Limit Cycle Research Articles

Tuning limit cycles with a noise: Survival and collapse.

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equationof motion for slow dynamics after appropriate averaging over the fast motion. The equationfor slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

Rich dynamics of a delayed Filippov avian-only influenza model with two-thresholds policy

A nonsmooth Filippov avian-only influenza model with threshold strategies of culling susceptible and/or infected birds under different conditions is proposed to control the spread of avian influenza. The time delay representing the incubation period of avian influenza is incorporated into our model to make it more realistic, which is different from the traditional Filippov model. The stability of various types of equilibria and the existence of Hopf bifurcation are researched. Moreover, the existence of the sliding mode and its dynamics are investigated by Filippov convexity method. The theoretical analysis and numerical simulations indicate that, according to the value of thresholds and time delay, all solutions eventually converge to the regular equilibrium, pseudoequilibrium or stable periodic solution. We also indicate that time delay has a great influence on the sliding mode. Due to the many factors involved, and our main purpose is to study the impact of time delay on system stability, we have only conducted a simple analysis of the impact of time delay on the sliding mode. Furthermore, the boundary bifurcation switching stable regular equilibrium or stable limit cycle to a stable pseudoequilibrium can be exhibited by numerical simulations. Finally, with the increase of time delay, the global bifurcations from grazing bifurcation to buckling bifurcation and then to cross bifurcation are obtained. Our results indicate that although Filippov control strategies can effectively control the number of infected birds in many cases, the existence of time delay may challenge influenza control by the emergence of buckling bifurcation and cross bifurcation.

On limit cycles of autonomous systems

We consider the problem of the existence of limit cycles for autonomous systems of differential equations. We present quite elementary considerations that can be useful in discussing qualitative issues that arise in the course of ordinary differential equations. We establish that any simple closed curve defined by the equation \(F(x,y)=1\) with a sufficiently general function \(F\) is a limit cycle for the corresponding autonomous system on the plane (and even for an infinite number of systems depending on the real parameter). These systems are written out explicitly. We analyze in detail several specific examples. Graphic illustrations are provided.

Multiparameter Bifurcation Analysis of Power Systems Integrating Large-Scale Solar Photovoltaic and Wind Farms Power Plants

In this paper, we propose a novel codimension-three-parameter bifurcation analysis of equilibria and limit cycles when integrating Renewable Energy Sources (RESs) power plants with an exponential static load model. The study investigates the effect of solar photovoltaic generation margin, wind power generation margin, and loading factor on the local bifurcation of the modified IEEE nine-bus system. The proposed technique considers the real case of the West System Coordination Council (WSCC), the western states of the USA, by using specific models of RES power plants and static loads. The proposed technique helps to create a set of linearly varying parameters for critical operating points of nonlinear systems. The study explores detailed voltage stability analysis through the examination of bifurcation diagrams. The Hopf, limit-induced, and saddle-node bifurcation branches are identified, defining the parameter space’s stable and unstable operational regions. Additionally, the stability regions surrounding the equilibrium points are diligently explored, clarifying the consequences that various bifurcations may exert on these regions. The study offered in this proposed work aids in determining the best ways to monitor and improve these margins while considering system variables and load design.

Data-Driven Bayesian Inference for Stochastic Model Identification of Nonlinear Aeroelastic Systems

The objective of this work is to propose a data-driven Bayesian inference framework to efficiently identify parameters and select models of nonlinear aeroelastic systems. The framework consists of the use of Bayesian theory together with advanced kriging surrogate models to effectively represent the limit cycle oscillation response of nonlinear aeroelastic systems. Three types of sampling methods, namely, Markov chain Monte Carlo, transitional Markov chain Monte Carlo, and the sequential Monte Carlo sampler, are implemented into Bayesian model updating. The framework has been demonstrated using a nonlinear wing flutter test rig. It is modeled by a two-degree-of-freedom aeroelastic system and solved by the harmonic balance methods. The experimental data of the flutter wing is obtained using control-based continuation techniques. The proposed methodology provided up to a 20% improvement in accuracy compared to conventional deterministic methods and significantly increased computational efficiency in the updating and uncertainty quantification processes. Transitional Markov chain Monte Carlo was identified as the optimal choice of sampling method for stochastic model identification. In selecting alternative nonlinear models, multimodal solutions were identified that provided a closer representation of the physical behavior of the complex aeroelastic system than a single solution.

