Limit Cycle Research Articles

Phase and amplitude responses for delay equationsusing harmonic balance.

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations(DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equationswith limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to external forcing. When combined with Floquet theory, the fundamental quantities for this reduction are phase and amplitude response functions. Here, we develop a framework for their construction that utilizes the method of harmonic balance.

RLC resonator with diode nonlinearity: Bifurcation comparison of numerical predictions and circuit measurements

A nonlinear RLC resonator is investigated experimentally and numerically using bifurcation analysis. The nonlinearity is due to the parallel combination of a semiconductor rectifier diode and a fixed capacitor. The diode’s junction capacitance, diffusion capacitance, and DC current–voltage relation each contribute to the nonlinearity. The closely related RL-diode resonator has been of interest for many years since its demonstration of period-doubling cascades to chaos. In this study, a direct comparison is made of dynamical regime maps produced from simulations and circuit measurements. The maps show the variety of limit cycles, their bifurcations, and regions of chaos over the 2D parameter space of the source voltage’s frequency and amplitude. The similar structures of the simulated and experimental maps suggest that the diode models commonly used in circuit simulators (e.g., SPICE) work well in bifurcation analyses, successfully predicting complex and chaotic dynamics detected in the circuit. These results may be useful for applications of varactor-loaded split ring resonators.

Bifurcations and mixed mode oscillations in a bi-stable plasma model with slow parametric excitation.

In this study, considering a bi-stable plasma model with slow parametric excitation, the bifurcation of periodic and chaotic responses as well as the resulting fast-slow motions is discussed analytically and numerically. For a nonautonomous fast sub-system, the generalized harmonic balancing method is utilized to obtain an averaged system. Bifurcation analysis about the averaged system shows that the critical manifolds form a S-shape structure. Meanwhile, supercritical and subcritical period doubling (PD) occurs on the upper branch simultaneously. As the frequency of the external excitation changes, bifurcation points on the limit cycle manifolds can present different relative locations. Moreover, an additional bi-stable structure induced by Cusp bifurcation emanates from the upper branch. On the other hand, the existence of a chaotic attractor and the corresponding boundary crisis phenomenon are verified using the Melnikov method and the basin of attraction. The structures of the numerical bifurcation diagram show good agreements with the analytical results. Considering two cases of low-frequency excitation, the corresponding fast-slow dynamics are discussed. It is found that, when the fast-slow flow passing the subcritical PD point, a low frequency with different magnitudes will lead to two patterns of bifurcation delay, i.e., the typical one and the excessive delay, which suppress the PD. As for the boundary crisis point, the slow passage effects show no distinct influence. Thus, three transition mechanisms based on two cases of the bifurcation structure are explained, including "fold of cycle-fold of cycle" type, "fold of cycle-delayed subcritical PD" type, and "fold of cycle-boundary crisis" type.

Prey group defense and hunting cooperation among generalist-predators induce complex dynamics: a mathematical study.

Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed 'bubble' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of "saltatory equilibria" in ecological systems and capture the dynamics observed for lion-herbivore interactions.

Design Polynomial IIR Digital Filters of the Integer Parameters Space Use to Compress Image Data

Polynomial IIR digital filters play a crucial role in the process of image data compression. The main purpose of designing polynomial IIR digital filters of the integer parameters space and introduce efficient filters to compress image data using a singular value decomposition algorithm. The proposed work is designed to break down the complex topic into bite-sized pieces of image data compression through the lens of compression image data using Infinite Impulse Response Filters. The frequency response of the filters is measured using a real signal with an automated panoramic measuring system developed in the virtual instrument environment. The analysis of the output signal showed that there are no limit cycles with a maximum radius of poles of 0.96 in the polynomial bandpass filters. Thus, all the functional requirements for the Integer Parameters Space of the proposed polynomial IIR digital filters were met. The results showed that the data compression and size reducing of an image file is processed without significantly impacting of visual quality. This is achieved by removing redundant or unnecessary information from the image while preserving the important details which removes unnecessary data to make the file smaller and more manageable.

