Limit Cycle Conditions Research Articles

Stability analysis for the phytoplankton-zooplankton model with depletion of dissolved oxygen and strong Allee effects

The photosynthetic activity of phytoplankton in the seas is responsible for an estimated 50–80 % of the world's oxygen generation. Both phytoplankton and zooplankton require some of this synthesized oxygen for cellular respiration. This study aims to better understand how the oxygen-phytoplankton dynamics are altered due to the Allee effect in phytoplankton development, particularly when considering the time-dependent oxygen generation rate. The dynamic analysis of the model is dedicated to finding the possible equilibrium points. The analysis reveals that three equilibrium points can be obtained. The stability study demonstrates that one of the equilibrium points is always stable. The remaining equilibrium points are stable under specific conditions. We also identify bifurcations originating from these equilibrium points, including transcritical, pitchfork, and Hopf bifurcation. We derive conditions for stable limit cycles (supercritical Hopf bifurcation) and, in some cases, establish the non-existence of solutions. Numerical simulations are performed to validate our theoretical findings. Furthermore, it is noted that the Allee threshold for the phytoplankton population (k0) significantly influences the overall dynamics of the system. When k0≤0.001, the population of plankton is at risk of extinction. On the other hand, when 0.001<k0≤0.01, the population of zooplankton is at risk of extinction. When 0.01<k0≤2, the solution reaches a stable condition of coexistence. Conversely, when k0≥2.1, the solution exhibits periodic attractor behaviour.

Investigation of the Existence of Limit Cycles in Multi Variable Nonlinear Systems with Special Attention to 3X3 Systems

The proposed work addresses the dynamics of a general system and explores the existence of limit cycles (LC) in multi-variable Non-linear systems with special attention to 3x3 nonlinear systems. It presents a simple, systematic analytical procedure as well as a graphical technique that uses geometric tools and computer graphics for the prediction of limit cycling oscillations in three-dimensional systems having both explicit and implicit nonlinear functions. The developed graphical method uses the harmonic balance/harmonic linearization for simplicity of discussion which provides a clear and lucid understanding of the problem and considers all constraints, especially the simultaneous intersection of two straight lines &amp; one circle for determination of limit cycling conditions. The method of analysis is made simpler by assuming the whole system exhibits the limit cycling oscillations predominantly at a single frequency. The discussions made either analytically/graphically are substantiated by digital simulation by a developed program as well as by the use of the SIMULINK Toolbox of MATLAB Software.

In-Depth Analysis of Smooth and Nonsmooth Bifurcations for an Open-Loop Boost Converter Feeding Constant Power Loads in Discontinuous Conduction Mode

This paper presents a study of the existence conditions of limit cycles and mechanisms when losing their stability in an open-loop DC–DC boost converter loaded with a constant power load and operating in discontinuous conduction mode. First, the existence conditions are determined. Then, boundaries associated with different kinds of instabilities such as the classical smooth fold and period-doubling bifurcations and nonsmooth border-collision bifurcations such as fold border-collision bifurcations are elucidated. To perform the stability analysis, an implicit 1D map for the system is derived. From this model, it is obtained that smooth period-doubling and fold bifurcations may occur at certain values of the circuit parameters. Some results from numerical simulations performed on the circuit-based switched model are provided showing consistency with the theoretical analysis from the derived model.

Stability and Hopf bifurcation of an epidemiological model with effect of delay the awareness programs and vaccination: analysis and simulation

The present research is concerned with studying the media coverage delay impact and vaccine impact to control outbreaks the infectious disease in the population, we suggested that our mathematical modeling divide the population into three classes: susceptible persons S(t), vaccinated person V(t) and infected persons I(t). The susceptible individuals are also divided due to media programs into awareness and unawareness of disease danger and its mode of transmission. We studied the influence of time delay in response to the media program on awareness of the epidemic. We first discussed the existence of the equilibrium points of the system and obtain sufficient conditions for the local asymptotic stability of all equilibrium points. Further, the existence of Hopf bifurcation is shown near endemic equilibrium. It is interesting to note that there exists at least one limit cycle around the unstable endemic equilibrium. In particular, sufficient conditions for a unique stable limit cycle have been presented. Finally, we verified the feasibility of the results by numerical simulation and give a brief conclusion to generalize that the delay effect of media and vaccination can cause unexpected dynamic predictions for interacting populations.

