitions. This study presents the use of the Python programming language for the modeling and visualization of non-dynamic systems and contributes to a deeper understanding of nonlinear oscillator dynamics.

Bursting dynamics induced by amplitude-modulated excitation in the composite nonlinear oscillator

Bursting dynamics induced by amplitude-modulated excitation in the composite nonlinear oscillator

Particle Size and Morphological Evaluation of Airborne Urban Dust Particles by Scanning Electron Microscopy and Bidimensional Empirical Mode Analysis

Airborne particles affect the health of the population. As particles decrease in size, they can penetrate deeper into the respiratory system, reaching the terminal bronchioles and alveoli. Particles as small as 0.1 µm in diameter may translocate into the bloodstream, potentially impacting various organs. Additionally, the smaller the particle size, the longer they remain suspended in the air, thereby increasing their deleterious damages. The aim of this work is to study the size distribution of airborne particles emitted from anthropogenic sources of air pollution, with a special emphasis on estimating the distribution of micro and nanoparticles considered the most harmful to health. The Bidimensional Empirical Mode Decomposition (BEMD) algorithm was used on micrographs of the particles obtained by Scanning Electron Microscopy (SEM). BEMD is a current empirical computational tool applied to image analysis that allows extracting non-linear heterogeneous oscillations of brightness. We studied ROFA (Residual Oil Fly Ash) from industrial sources and DEP (Diesel Exhaust Particles) from vehicular emissions as airborne particles. After collecting the particles on filters, micrographs were taken using SEM at different magnifications to which the BEMD algorithm was applied. Particle size and asymmetry distributions were obtained for each mode, allowing the identification of the most deleterious particles. The methodology employed herein is relatively simple and effective for inferring the impact of airborne particulate matter on health and the environment.

Nonlinearity-induced symmetry breaking in a system of two parametrically driven Kerr-Duffing oscillators

Abstract We study the classical dynamics of a system comprising a pair of Kerr-Duffing nonlinear oscillators, which are coupled through a nonlinear interaction and subjected to a parametric drive. Using the rotating wave approximation (RWA), we analyze the steady-state solutions for the amplitudes of the two oscillators. For the case of almost identical oscillators, we investigate separately the cases in which only one oscillator is parametrically driven and in which both oscillators are simultaneously driven. In the latter regime, we demonstrate that even when the parametric drives acting on the two oscillators are identical, the system can transition from a stable symmetric solution to a broken-symmetry solution as the detuning is varied.

Adaptive control of cardiac rhythms

Cardiac rhythms are related to heart electrical activity, being the essential aspect of the cardiovascular physiology. Usually, these rhythms are represented by electrocardiograms (ECGs) that are useful to detect cardiac pathologies. Essentially, the heart activity starts in the sinoatrial node (SA) node, the natural pacemaker, propagating to the atrioventricular node (AV), and finally reaching the His-Purkinje complex (HP). This paper investigates the control of cardiac rhythms in order to induce normal rhythms from pathological responses. A mathematical model that presents close agreement with experimental measurements is employed to represent the heart functioning. The adopted model comprises a network of three nonlinear oscillators that represent each one of the cardiac nodes, connected by delayed couplings. The pathological behavior is induced by an external stimulus in the SA node. An adaptive controller is proposed acting in the SA node considering an strategy based on the signal obtained by the natural pacemaker and its regularization. The incorporation of adaptive compensation in a Lyapunov-based control scheme allows the compensation for the unknown dynamics. The controller ability to deal with interpatient variability is evaluated by assuming that the heart model is not available to the controller design, being used only in the simulator to assess the control performance. Results show that the adaptive term can reduce the control effort by around 3% while reducing the tracking error by 20%, when compared to the conventional feedback approach. Additionally, the controller can avoid abnormal rhythms, turning the ECG closer to the expected normal behavior and preventing critical cardiac responses. Therefore, this work demonstrates that an adaptive controller can be used to regulate the ECG signal without prior information about the system and disregarding inter- and intrapatient variability.