On Stability Discrimination of Limit Cycles for Piecewise Smooth Systems

On Stability Discrimination of Limit Cycles for Piecewise Smooth Systems

On the Number of Limit Cycles Bifurcating from the Linear Center with a Cubic Switching Curve

On the Number of Limit Cycles Bifurcating from the Linear Center with a Cubic Switching Curve

Nonlinear State-Space Model of Self-excited forces for Bluff Body Aeroelasticity

This paper introduces a novel state-space model of nonlinear self-excited forces designed to capture amplitude dependency and unsteady effect in bluff body aeroelasticity. In the present form, this model represents a theoretical extension of the classical linear state-space model into the nonlinear regime, particularly relevant when a bluff section undergoes large amplitude oscillations. The proposed model incorporates additional state variables to approximate nonlinear convolution-based indicial functions, thereby providing a valuable tool for estimating nonlinear wind-induced instabilities, including nonlinear flutter limit cycle oscillation (LCO), vortex-induced vibration (VIV), and unsteady galloping. Furthermore, we establish the analytical foundation for identifying nonlinear parameters through the equivalent linearization of nonlinear transfer functions and the development of a numerical solving algorithm for the nonlinear governing equation. The feasibility of this model is rigorously validated through experimental results pertaining to nonlinear flutter, unsteady galloping, and VIV of typical bluff sections. Additionally, this model serves as the cornerstone for a nonlinear analytical framework in the time domain, facilitating the integration of both aerostatic and structural nonlinearities into comprehensive structural analyses.

Garnet-Based Solid-State Li Batteries with High-Surface-Area Porous LLZO Membranes.

Rechargeable garnet-based solid-state Li batteries hold immense promise as nonflammable, nontoxic, and high energy density energy storage systems, employing Li7La3Zr2O12 (LLZO) with a garnet-type structure as the solid-state electrolyte. Despite substantial progress in this field, the advancement and eventual commercialization of garnet-based solid-state Li batteries are impeded by void formation at the LLZO/Li interface at practical current densities and areal capacities beyond 1 mA cm-2 and 1 mAh cm-2, respectively, resulting in limited cycling stability and the emergence of Li dendrites. Additionally, developing a fabrication approach for thin LLZO electrolytes to achieve high energy density remains paramount. To address these critical challenges, herein, we present a facile methodology for fabricating self-standing, 50 μm thick, porous LLZO membranes with a small pore size of ca. 2.3 μm and an average porosity of 51%, resulting in a specific surface area of 1.3 μm-1, the highest reported to date. The use of such LLZO membranes significantly increases the Li/LLZO contact area, effectively mitigating void formation. This methodology combines two key elements: (i) the use of small pore formers of ca. 1.5 μm and (ii) the use of ultrafast sintering, which circumvents ceramics overdensification using rapid heating/cooling rates of ca. 50 °C per second. The fabricated porous LLZO membranes demonstrate exceptional cycling stability in a symmetrical Li/LLZO/Li cell configuration, exceeding 600 h of continuous operation at a current density of 0.1 mA cm-2.

Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system

Abstract The fractional-order controllers (FOCs) have recently had a significant impact on control applications. However, they still need further research for feedback systems with hard nonlinearities, such as dead zones. The above compelling evidence motivates the design of a new robust FOC to avoid limit cycles caused by dead zones in the control loops. The proposed FOC consists of the cascade of two shifted in frequency, fractional-order lead networks. They provide high-value and sufficiently flat phase leads in sufficiently large frequency intervals. In this way, the linear part of the control loop can be easily shaped to achieve avoidance of limit cycles. The article applies classical concepts, such as the Nyquist plot and describing function method, to derive guidelines for designing the free parameters of the FOC. Moreover, a realization algorithm and a parameter setting procedure make the new FOC easily implementable in engineering practice.

