Dynamics Of Nonlinear Oscillator Research Articles

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.

Learning spiking neuronal networks with artificial neural networks: neural oscillations.

First-principles-based modelings have been extremely successful in providing crucial insights and predictions for complex biological functions and phenomena. However, they can be hard to build and expensive to simulate for complex living systems. On the other hand, modern data-driven methods thrive at modeling many types of high-dimensional and noisy data. Still, the training and interpretation of these data-driven models remain challenging. Here, we combine the two types of methods to model stochastic neuronal network oscillations. Specifically, we develop a class of artificial neural networks to provide faithful surrogates to the high-dimensional, nonlinear oscillatory dynamics produced by a spiking neuronal network model. Furthermore, when the training data set is enlarged within a range of parameter choices, the artificial neural networks become generalizable to these parameters, covering cases in distinctly different dynamical regimes. In all, our work opens a new avenue for modeling complex neuronal network dynamics with artificial neural networks.

Analytical phase reduction for weakly nonlinear oscillators

Phase reduction is a dimensionality reduction scheme to describe the dynamics of nonlinear oscillators with a single phase variable. While it is crucial in synchronization analysis of coupled oscillators, analytical results are limited to few systems. In this work, we analytically perform phase reduction for a wide class of oscillators by extending the Poincaré–Lindstedt perturbation theory. We exemplify the utility of our approach by analyzing an ensemble of Van der Pol oscillators, where the derived phase model provides analytical predictions of their collective synchronization dynamics.

Dynamics of Nonlinear Oscillators with Discontinuous Nonlinearities Subjected to Harmonic and Stochastic Excitations

Dynamics of Nonlinear Oscillators with Discontinuous Nonlinearities Subjected to Harmonic and Stochastic Excitations

Resonance synchronisation between memristive oscillators and network without variable coupling

Continuous energy pumping and exchange along the coupling channel can balance the energy release between nonlinear oscillators for reaching complete synchronisation. When external stimulus is applied, energy is injected and encoded for regulating the dynamics of nonlinear oscillators and circuits. In this paper, the synchronisation between memristive Rossler oscillators is investigated by reactivating one memristive variable, and external stimuli are changed to detect the occurrence of synchronisation without direct variable coupling. In the presence of periodical stimulus, stochastic switch and feedback on the memristive variable can induce synchronisation between two memristive oscillators and chain network composed of memristive oscillators. In the presence of noise, stochastic feedback and disturbance on the memristive variable can keep synchronisation stable between two oscillators, and complete synchronisation is realised. In addition, the synchronisation factor and spatial patterns are calculated to confirm the occurrence of synchronisation between more chaotic oscillators when memristive function is activated even when no coupling channels are switched on.

Duffing oscillation and jump resonance: Spectral hysteresis and input-dependent resonance shift

The purpose of this paper is to present experimental results for jump resonance observed in a vibrating cantilever beam of a macro-scale atomic force microscope (AFM), and to connect these results with the well-known Duffing nonlinear oscillator dynamics. The paper gives a review of the theoretical analyses of such behaviors, as well as detailed experimental results for nonlinear frequency-domain jump resonance in the vibrating AFM probe. The experimental results clearly show the model-predicted input-dependent resonant frequency shift. A real-time simulation model of the experimental system is also presented as a useful tool to understand the nonlinear dynamic structure, as an aid in identifying the Duffing equation parameters, and allow simulation studies of ideas such as controlling the nonlinear system behavior. We also present a feedback linearization algorithm to compensate for the cubic stiffness internal to the Duffing nonlinear oscillator. The simple experimental system described herein can be easily and inexpensively constructed to allow students and researchers to conduct their own experiments with such nonlinear dynamics.

A Novel Hardware-Efficient Central Pattern Generator Model Based on Asynchronous Cellular Automaton Dynamics for Controlling Hexapod Robot

This paper presents a novel hardware-efficient central pattern generator (CPG) model to realize a bio-inspired gait of a hexapod robot. The CPG model consists of a network of cellular automaton (CA) oscillators; thus, it can be implemented as a network of sequential logic circuits. Detailed analyses of nonlinear oscillation dynamics show that the oscillator that is driven by multiple asynchronous clocks is more suitable to realize the gait of the robot than an oscillator that is driven by a single clock or multiple synchronous clocks. Moreover, detailed analyses of nonlinear network dynamics show that the clocks among the CA oscillators should be asynchronous to appropriately realize the gait. Using the analyses, systematic procedures to design the CPG model are proposed. The proposed CPG model is implemented in a field programmable gate array (FPGA); our experiments validate that the CPG model implemented in an FPGA can realize the bio-inspired gait of a hardware robot. Further, we show that the proposed CPG model utilizes fewer circuit elements and lower power than a conventional CPG model.

Explosive death in nonlinear oscillators coupled by quorum sensing.