L-H Transition Initiated by Limit Cycle Oscillation Transition in the Globus-M Tokamak

The intermediate mode between the ohmic, or low confinement (L-mode), and the increased confinement (H-mode) regimes, or the so called I-phase, which is characterized by the existence of zonal flows in the form of limit cycle oscillations (LCOs), was observed on the Globus-M tokamak. Depending on the LCO frequency, the I-phase resulted in either a transition to H-mode or back to L-mode. The possibility of L-I-H transition initiation induced by LCOs and the effect of LCO frequency were studied by means of numerical modeling of the density profile evolution, taking into account turbulence suppression by the inhomogeneous radial electric field. The modeling results show that lower LCO frequency could be a factor facilitating the L-H transition, whereas higher frequency LCOs are more likely to cause the backward transition to L-mode. The results are in qualitative agreement with the results of the studies of geodesic acoustic mode (GAM)–initiated L-H transition in the TUMAN-3M tokamak, where a lower GAM frequency was found to be beneficial for L-H transition initiation.

Quartic Rigid Systems in the Plane and in the Poincaré Sphere

We consider the planar family of rigid systems of the form x′=-y+xP(x,y),y′=x+yP(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x'=-y+xP(x,y), y'=x+yP(x,y)$$\\end{document}, where P is any polynomial with monomials of degree one and three. This is the simplest non-trivial family of rigid systems with no rotatory parameters. The family can be compactified to the Poincaré sphere such that the vector field along the equator is not identically null . We study the centers, singular points and limit cycles of that family on the plane and on the sphere.

Understanding and Mitigating Lattice Collapse Degradation in Layered Oxide Materials for Sodium-Ion Battery Anode.

Na2Ti3O7 has attracted significant attention due to its ecofriendliness and cost-effectiveness for sodium-ion batteries. However, their limited cycling stability hampers their practical applications. Herein, we elucidate a mechanism of structural degradation caused by the heterogeneous phase transition in the Na2Ti3O7 anode using aberration-corrected (scanning) transmission electron microscopy (S)TEM and in situ TEM. It is found that the unevenly distributed phase transition results in the accumulation of strain, which promotes the growth of microcracks and eventually leads to structural decomposition and electrochemical failure. Motivated by this degradation mechanism, nanowires were proposed, and the structural stability is thus improved with the lattice strain effectively released. These findings deepen our understanding of ion transport and degradation mechanisms in intercalated layered electrode materials while emphasizing the significance of the material structure engineered for improving electrode performance.

Universal Phenomenology at Critical Exceptional Points of Nonequilibrium O(N) Models

In thermal equilibrium the dynamics of phase transitions is largely controlled by fluctuation-dissipation relations: On the one hand, friction suppresses fluctuations, while on the other hand, the thermal noise is proportional to friction constants. Out of equilibrium, this balance dissolves and one can have situations where friction vanishes due to antidamping in the presence of a finite noise level. We study a wide class of O(N) field theories where this situation is realized at a phase transition, which we identify as a critical exceptional point. In the ordered phase, antidamping induces a continuous limit cycle rotation of the order parameter with an enhanced number of 2N−3 Goldstone modes. Close to the critical exceptional point, however, fluctuations diverge so strongly due to the suppression of friction that in dimensions d<4 they universally either destroy a preexisting static order or give rise to a fluctuation-induced first-order transition. This is demonstrated within a full resummation of loop corrections via Dyson-Schwinger equations for N=2, and a generalization for arbitrary N, which can be solved in the long wavelength limit. We show that in order to realize this physics it is not necessary to drive a system far out of equilibrium: Using the peculiar protection of Goldstone modes, the transition from an xy magnet to a ferrimagnet is governed by an exceptional critical point once weakly perturbed away from thermal equilibrium. Published by the American Physical Society 2024

Hopf bifurcation and limit cycle of the two‐variable Oregonator model for Belousov–Zhabotinsky reaction

This paper is concerned with a two‐variable Oregonator model for Belousov–Zhabotinsky Reaction. Llibre and Oliveira (2022) proved that Oregonator model with an unstable node or focus has at least one limit cycle in the positive quadrant, and the system has a unique stable limit cycle for some values of the parameters. It is shown in the present paper that the positive equilibrium is not a center, and that if the system has a positive equilibrium which is a weak focus, then its order is at most 2. There exist some parameter values such that system has small limit cycle around the positive equilibrium.