Turing Instabilities and Rotating Spiral Waves in Glycolytic Processes.

We study single-frequency oscillations and pattern formation in the glycolytic process modeled by a reduction in the well-known Sel'kov's equations (Sel'kov in Eur J Biochem 4:79, 1968), which describe, in the whole cell, the phosphofructokinase enzyme reaction. By using averaging theory, we establish the existence conditions for limit cycles and their limiting average radius in the kinetic reaction equations. We analytically establish conditions on the model parameters for the appearance of unstable nonlinear modes seeding the formation of two-dimensional patterns in the form of classical spots and stripes. We also establish the existence of a Hopf bifurcation, which characterizes the reaction dynamics, producing glycolytic rotating spiral waves. We numerically establish parameter regions for the existence of these spiral waves and address their linear stability. We show that as the model tends toward a suppression of the relative source rate, the spiral wave solution loses stability. All our findings are validated by full numerical simulations of the model equations. Finally, we discuss in vitro evidence of spatiotemporal activity patterns found in glycolytic experiments, and propose plausible biological implications of our model results.

Mathematical model with time‐delay and delayed controller for a bioreactor

In this paper, a fractional Lotka–Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time‐delay due to the dead‐time (non‐trivial) that the microbe needed to metabolize the substrate. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators, or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller , which is shown to fail to stabilize the obtained model under the proposed analysis. Another suitable choice is the use of a delayed controller which successfully stabilizes the model even when it is unstable. The delay in the feedback control is due to the dead‐time necessary to obtain the measurement of the biomass in the bioreactor by dry weight. Finally, the proposed theoretical results are corroborated through numerical simulations.

A Study of the Lyapunov Stability of an Self-Excited Induction Generator

The self-excited induction generators (SEIGs) can provide attractive renewable energy sources. Since it can be used in an open-loop fashion which includes no control closed-loop, its inherent stability can be counted on to allow operation over a wide range of operating conditions. Unlike classical arguments based on transient and steady-state analysis, it is proposed rigorous holistic analytical stability arguments based on the full nonlinear transient state-space model of a SEIG under two-phase stationary reference frame using dynamical systems theory. The conditions for critical transient stability of the unloaded SEIG are obtained in the sense of Lyapunov. Analytic formulas from the limit cycle conditions give the values of the range of capacitance, the range of rotor speed and the stator frequency for self-excitation. Analytic formulas from steady-state conditions give all the steady-state parameter and state values. Thus, the analytical results yield a set of simple and universal analytic formulas that can predict and evaluate the transient and steady-state stability characteristics of the SEIG. Good agreement between the experimental results and computed results validates the analytical results.

On dynamics in a medium-term Keynesian model

&lt;p style='text-indent:20px;'&gt;This paper rigorously examines the (in)stability of limit cycles generated by Hopf bifurcations in a medium-term Keynesian model. The bifurcation equation of the model is derived and the conditions for stable and unstable limit cycles are presented. Numerical simulations are performed to illustrate the analytical results.&lt;/p&gt;

Modeling and approximate analytical solution of nonlinear behaviors for a self-excited electrostatic actuator

Analytical modeling and solution of nonlinear vibration behaviors are vitally important for the optimization of vibratory actuators. In this paper, the nonlinear vibration characteristics of a self-excited electrostatic actuator are investigated. The actuator consists of two plate electrodes and a metal cantilever, which is fixed on an isolated insulating base and placed in the middle of the electrodes. When a DC voltage is applied to the electrodes, the cantilever is excited into reciprocating vibration autonomously and collides with the electrodes with strongly nonlinear features. During modeling, the collision is considered as a transient process and only the velocities of the cantilever before and after the collisions are concerned. By using the phase plane method, the limit cycle and boundary conditions of the self-excited oscillation of the cantilever are obtained and the strongly nonlinear problem of the entire system is converted into a weakly nonlinear problem of the cantilever moving between the electrodes. The nonlinear equations are then solved by using the averaging method, and the approximate analytical results demonstrate good consistency with the numerical and experimental results. Besides, the approximate analytical results are also adopted in the optimization of the actuator for enhanced power density and efficiency. The modeling and solution methods provide a framework for further optimization of the electrostatic actuator in the application field of centimeter-scale robotics.