Many biological and chemical systems exhibit collective behavior in response to the change in their population density. These elements or cells communicate with each other via dynamical agents or signaling molecules. In this work, we explore the dynamics of nonlinear oscillators, specifically Stuart-Landau oscillators and Rayleigh oscillators, interacting globally through dynamical agents in the surrounding environment modeled as a quorum sensing interaction. The system exhibits the typical continuous second-order transition from oscillatory state to death state, when the oscillation amplitude is small. However, interestingly, when the amplitude of oscillations is large we find that the system shows an abrupt transition from oscillatory to death state, a transition termed "explosive death." So the quorum-sensing form of interaction can induce the usual second-order transition, as well as sudden first-order transitions. Further, in the case of the explosive death transitions, the oscillatory state and the death state coexist over a range of coupling strengths near the transition point. This emergent regime of hysteresis widens with increasing strength of the mean-field feedback, and is relevant to hysteresis that is widely observed in biological, chemical, and physical processes.

Dynamics of nonlinear oscillator with transient feedback

The feedback is an integral part of many natural as well as engineering systems. Here, we explore the effect of transient feedback on the dynamics of a nonlinear oscillator. Transient feedback is active in either a certain state-space or for a given time interval otherwise inactive. The transient feedback suppresses the oscillations and stabilizes the system to a steady state. The transition from oscillatory state to steady state is possible even with the transient spatial or temporal feedback. Chaotic and limit cycle oscillators are studied as a prototype model.

The analytical method of studying subharmonic periodic orbits for planar piecewise-smooth systems with two switching manifolds

Piecewise linear approach is one of important methods to study nonlinear dynamics of oscillators with complex nonlinear restoring forces, so piecewise-smooth systems with multiple switching manifolds sometimes are ideal mathematical models to analyze the nonlinear dynamics of nonlinear oscillators. In this paper, inspired by the work presented by Cao et al. (Philos Trans R Soc A 366(1865):635–652, 2008) and Granados et al. (SIAM J Appl Dyn Syst 11(3):801–830, 2012), we want to formulate an analytical method for studying subharmonic periodic orbits for piecewise-smooth systems with two switching manifolds. We will define what are subharmonic orbits for this class of piecewise-smooth systems and develop a Melnikov-type analytical method to detect the existence of subharmonic orbits. In order to obtain this objective, we assume that the plane is divided into three zones by the two switching manifolds which may not be symmetric, and the dynamics in each zone is decided by a smooth system. Furthermore, we suppose that the unperturbed system is a piecewise-defined Hamiltonian system and possesses a pair of piecewise-smooth homoclinic orbits crossing the two switching manifolds transversally exactly twice and connecting origin to itself. The region outside the pair of homoclinic orbits is fully covered by periodic orbits crossing respectively the two switching manifolds transversally exactly twice. Finally, we want to study the persistence of those continuum of periodic orbits outside the pair of homoclinic orbits when a small time-periodic perturbation is considered. By choosing an appropriate Poincare section, constructing a Poincare map and applying perturbation techniques, we obtain the Melnikov function of subharmonic orbits for the class of planar piecewise-smooth systems and employ it to study the existence of subharmonic periodic motions for a piecewise-linear oscillator. Numerical simulations are also shown the effectiveness of the analytical method to detect the parameters and initial conditions for the existence of subharmonic orbits in piecewise-smooth oscillators.

Dynamics of nonlinear oscillators with time-varying conjugate coupling

Dynamics of nonlinear oscillators with time-varying conjugate coupling

Phase oscillator model for noisy oscillators

The Kuramoto model has become a paradigm to describe the dynamics of nonlinear oscillator under the influence of external perturbations, both deterministic and stochastic. It is based on the idea to describe the oscillator dynamics by a scalar differential equation, that defines the time evolution for the phase of the oscillator. Starting from a phase and amplitude description of noisy oscillators, we discuss the reduction to a phase oscillator model, analogous to the Kuramoto model. The model derived shows that the phase noise problem is a drift-diffusion process. Even in the case where the expected amplitude remains unchanged, the unavoidable amplitude fluctuations do change the expected frequency, and the frequency shift depends on the amplitude variance. We discuss different degrees of approximation, yielding increasingly accurate phase reduced descriptions of noisy oscillators.

Global bifurcations of symmetric cross‐ply composite laminated plates with 1:2 internal resonance

AbstractThis paper investigates the global bifurcations and chaotic dynamics for nonlinear oscillations of symmetric cross‐ply composite laminated plates in case of 1:2 internal resonance. The higher‐dimensional Melnikov method and its extensions developed by Kovai and Wiggins is employed to analyze the global bifurcations for composite laminated plates. The explicitly sufficient conditions of the existence of Silnikov‐type homoclinic orbits in perturbed phase space are gained, which may lead to chaotic motions for composite laminated plates. Finally, numerical results obtained by fourth‐order Runge‐Kutta method also indicate that there exist the jumping phenomena and chaotic responses for the nonlinear composite laminated plates, which agree with theoretic predictions.

Oscillation death and revival by coupling with damped harmonic oscillator.

Dynamics of nonlinear oscillators augmented with co- and counter-rotating linear damped harmonic oscillator is studied in detail. Depending upon the sense of rotation of augmenting system, the collective dynamics converges to either synchronized periodic behaviour or oscillation death. Multistability is observed when there is a transition from periodic state to oscillation death. In the periodic region, the system is found to be in mixed synchronization state, which is characterized by the newly defined "relative phase angle" between the different axes.