Filtering induced explosive death in coupled FitzHugh–Nagumo neurons: Theory and experiment

Explosive emergent behavior governed by discontinuous and irreversible phase transition is in the center of recent research. In this paper we show that explosive amplitude death (AD) can be induced in a system of coupled oscillators if diffusion is accompanied by low-pass filtering in the mutual coupling path. Here we consider a ring network of diffusively coupled FitzHugh–Nagumo oscillators, and through theoretical and experimental investigations explore how the low-pass filter induces and controls the transition to explosive AD state. We derive the conditions of getting AD in the network through linear stability analysis. Using one and two parameter bifurcation analysis we find the explicit parametric zone of explosive death in the limiting case of two neurons, which reveals that the explosive death arises through the saddle–node bifurcation of limit cycle. In the strong coupling limit, we observe an interesting semi-explosive death scenario, where both continuous and discontinuous transitions are present. Finally, using electronic hardware circuits, we demonstrate the first experimental observation of explosive amplitude death that establishes the robustness of the scenario in a practical setup.

An upper bound for the number of small-amplitude limit cycles in non-smooth Liénard system

The non-smooth Liénard system, a well-known nonlinear model, appears in a natural way in physics, chemistry, biology, and so on, in which periodic phenomena play a relevant role. In this paper, we investigate the small-amplitude limit cycles generated by the Hopf bifurcation of the non-smooth Liénard system. By utilizing the Picard–Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles for a generic non-smooth Liénard system. Finally, an example is given to illustrate the efficiency of the theoretical results.

Samuelson's last macroeconomic model: Secular stagnation and endogenous cyclical growth

On the occasion of the centennial of his mentor Alvin Hansen, Paul Samuelson published in 1988 a modified version of his seminal 1939 multiplier-accelerator model to address aspects of Hansen's secular stagnation hypothesis. The “Keynes-Hansen-Samuelson” model (or KHS, as he called it) was built to analyse the effects of population growth on the economy's trajectory. Several changes were then made. Instead of difference equations and a tight accelerator, as in his 1939 model, Samuelson deployed differential equations and a flexible accelerator to produce a nonlinear limit cycle. Despite Samuelson's strong claims for the analytical contributions of his 1988 paper, it has – in contrast with the 1939 model – received only scant attention by macroeconomists and historians of economics alike. Samuelson's 1988 paper was his last published macroeconomic model, based on his long-established tradition of non-optimising macro-dynamics. Our paper provides a close reading of that article and some analytical results that shed new light on the formal aspects of Samuelson's 1988 model. We also discuss how it historically links up with business cycle models advanced by John Hicks, Nicholas Kaldor, Roy Harrod and Richard Goodwin and examine how far Samuelson's use of the term secular stagnation differs from Larry Summers's recent reconstruction of it.

A Complex Dynamic of an Eco-Epidemiological Mathematical Model with Migration

In this paper, we propose an eco-epidemiological mathematical model in order to describe the effect of migration on the dynamics of a prey–predator population. The functional response of the predator is governed by the Holling type II function. First, from the perspective of mathematical results, we develop results concerning the existence, uniqueness, positivity, boundedness, and dissipativity of solutions. Besides, many thresholds have been computed and used to investigate the local and global stability results by using the Routh–Hurwitz criterion and Lyapunov principle, respectively. We have also established the appearance of limit cycles resulting from the Hopf bifurcation. Numerical simulations are performed to explore the effect of migration on the dynamic of prey and predator populations.

Global planar dynamics with a star node and contracting nonlinearity

This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the existence of an invariant circle and allows us to obtain a classification through a complete invariant for the dynamics, extending previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to some classes of examples: definite nonlinearities, Z2⊕Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {Z}}_2\\oplus {\ extbf {Z}}_2$$\\end{document} symmetric systems and nonlinearities of degree 3, for which we provide complete sets of phase-portraits.