Approximate Limit Cycles for Vortex-Induced Vibration of a Sprung Cylinder

AbstractThe homotopy analysis method (HAM) is an analytical approximate method to solve nonlinear problems, which is employed to give series solutions of a sprung cylinder's vortex-induced vibration. The wake flow is modeled by a classical van der Pol oscillation coupled with a cylinder by an acceleration term. The frequency and initial conditions of all possible limit cycles are obtained as Maclaurin series of an embedding parameter. A series of algebraic equations for eliminating secular terms are derived and solved to obtain the priori unknown coefficients, such as the initial conditions and frequency. The validity and efficiency of the HAM are conducted by numerical integration solutions and the harmonic balance method (HBM). The influence of fluid velocity on the frequencies and amplitudes of the limit cycles are obtained very accurately compared with numerical ones.

Infinite boundary conditions for response functions and limit cycles within the infinite-system density matrix renormalization group approach demonstrated for bilinear-biquadratic spin-1 chains

Response functions $\langle A_x(t) B_y(0)\rangle$ for one-dimensional strongly correlated quantum many-body systems can be computed with matrix product state (MPS) techniques. Especially, when one is interested in spectral functions or dynamic structure factors of translation-invariant systems, the response for some range $|x-y|<\ell$ is needed. We demonstrate how the number of required time-evolution runs can be reduced substantially: (a) If finite-system simulations are employed, the number of time-evolution runs can be reduced from $\ell$ to $2\sqrt{\ell}$. (b) To go beyond, one can employ infinite MPS (iMPS) such that two evolution runs suffice. To this purpose, iMPS that are heterogeneous only around the causal cone of the perturbation are evolved in time, i.e., the simulation is done with infinite boundary conditions. Computing overlaps of these states, spatially shifted relative to each other, yields the response functions for all distances $|x-y|$. As a specific application, we compute the dynamic structure factor for ground states of bilinear-biquadratic spin-1 chains with very high resolution and explain the underlying low-energy physics. To determine the initial uniform iMPS for such simulations, infinite-system density-matrix renormalization group (iDMRG) can be employed. We discuss that, depending on the system and chosen bond dimension, iDMRG with a cell size $n_c$ may converge to a non-trivial limit cycle of length $m$. This then corresponds to an iMPS with an enlarged unit cell of size $m n_c$.

Non-linear dynamics of thermoacoustic eigen-mode interactions

The natural eigen-modes of a combustion system are a strong function of its geometric design, temperature profile, and boundary conditions. Many practical systems have multiple, closely-spaced natural mode frequencies. While several prior studies have noted the nonlinear interactions that occur when multiple, linearly unstable modes are present, the nature of these interactions for closely-spaced modes has not been analytically treated. This paper treats this problem theoretically by utilizing a Galerkin expansion of the Euler equations, and the method of averaging to derive a set of amplitude equations for the modal amplitudes. In cases where the frequency spacing is large, many of the oscillatory terms have short time-scales and average out. However, in the case of closely-spaced modes, terms oscillating at the frequency difference between the closely-spaced frequencies correspond to long time-scales and, thus, remain after the averaging process. Results show that frequency spacing between the modes has significant impacts on limit-cycle amplitudes and their stability, even in the case of a static non-linearity. In addition, the conditions under which both modes can co-exist are a strong function of the frequency spacing. There are also certain conditions where one mode is completely suppressed by the other, even if both modes have positive linear growth rates; non-intuitively, the mode that is suppressed could be the one with the larger growth rate. The conditions under which one mode is suppressed are also a strong function of the frequency spacing parameter. Finally, for conditions in which one mode is suppressed, the limit cycle amplitude of the other mode is independent of the frequency spacing. Taken together these results show the difficulty in experimentally inferring the linear stability/instability of (non-dominant) combustor modes based upon “steady state” data taken during limit cycle conditions.