Nonlinear optical oscillation dynamics in high-Q lithium niobate microresonators.

Recent advance of lithium niobate microphotonic devices enables the exploration of intriguing nonlinear optical effects. We show complex nonlinear oscillation dynamics in high-Q lithium niobate microresonators that results from unique competition between the thermo-optic nonlinearity and the photorefractive effect, distinctive to other device systems and mechanisms ever reported. The observed phenomena are well described by our theory. This exploration helps understand the nonlinear optical behavior of high-Q lithium niobate microphotonic devices which would be crucial for future application of on-chip nonlinear lithium niobate photonics.

Phase-flip and oscillation-quenching-state transitions through environmental diffusive coupling.

We study the dynamics of nonlinear oscillators coupled through environmental diffusive coupling. The interaction between the dynamical systems is maintained through its agents which, in turn, interact globally with each other in the common dynamical environment. We show that this form of coupling scheme can induce an important transition like phase-flip transition as well transitions among oscillation quenching states in identical limit-cycle oscillators. This behavior is analyzed in the parameter plane by analytical and numerical studies of specific cases of the Stuart-Landau oscillator and van der Pol oscillator. Experimental evidences of the phase-flip transition and quenching states are shown using an electronic version of the van der Pol oscillators.

Noise Induces the Population-Level Entrainment of Incoherent, Uncoupled Intracellular Oscillators

Intracellular oscillators entrain to periodic signals by adjusting their phase and frequency. However, the low copy numbers of key molecular players make the dynamics of these oscillators intrinsically noisy, disrupting their oscillatory activity and entrainment response. Here, we use a combination of computational methods and experimental observations to reveal a functional distinction between the entrainment of individual oscillators (e.g., inside cells) and the entrainment of populations of oscillators (e.g., across tissues). We demonstrate that, in the presence of intracellular noise, weak periodic cues robustly entrain the population averaged response, even while individual oscillators remain un-entrained. We mathematically elucidate this phenomenon, which we call stochastic population entrainment, and show that it naturally arises due to interactions between intrinsic noise and nonlinear oscillatory dynamics. Our findings suggest that robust tissue-level oscillations can be achieved by a simple mechanism that utilizes intrinsic biochemical noise, even in the absence of biochemical couplings between cells.

Synthesizing Virtual Oscillators to Control Islanded Inverters

Virtual oscillator control (VOC) is a decentralized control strategy for islanded microgrids where inverters are regulated to emulate the dynamics of weakly nonlinear oscillators. Compared to droop control, which is only well defined in sinusoidal steady state, VOC is a time-domain controller that enables interconnected inverters to stabilize arbitrary initial conditions to a synchronized sinusoidal limit cycle. However, the nonlinear oscillators that are elemental to VOC cannot be designed with conventional linear-control design methods. We address this challenge by applying averaging- and perturbation-based nonlinear analysis methods to extract the sinusoidal steady-state and harmonic behavior of such oscillators. The averaged models reveal conclusive links between real- and reactive-power outputs and the terminal-voltage dynamics. Similarly, the perturbation methods aid in quantifying higher order harmonics. The resultant models are then leveraged to formulate a design procedure for VOC such that the inverter satisfies standard ac performance specifications related to voltage regulation, frequency regulation, dynamic response, and harmonic content. Experimental results for a single-phase 750 VA, 120 V laboratory prototype demonstrate the validity of the design approach. They also demonstrate that droop laws are, in fact, embedded within the equilibria of the nonlinear-oscillator dynamics. This establishes the backward compatibility of VOC in that, while acting on time-domain waveforms, it subsumes droop control in sinusoidal steady state.

Phase transition in multimode nonlinear parity-time-symmetric waveguide couplers.

Parity-time-symmetric (-symmetric) optical waveguide couplers offer new possibilities for fast, ultracompact, configurable, all-optical signal processing. Here, we study nonlinear properties of finite-size multimode -symmetric couplers and predict the nonlinear oscillatory dynamics that can be controlled by three parameters: input light intensity, gain and loss amplitude, and input beam profile. Moreover, we show that this dynamics is driven by a transition triggered by nonlinearity in these structures, and we demonstrate that with the increase of the number of dimers in the system, the transition threshold decreases and converges to the value corresponding to an infinite array. Finally, we present a variety of periodic intensity patterns that can be formed in these couplers depending on the initial excitation.

Random Perturbations of Periodically Driven Nonlinear Oscillators

This paper develops a unified approach to study the dynamics of nonlinear oscillators excited by both periodic and random per- turbations. This study is motivated by problems that range from nonlinear energy harvesting to ship capsizing in random seas. The near resonant dynamics of such systems, in the presence of weak noise, is not well understood. Nonlinear systems driven by sufficiently strong periodic parametric excitation often display a range of phenomena from period doubling to chaos. In the presence of weak noise there are transitions between the domains of attraction of the stable periodic orbits. The effects of noisy perturbations on the passage of trajectories through the resonance zones is studied in depth using the large deviation theory.