Four Limit Cycles of Three-Dimensional Discontinuous Piecewise Differential Systems Having a Sphere as Switching Manifold

Because of their applications, the study of piecewise-linear differential systems has become increasingly important in recent years. This type of system already exists to model many different natural phenomena in physics, biology, economics, etc. As is well known, the study of the qualitative theory of piecewise differential systems focuses mainly on limit cycles. Most papers studying the problem of existence and the maximum number of limit cycles of piecewise differential systems have precisely considered planar systems. However, few papers have examined this problem in [Formula: see text]. In this paper, our main goal is to examine a class of discontinuous piecewise differential systems in [Formula: see text], where we consider the unit sphere as the separation surface that divides the entire space into two regions, each one has a linear vector field analogous to planar center. In general, it is hard to determine an exact upper bound for the number of limit cycles that a class of differential systems can exhibit. We prove that this class of differential systems can have at most four limit cycles. We show that there are examples of such differential systems with exactly 1, 2, 3 and 4 limit cycles.

Stability, period and chaos of the evolutionary game strategy induced by time-delay and mutation feedback

In the classical evolutionary game theory, mutation is usually considered as a constant, however strategy mutation is affected by strategies in the real game process. Therefore, the main purpose of this paper is to study the effects of mutation feedback and time delays on strategy dynamics, where mutation is a linear feedback related to strategy. First, we construct a co-evolutionary game model with two time delays induced by mutation feedback and analyze the existence and stability of the equilibria of the non-time delay system. The conditions for the coexistence of the two strategies and the mutation rate are obtained. Second, Sotomayor’s theorem is used to explore the transcritical bifurcation of the system. Then, the existence of Hopf bifurcation is investigated by using feedback delay and payoff delay as bifurcation parameters in the time-delay system. Furthermore, we discuss the direction of the Hopf bifurcation, stability and periodic change of the periodic solution in detail. Finally, a series of numerical simulations are used to describe the theoretical analysis. The main results are as follows: (i) Mutation causes negative feedback to cooperative strategy. (ii) As the time delay increases, the stable equilibrium point becomes unstable, and which branches a stable limit cycle. When the time delay continues to increase sufficiently, the stable limit cycle becomes unstable and produces irregular oscillation and chaos. (iii) When the two time delays are large enough, the coexistence of the two strategies becomes that defective strategy is dominant, and the mutation rate also reaches the maximum.

How Drivers Lose Control of the Car

<div>After a severe lane change, a wind gust, or another disturbance, the driver might be unable to recover the intended motion. Even though this fact is known by any driver, the scientific investigation and testing on this phenomenon is just at its very beginning, as a literature review, focusing on SAE Mobilus<sup>®</sup> database, reveals. We have used different mathematical models of car and driver for the basic description of car motion after a disturbance. Theoretical topics such as nonlinear dynamics, bifurcations, and global stability analysis had to be tackled. Since accurate mathematical models of drivers are still unavailable, a couple of driving simulators have been used to assess human driving action. Classic unstable motions such as Hopf bifurcations were found. Such bifurcations seem almost disregarded by automotive engineers, but they are very well-known by mathematicians. Other classic unstable motions that have been found are “unstable limit cycles.” The driving simulator results have been reproduced by experimental tests on track. We have assessed that the driver’s steering action can make the car motion unstable if a proper disturbance has acted. The delay of the driver’s steering action is the primary cause for the generation of limit cycles. Future automated vehicles should be conceived by focusing on the addressed phenomenon.</div>

Effects of whaling and krill fishing on the whale-krill predation dynamics: bifurcations in a harvested predator-prey model with Holling type I functional response.

In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill-whale interaction we revisit a harvested predator-prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region , the model exhibits degenerate Bogdanov-Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region , the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of , the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line under some specific parameter conditions.

Dynamics of a Class of Prey–Predator Models with Singular Perturbation and Distributed Delay

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter [Formula: see text], the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results.