Value Of Bifurcation Parameter Research Articles

Maximal canards in a slow–fast Rosenzweig–MacArthur model with intraspecific competition among predators

Using geometric singular perturbation theory, this paper investigates the canard phenomenon of a slow–fast Rosenzweig–MacArthur model. The model incorporates intraspecific competition among predators, assuming predator reproduction occurs much slower than prey. We demonstrate the occurrence of maximal canards between attracting and repelling slow manifolds as a bifurcation parameter varies. Additionally, we derive an analytic expression to approximate the bifurcation parameter value at which a maximal canard occurs. The method employed for this analysis relies on the blowup method. This involves finding a quasihomogeneous blowup map to desingularize the nonnormally hyperbolic point. Subsequently, charts are utilized to express the blowup in local coordinates, calculate local data, investigate the dynamics of the blown-up vector fields, and establish connections across charts. Furthermore, we provide numerical simulations to illustrate the canard explosion phenomenon in the model. Through parameter variation, we observe that the model transitions from a small amplitude limit cycle to small amplitude canard cycles (canards without a head), then to large amplitude canard cycles (canards with a head), and finally to a large amplitude relaxation cycle.

Exploring chaos and ergodic behavior of an inductorless circuit driven by stochastic parameters

There exist extensive studies on periodic and random perturbations of various smooth maps investigating their dynamics. Unlike smooth maps, non-smooth maps are yet to be studied extensively under a stochastic regime. This paper presents a stochastic piecewise-smooth map derived from a simple inductorless switching circuit. The stochasticity is introduced in parameter values. The distribution of the parameter values is bounded and randomly selected from uniform and triangular distributions and ranges between high and low bifurcation parameter values of the deterministic map. Due to this inherent stochasticity in parameter values, the time evolution of the state variable cannot be predicted at a specific time instant. We observe that the state variable exhibits completely ergodic behavior when the minimum value of the parameter is the same as the minimum bifurcation parameter of the deterministic system. However, the ensemble average of the state variable converges to a fixed value. The system demonstrates nonchaotic behavior for a particular range of parameter values but the deterministic map in that bifurcation range shows interplay between chaos and periodic orbits. The values of Lyapunov exponents decrease monotonically with increased asymmetry of the distribution from which the bifurcation parameter values are chosen. We determine the probability density function of the stochastic map and verify its invariance under initial conditions. The most noteworthy result is the disappearance of chaotic behavior when the lower range of the distribution is varied while maintaining a fixed upper threshold for a particular distribution, even though the deterministic map exhibits an array of periodic and chaotic behaviors within the range. As the period-incrementing cascade with chaotic inclusion only occurs in nonsmooth maps, this paper numerically shows the stochasticity of a piecewise-smooth map obtained from a practical system for the first time where randomness is introduced in the parameter space.

The stochastic stability and H∞‐fuzzy control of stochastic bifurcation of a doubly‐fed induction generator

AbstractConsidering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

Bifurcación de Hopf en un sistema autónomo presa-predador con crecimiento logístico y respuesta funcional de Holling tipo II

The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis, values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation. The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.

Exploration on dynamics in a ratio-dependent predator-prey bioeconomic model with time delay and additional food supply.

In this manuscript, a novel ratio-dependent predator-prey bioeconomic model with time delay and additional food supply is investigated. We first change the bioeconomic model into a normal version by virtue of the differential-algebraic system theory. The local steady-state of equilibria and Hopf bifurcation could be derived by varying time delay. Later, the formulas of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions are obtained by the normal form theory and the center manifold theorem. Moreover, employing the Pontryagin's maximum principle and considering the instantaneous annual discount rate, the optimal harvesting problem of the model without time delay is analyzed. Finally, four numeric examples are carried out to verify the rationality of our analytical findings. Our analytical results show that Hopf bifurcation occurs in this model when the value of bifurcation parameter, the time delay of the maturation time of prey, crosses a critical value.

Stability and bifurcation analysis in a diffusive predator–prey model with delay and spatial average

In this paper, we study the effect of spatial average and time delay on the dynamics of a diffusive predator–prey model under the Neumann boundary condition. Compared to the model without spatial average, the delay‐induced Hopf bifurcation at the first critical value of delay is nonhomogeneous due to the joint effects of spatial average and delay, and spatiotemporal patterns arise. Also, we show that the spatially homogeneous and nonhomogeneous periodic patterns may switch for different bifurcation parameter values. Moreover, a double Hopf bifurcation occurs at the intersection point of the homogeneous and nonhomogeneous Hopf bifurcation curves, and spatially nonhomogeneous quasi‐periodic patterns can be observed via numerical simulations near the double Hopf bifurcation point. The normal form algorithm for the spatially nonhomogeneous/homogeneous Hopf bifurcation is derived for a general reaction‐diffusion system with spatial average and delay.

Steady‐state bifurcation and Hopf bifurcation in a cross‐diffusion prey–predator system with Ivlev functional response

In this paper, we investigate bifurcations of stationary solutions of a cross‐diffusion prey–predator system with Ivlev functional response and Neumann boundary conditions. First, we analyze the local stability and existence of a Hopf bifurcation at a coexistence stationary solution. We show that the bifurcating periodic solutions are asymptotically orbitally stable and the bifurcation direction is supercritical when the ratio of the conversion of prey captured by predator to the death rate of predator is in the interval . Next, we derive sufficient conditions for the existence of a steady‐state bifurcation from a simple eigenvalue. We establish the existence of two intersecting curves of steady‐state solutions near the coexistence stationary solution. To illustrate our theoretical results, we give some numerical examples. From numerical simulations we observe that the coexistence stationary solution loses its stability via the Hopf bifurcation and a periodic solution emerges after a short time. Also by taking the bifurcation parameter value in the stable region, the effect of the initial condition disappears over time and the solution returns to the coexistence stationary solution.