Bistability in the sunspot cycle

A direct dynamical test of the sunspot cycle is carried out to indicate that a stochastically forced nonlinear oscillator characterizes its dynamics. The sunspot series is then decomposed into its eigen time-delay coordinates. The relevant analysis reveals that the sunspot series exhibits bistability, with the possibility of modeling the solar cycle as a stochastically and periodically forced bistable oscillator, accounting for poloidal and toroidal modes of the solar magnetic field. Such a representation enables us to conjecture stochastic resonance as the key mechanism in amplifying the planetary influence on the Sun, and that extreme events, due to turbulent convection noise inside the Sun, dictate crucial phases of the sunspot cycle, such as the Maunder minimum.

Optimal synchronization to a limit cycle.

In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic trajectory-the limit cycle-after infinite time. Here, we drive the van der Pol oscillator with an external time-dependent force to reach the limit cycle in a given finite time. Specifically, we are interested in minimizing the non-conservative contribution to the work when driving the system from a given initial point on the phase plane to any final point belonging to the limit cycle. There appears a speed-limit inequality, which expresses a trade-off between the connection time and cost-in terms of the non-conservative work. We show how the above results can be generalized to the broader family of non-linear oscillators given by the Liénard equation. Finally, we also look into the problem of minimizing the total work done by the external force.

Approximate Analytic Frequency of Strong Nonlinear Oscillator

In this paper, a new analytic expression for the frequency of vibration of a strong nonlinear polynomial-type oscillator is introduced. The method for frequency calculation is based on the transformation of the nonlinear oscillators into linear ones using the equality of their amplitudes and periods of vibration. The frequency of the linear oscillator is assumed to be the sum of frequencies corresponding to each nonlinearity in the original oscillator separately, i.e., the sum of frequencies of truly nonlinear oscillators. The obtained frequency is a complex function of amplitude, coefficient and order of nonlinearity. For simplification, the frequencies of the truly nonlinear oscillators are modified as power order functions of the exact frequency of the cubic oscillator which is linearly dependent on the amplitude of vibration. In this paper, the approximate frequency expression is developed for the harmonic any-order nonlinear oscillator and oscillators with the sum of polynomial nonlinearities. The accuracy of the obtained frequencies is tested on the examples of non-integer order nonlinear oscillators and also on a quadratic-cubic oscillator. The difference between the analytical and exact, numerically obtained results is negligible. The suggested approximate frequency expression has a simple algebraic form and is suitable for application by engineers and technicians.

Dynamics of the Classical Counterpart of a Quantum Nonlinear Oscillator with Parametric Dissipation

Dynamics of the Classical Counterpart of a Quantum Nonlinear Oscillator with Parametric Dissipation

A Reconfigurable, Nonlinear, Low-Power, VCO-Based ADC for Neural Recording Applications.

Neural recording systems play a crucial role in comprehending the intricacies of the brain and advancing treatments for neurological disorders. Within these systems, the analog-to-digital converter (ADC) serves as a fundamental component, converting the electrical signals from the brain into digital data that can be further processed and analyzed by computing units. This research introduces a novel nonlinear ADC designed specifically for spike sorting in biomedical applications. Employing MOSFET varactors and voltage-controlled oscillators (VCOs), this ADC exploits the nonlinear capacitance properties of MOSFET varactors, achieving a parabolic quantization function that digitizes the noise with low resolution and the spikes with high resolution, effectively suppressing the background noise present in biomedical signals. This research aims to develop a reconfigurable, nonlinear voltage-controlled oscillator (VCO)-based ADC, specifically designed for implantable neural recording systems used in neuroprosthetics and brain-machine interfaces. The proposed design enhances the signal-to-noise ratio and reduces power consumption, making it more efficient for real-time neural data processing. By improving the performance and energy efficiency of these devices, the research contributes to the development of more reliable medical technologies for monitoring and treating neurological disorders. The quantization step of the ADC spans from 44.8 mV in the low-amplitude range to 1.4 mV in the high-amplitude range. The circuit was designed and simulated utilizing a 180 nm CMOS process; however, no physical prototype has been fabricated at this stage. Post-layout simulations confirm the expected performance. Occupying a silicon area is 0.09 mm2. Operating at a sampling frequency of 16 kS/s and a supply voltage of 1 volt, this ADC consumes 62.4 µW.