Optimal control of oscillatory neuronal models with applications to communication through coherence

Macroscopic oscillations in the brain are involved in various cognitive and physiological processes, yet their precise function is not completely understood. Communication through coherence (CTC) theory proposes that these rhythmic electrical patterns might serve to regulate the information flow between neural populations. Thus, to communicate effectively, neural populations must synchronize their oscillatory activity, ensuring that input volleys from the presynaptic population reach the postsynaptic one at its maximum phase of excitability. We consider an Excitatory–Inhibitory (E–I) network whose macroscopic activity is described by an exact mean-field model. The E–I network receives periodic inputs from either one or two external sources, for which effective communication will not be achieved in the absence of control. We explore strategies based on optimal control theory for phase–amplitude dynamics to design a periodic control that sets the target population in the optimal phase to synchronize its activity with a specific presynaptic input signal and establish communication. The control mechanism resembles the role of a higher cortical area in the context of selective attention. To design the control, we use the phase–amplitude reduction of a limit cycle and leverage recent developments in this field in order to find the most effective control strategy regarding a defined cost function. Furthermore, we present results that guarantee the local controllability of the system close to the limit cycle.

MOFs-derived CoMn-layered double hydroxide array anchored to Ti3C2Tx MXene nanosheets as efficent catalysts for rechargeable lithium-oxygen batteries

Rechargeable lithium-oxygen batteries (LOBs) are acknowledged as potential candidates for advanced energy storage devices as a result of their considerable theoretical energy densities. Nevertheless, they encounter significant challenges throughout discharging and recharging phases, including the difficult decomposition of Li2O2 products, sluggish reaction kinetics, and limited cycle stability. In this study, an efficient catalyst was synthesized by in situ growth of metal–organic frameworks (MOFs)-derived cobalt manganese layered double hydroxide (CoMn-LDH) array on Ti3C2Tx MXene nanosheets (CoMn-LDH@Ti3C2Tx). The LOBs with CoMn-LDH@Ti3C2Tx cathode demonstrated high discharge specific capacity of 27.61 Ah g−1 at 100 mA g−1 and an extended cycle stability of over 224 cycles at 500 mA g−1. Density functional theory (DFT) computations showed that CoMn-LDH@Ti3C2Tx facilitates electron redistribution, optimising the adsorption energy of intermediates, thereby lowering the energy barriers for oxygen reduction reactions (ORR) and oxygen evolution reactions (OER). Consequently, this work presents a multi-level nanostructured catalyst preparation strategy derived from MOFs, offering new insights for the fabrication of efficient electrocatalytic materials.

Finite‐time control of DC–AC boost converter: The limit cycle approach

AbstractThis paper proposes a novel finite‐time limit cycle controller for DC to AC conversion in power boost converters. The stable limit cycle is categorized as a positive limit set. Achieving the control law, based on the concepts of set stability and finite‐time stability, is a challenging problem. To fulfill the control objective, a novel sliding manifold is suggested concerning the structure of the wanted limit cycle which facilitates conditions for the development of finite‐time stability for the limit cycle. By utilizing the sliding mode control method, a control strategy is proposed that guarantees the desired stable limit cycle is created in the phase plane of the closed‐loop system and the phase trajectories reach it in a finite time and remain on it for all future times. This ensures the finite‐time generation of the biased sinusoidal oscillations in the output of the power boost converter from a DC input source. The simulation results of a practical boost converter have validated the effectiveness and feasibility of the presented algorithm.

Higher-Order Network Interactions Through Phase Reduction for Oscillators with Phase-Dependent Amplitude

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.

On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil

The limit cycle oscillation (LCO) in the transonic buffet on the fixed supercritical airfoil OAT15A under Ma = 0.73, AoA = 3.5° and Re≈3×106, is successfully simulated by means of the Reynolds Stress Model. Further, the buffet lock-in phenomena under prescribed pitch conditions near the buffet onset are also studied by evaluating the modified energy exchange based on the pitching component of the moment coefficients as well as the normalized relative phase map between the pitching component of the moment coefficients and the airfoil’s angular velocity. The zero energy transfer branches in the modified energy map fail to indicate the lock-in boundaries, while the normalized phase map generally outlines the lock-in boundaries for small pitch amplitudes near the buffet onset, which suggests that the lock-in occurs where the moment is in phase with the angular velocity at small pitch amplitudes near the buffet onset. For pitch amplitudes larger than 0.4°, the lock-in onset deviates from the phase shift, possibly due to the fact the instantaneous angle of attack can be lower than the buffet onset where the buffet phenomena may vanish.

At Most Five Limit Cycles in a Class of Discontinuous Piecewise Linear Systems with Three Zones

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems.