Chattering Analysis of HOSM Controlled Systems: Frequency Domain Approach

In this article, a methodology of chattering analysis of Sliding Mode/Higher Order Sliding Mode (SM/HOSM) control systems in the frequency domain is presented. A numerical method for computing the Describing Functions (DFs) of HOSM control algorithms is given. The algorithm for predicted chattering parameters in dynamically perturbed system via the Describing Function-Harmonic Balance (DF-HB) technique is proposed. The stability conditions for limit cycles in dynamically perturbed HOSM control systems are presented. The concepts of Practical Stability Phase Margin ( PSPM ) and Practical Stability Gain Margin ( PSGM ) as the robustness metrics to unmodeled dynamics in HOSM control systems are introduced. The methodologies for computing PSPM and PSGM via DF-HB technique are provided. The accuracy of the proposed chattering analysis techniques is confirmed via computer simulations.

Application of a high-order CFD harmonic balance method to nonlinear aeroelasticity

An Aeroelastic-Harmonic Balance (A-HB) formulation of the Euler flow equations using a high-order spatial discretization scheme coupled with structural dynamic equations is proposed. The main objective of this new approach is to dramatically reduce the computational cost required to predict unsteady, periodic problems such as limit cycle oscillations (LCO). To this end, a new solver based on the Monotonicity Preserving limiter together with the AUSM+-up flux function is developed for the harmonic balance equations. The use of high-order CFD schemes allows the reduction of the number of degrees of freedom required to achieve a given desired accuracy, with respect to lower order schemes. In this paper, the reduction in degrees of freedom of the fluid system is exploited in the context of a CFD based Harmonic-Balance framework using a frequency updating procedure to determine the limit cycle conditions. The standard A-HB methodology has shown over one order of magnitude speed-up over time-marching methods; by employing the proposed high-order scheme in conjunction with coarser grids, the LCO computational time is halved without compromising accuracy.

Mathematical modelling and analysis of two-component system with Caputo fractional derivative order

A class of generic spatially extended fractional reaction-diffusion systems that modelled predator-prey interactions is considered. The first order time derivative is replaced with the Caputo fractional derivative of order γ ∈ (0, 1). The local analysis where the equilibrium points and their stability behaviours are determined is based on the adoption of qualitative theory for dynamical systems ordinary differential equations. We derived conditions for Hopf bifurcation analytically. Most significantly, existence conditions for a unique stable limit cycle in the phase plane are determined analytically. Our analytical findings are in agreement with the numerical results presented in one and two dimensions. The system of fractional nonlinear reaction-diffusion equations has demonstrated the usefulness of understanding the dynamics of nonlinear phenomena.

Flame Describing Function analysis of spinning and standing modes in an annular combustor and comparison with experiments

This article reports a numerical analysis of combustion instabilities coupled by a spinning mode or a standing mode in an annular combustor. The method combines an iterative algorithm involving a Helmholtz solver with the Flame Describing Function (FDF) framework. This is applied to azimuthal acoustic coupling with combustion dynamics and is used to perform a weakly nonlinear stability analysis yielding the system response trajectory in the frequency-growth rate plane until a limit cycle condition is reached. Two scenarios for mode type selection are tentatively proposed. The first is based on an analysis of the frequency growth rate trajectories of the system for different initial solutions. The second considers the stability of the solutions at limit cycle. It is concluded that a criterion combining the stability analysis at the limit cycle with the trajectory analysis might best define the mode type at the limit cycle. Simulations are compared with experiments carried out on the MICCA test facility equipped with 16 matrix burners. Each burner response is represented by means of a global FDF and it is considered that the spacing between burners is such that coupling with the mode takes place without mutual interactions between adjacent burning regions. Depending on the nature of the mode being considered, two hypotheses are made for the FDFs of the burners. When instabilities are coupled by a spinning mode, each burner features the same velocity fluctuation level implying that the complex FDF values are the same for all burners. In case of a standing mode, the sixteen burners feature different velocity fluctuation amplitudes depending on their relative position with respect to the pressure nodal line. Simulations retrieve the spinning or standing nature of the self-sustained mode that were identified in the experiments both in the plenum and in the combustion chamber. The frequency and amplitude of velocity fluctuations predicted at limit cycle are used to reconstruct time resolved pressure fluctuations in the plenum and chamber and heat release rate fluctuations at two locations. For the pressure fluctuations, the analysis provides a suitable estimate of the limit cycle oscillation and suitably retrieves experimental data recorded in the MICCA setup and in particular reflects the difference in amplitude levels observed in these two cavities. Differences in measured and predicted amplitudes appear for the heat release rate fluctuations. Their amplitude is found to be directly linked to the rapid change in the FDF gain as the velocity fluctuation level reaches large amplitudes corresponding to the limit cycle, underlying the need of FDF information at high modulation amplitudes.