A universal application of the general integral transform: New technique for the analysis of nonlinear oscillator systems

The wide applicability of a nonlinear oscillator is the main reason why it has gained so much attention from mathematicians and scientists, while it is difficult to be solved both numerically and theoretically. Variational iteration method is one of the numerical approaches which are used to investigate these nonlinear systems. In this paper, we present a nonlinear oscillator arising in a micro-electromechanical system as an example and derive its analytical solution by He’s frequency formula method and variational iteration method coupled with a new general integral transformation. The Lagrange multiplier is formulated and the new iterative format for the correction functional of the variational iteration method is given by the general integral transformation. The approximate nonlinear frequencies are achieved by this new technique and compared with the numerical ones obtained by the Runge–Kutta method. The new general integral transformation has the same essential qualities as those for Laplace and Fourier transform, and it becomes a promising tool to nonlinear problems when coupled with the variational iteration method.

Numerical analysis of fractional nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass

This paper focuses on the numerical investigation of the fractional nonlinear oscillator for a restrained cantilever beam with an intermediate lumped mass. A numerical approach based upon the fractional complex transformation and the global residue harmonic balance method (FCT-GRHBM) is suggested for approximately solving this fractional oscillation system of fifth-order nonlinearity. The approximated fractional periodic solutions and frequencies are provided by FCT-GRHBM, which are not touched in the existing literature. Numerical results and sensitive analysis with respect to different parameters are shown to confirm its efficiency.

Impact of compliant electrodes on the dynamics of electromagnetoactive membranes

The dynamics of electromagnetoactive polymer (EMAP) membranes have attracted much attention recently because of their wide range of modern robotic applications. Such applications majorly centered on how the dynamics of this novel class of membranes are affected by the mechanical behavior of the compliant electrode. This article presents the dynamic modeling and analysis of EMAP membranes, examining how the inertia of the electrode, coupled with its inherent viscoelastic properties, impacts its dynamic performance. Both the compression and suspension stages of the membrane are covered here in broad terms. An Euler–Lagrange equation of motion is implemented to deduce the governing dynamic model equation of the membrane system. The findings of the model solutions provide preliminary insights to characterize the dynamic response, instability analysis, periodic behavior, and resonance properties across varying parameters such as inertia, electric field, magnetic field, and prestress. Moreover, the study also evaluates the periodicity and stability of the nonlinear oscillations using Poincaré maps and phase portraits, facilitating an assessment of quasi-periodic to periodic transitions.

A New Paradigm for Active–Break Cycle in the Indian Summer Monsoon

Abstract A multichannel singular spectrum analysis of 121 years of daily rainfall over India and 43 years of outgoing longwave radiation and horizontal wind has revealed two dominant intraseasonal oscillations (ISOs) that describe the space–time variability of monsoon rainfall. The two nonlinear oscillations are not perfectly periodic but exhibit broadband spectra centered at 45 and 28 days. These modes have coherent and near-regular spatial structure and temporal variations. This study posits that the two nonlinear oscillations provide a consistent and comprehensive framework to study the variability and predictability of intraseasonal rainfall variations. The active and break cycles of monsoon rainfall over India have been traditionally defined by using an arbitrarily chosen duration of persistence of arbitrarily chosen amount of rainfall averaged over an arbitrarily chosen area. This study shows that the phases of the two objectively derived modes of variability provide an objective method for defining the active and break cycles. The two ISOs are further shown to make negligible contribution to the seasonal mean rainfall and its interannual variability. This study proposes that the investigations of the origins of these nonlinear modes, and modulations of their amplitudes and phases, provide a new paradigm for future research on validation and predictability of intraseasonal variations in climate models.