Transition to oscillatory instability of lean methane–air flames in microchannels

Micro-flow reactors and porous media burners are prone to instability in the form of flame with repetitive extinction and ignition (FREI) near the region of a temperature gradient. Although significant progress has been made in this field, there is still a lack of understanding regarding the nature of such stable-to-unstable transition. Some studies reported a smooth and continuous instability transition in microchannels, interpreting this phenomenon as a supercritical bifurcation. Meanwhile, others have observed that the transition is subcritical, with rapid, stepwise evolution of the stable flame to FREI. This study aims to rigorously determine the transition character theoretically and numerically using a two-dimensional model with a precise resolution of the bifurcation parameter (flow velocity) near the critical point and gradual wall temperature ramp. The results indicate that the transition is a subcritical bifurcation with strong hysteresis. However, the amplitude of the limit cycle born in the bifurcation point could be small compared to FREI-like oscillations. Moreover, further development of the instability depends significantly on the channel width. In relatively wide channels, the stable regime transforms immediately into the extinction/ignition one with rapid growth in amplitude. In moderate channels, complex transitional dynamics were observed with an evident pulsating regime, which evolved into FREI through mixed-mode oscillations. In narrow channels, the amplitude of transitional pulsating flame becomes comparable to the fully-developed extinction/ignition cycles, and the bifurcation diagram has a continuous character with no significant jumps or discontinuities. Such qualitative variation of transition character provides a possible explanation for contradictory interpretations presented in the literature.Novelty and Significance StatementCurrently, there is no consensus about the character of transition between the stable and extinction/ignition regimes in microchannels. Some studies have reported a continuous instability transition with an evident pulsating regime, interpreting it as a supercritical bifurcation, while others have observed subcritical bifurcation with a rapid transition. The novelty of this research lies in the rigorous determination of the bifurcation type and further instability development in microchannels of different widths based on an accurate simulation of flame dynamics near the critical point with very small steps of the flow velocity variation, along with the bifurcation theory. The study contributes to understanding the instability transition nature and may explain the different interpretations of the transitional phenomenon found in the literature. The results offer valuable insights for designing micro-scale and porous media combustion devices.

High-entropy doping promising ultrahigh-Ni Co-free single-crystalline cathode toward commercializable high-energy lithium-ion batteries.

The development of advanced layered Ni-rich cathodes is essential for high-energy lithium-ion batteries (LIBs). However, the prevalent Ni-rich cathodes are still plagued by inherent issues of chemomechanical and thermal instabilities and limited cycle life. For this, here, we introduce an efficient approach combining single-crystalline (SC) design with in situ high-entropy (HE) doping to engineer an ultrahigh-Ni cobalt-free layered cathode of LiNi0.88Mn0.03Mg0.02Fe0.02Ti0.02Mo0.02Nb0.01O2 (denoted as HE-SC-N88). Thanks to the SC- and HE-doping merits, HE-SC-N88 is featured with a grain-boundary-free and stabilized structure with minimal lattice strain, preventing mechanical degradation, reducing surface parasitic reactions, and mitigating oxygen loss. Accordingly, our HE-SC-N88 cathode demonstrates exceptional electrochemical properties particularly with prolonged cycling stability under strenuous conditions in both half and full cells, and the delayed O loss-induced phase transitions upon heating. More meaningfully, our design of HE doping in redefining the ultrahigh-Ni Co-free SC cathodes will make a tremendous progress toward industrial application of next-generation LIBs.

Limit cycles of piecewise smooth differential systems of the type nonlinear centre and saddle

Piecewise linear differential systems separated by two parallel straight lines of the type of centre-centre-Hamiltonian saddle and the centre-Hamiltonian saddle-Hamiltonian saddle can have at most one limit cycle and there are systems in these classes having one limit cycle. In this paper, we study the limit cycles of a piecewise smooth differential system separated by two parallel straight lines formed by nonlinear centres and a Hamiltonian saddle.

A necessary and sufficient condition on algebraic limit cycles of a hybrid van der Pol-Rayleigh oscillator

We prove in this paper that a hybrid van der Pol-Rayleigh oscillator x˙=y,y˙=−x−αy−βy3−x2y with (α,β)∈R2 has no irreducible invariant algebraic curves in C[x,y] except for (α,β)=(0,−1), or α2=−β−1/β+2, β≠0 or α≠0, β=1, and that the oscillator has only one irreducible invariant algebraic curve in R[x,y] when (α,β)=(0,−1), or α2=−β−1/β+2, β<0, or α∈R, β=1. These together with that in Chen et al. (2021) [7] ensure that the hybrid van der Pol-Rayleigh oscillator has an algebraic limit cycle if and only if β=1 and α<0. This result solves the conjecture posed in that paper.