A Weakly Nonlinear Approach Based on a Distributed Flame Describing Function to Study the Combustion Dynamics of a Full-Scale Lean-Premixed Swirled Burner

Modern combustion chambers of gas turbines for power generation and aero-engines suffer of thermo-acoustic combustion instabilities generated by the coupling of heat release rate fluctuations with pressure oscillations. The present article reports a numerical analysis of limit cycles arising in a longitudinal combustor. This corresponds to experiments carried out on the longitudinal rig for instability analysis (LRIA) test facility equipped with a full-scale lean-premixed burner. Heat release rate fluctuations are modeled considering a distributed flame describing function (DFDF), since the flame under analysis is not compact with respect to the wavelengths of the unstable modes recorded experimentally. For each point of the flame, a saturation model is assumed for the gain and the phase of the DFDF with increasing amplitude of velocity fluctuations. A weakly nonlinear stability analysis is performed by combining the DFDF with a Helmholtz solver to determine the limit cycle condition. The numerical approach is used to study two configurations of the rig characterized by different lengths of the combustion chamber. In each configuration, a good match has been found between numerical predictions and experiments in terms of frequency and wave shape of the unstable mode. Time-resolved pressure fluctuations in the system plenum and chamber are reconstructed and compared with measurements. A suitable estimate of the limit cycle oscillation is found.

Prey-predator dynamics with prey refuge providing additional food to predator

The impacts of additional food for predator on the dynamics of a prey-predator model with prey refuge are investigated. The equilibrium points and their stability behaviours are determined. Hopf bifurcation conditions are derived analytically. Most significantly, existence conditions for unique stable limit cycle in the phase plane are shown analytically. The analytical results are in well agreement with the numerical simulation results. Effects of variation of refuge level as well as the variation of quality and quantity of additional food on the dynamics are reported with the help of bifurcation diagrams. It is found that high quality and high quantity of additional food supports oscillatory coexistence of species. It is observed that predator extinction possibility in high prey refuge ecological systems may be removed by supplying additional food to predator population. The reported theoretical results may be useful to conservation biologist for species conservation in real world ecological systems.

Build‐up steady‐state analysis of wind‐driven self‐excited induction generators

For the contradiction between the increasing importance of the application of induction generator in the new energy power generation such as wind energy and the relative laggard of research on its comprehensive performance, the build-up steady-state analysis is performed for the wind-driven self-excited induction generator (SEIG) system based on the unloaded transient equivalent circuits and its mathematic model of SEIG in the two-phase stationary reference frame. Thus, it obtains the build-up steady-state conditions represented by the unloaded transient mathematic model. On the basis of the result obtained for the build-up steady-state limit cycle condition, the analytic solution for the build-up steady-state operating point condition and the analytic formulas for the limit magnetising induction are further obtained. Example analysis is to test out the only set of analytic solution for the build-up steady-state condition. Thus, the analytic solution for the build-up steady-state condition produces the formulas of the torture and the power factor, which can be applicable to further analysis for the system dynamics and build-up steady-state operating performance, showing practicability and referenced value in engineering.

Chaos in Essentially Singular 3D Dynamical Systems with Two Quadratic Nonlinearities

A new class of 3D autonomous quadratic systems, the dynamics of which demonstrate a chaotic behavior, is found. This class is a generalization of the well-known class of Lorenz-like systems. The existence conditions of limit cycles in systems of the mentioned class are found. In addition, it is shown that, with the change of the appropriate parameters of systems of the indicated class, chaotic attractors different from the Lorenz attractor can be generated (these attractors are the result of the cascade of limit cycles bifurcations). Examples are given.