Stability analysis of a predator-prey model with stage structure

Many scholars have explored the population dynamics system, functional response, time-delay response, and so on. Based on the existing research, this paper studies a class of predator-prey models with stage structure, time delay, and Holling type-II functional response function. The stability of that model at the positive equilibrium point, the sufficient condition for stability, and the existence of Hopf bifurcation are discus. The model was numerically simulated by taking appropriate parameters and different time delay values, variation diagram of each component, and solution curves were given near the critical value. The results show that the stability of the system will change with the variation of the bifurcation parameter value, and Hopf bifurcation will occur.

Analytical prediction of delayed Hopf bifurcations in a simplified stochastic model of reed musical instruments

This paper investigates the dynamic behavior of a simplified single reed instrument model subject to a stochastic forcing of white noise type when one of its bifurcation parameters (the dimensionless blowing pressure) increases linearly over time and crosses the Hopf bifurcation point of its trivial equilibrium position. The stochastic slow dynamics of the model is first obtained by means of the stochastic averaging method. The resulting averaged system reduces to a non-autonomous one-dimensional Itô stochastic differential equation governing the time evolution of the mouthpiece pressure amplitude. Under relevant approximations the latter is solved analytically treating separately cases where noise can be ignored and cases where it cannot. From that, two analytical expressions of the bifurcation parameter value for which the mouthpiece pressure amplitude gets its initial value back are deduced. These special values of the bifurcation parameter characterize the effective appearance of sound in the instrument and are called deterministic dynamic bifurcation point if the noise can be neglected and stochastic dynamic bifurcation point otherwise. Finally, for illustration and validation purposes, the analytical results are compared with direct numerical integration of the model in both deterministic and stochastic situations. In each considered case, a good agreement is observed between theoretical results and numerical simulations, which validates the proposed analysis.

On the Search for Brain Bifurcation Parameters: Lessons From FMRI Studies on Visual Illusions

Data from three functional magnetic resonance imaging (fMRI) studies that involved in total about 100 participants and showed that the strength of several visual illusions such as the Ebbinghaus, Ponzo, and Muller-Lyer illusions depends on neuroanatomical subject measures such as visual cortex surface area and parahippocampal cortex gray matter volume were evaluated using a dynamical systems perspective to determine brain bifurcation parameters. Bifurcation parameters that involved power laws and captured relational dependencies were fitted separately to the three fMRI studies. The bifurcation parameter hypothesis that states that such parameters show unique quantities and are no longer correlated to structural systems properties was tested. The power law exponents and mean bifurcation parameter values were determined. For all three studies and three illusion types, the bifurcation parameter hypothesis was supported. Accordingly, the constructed parameters characterized the reactions of the participants under the Ebbinghaus, Ponzo, and Muller-Lyer illusions in terms of unique threshold values that no longer depended on neuroanatomical subject measures. Power law exponents in the range from 1 to 7 were found. The fMRI data describing gray matter volume of certain active regions in the parahippocampal cortex showed some interesting relationship between the mean bifurcation parameter values.

Predicting amplitude death with machine learning.

In nonlinear dynamics, a parameter drift can lead to a sudden and complete cessation of the oscillations of the state variables-the phenomenon of amplitude death. The underlying bifurcation is one at which the system settles into a steady state from chaotic or regular oscillations. As the normal functioning of many physical, biological, and physiological systems hinges on oscillations, amplitude death is undesired. To predict amplitude death in advance of its occurrence based solely on oscillatory time series collected while the system still functions normally is a challenge problem. We exploit machine learning to meet this challenge. In particular, we develop the scheme of "parameter-aware" reservoir computing, where training is conducted for a small number of bifurcation parameter values in the oscillatory regime to enable prediction upon a parameter drift into the regime of amplitude death. We demonstrate successful prediction of amplitude death for three prototypical dynamical systems in which the transition to death is preceded by either chaotic or regular oscillations. Because of the completely data-driven nature of the prediction framework, potential applications to real-world systems can be anticipated.

Stability and Bifurcation Analysis of a Prey–Predator Model

The purpose of the present paper is to study the stability of a prey–predator model using KCC theory. The KCC theory is based on the assumption that the second-order dynamical system and geodesics equation, in associated Finsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theory and linear stability of the model are discussed in detail. Further, the effect of parameters on stability and the presence of chaos in the model are investigated. The critical values of bifurcation parameters are found and their effects on the model are investigated. The numerical examples of particular interest are compared to the results of Jacobi stability and linear stability and it is found that Jacobi stability on the basis of KCC theory is global than the linear stability.

Algorithm for predicting of the transition of a key point of geodynamic control to the risk zone

The article discusses the algorithm for assessing changes in the level of risk of karst hazard of the territory based on the forecasting of the number of holes depending on the water level. Based on the results of the analysis, the values of bifurcation parameters are determined, the transition through which sharply increases the formation of new holes. The article provides the developed block diagram of the neural network for assessing the dynamics of the occurrence of holes, as well as an algorithm for generating a predictive estimate of the number of holes. An analysis of the occurrence of the number of holes is carried out on the basis of water level data in the Oka River. The results of spline interpolation of the data are presented in the dependence on the number of holes on the dynamics of the water level in the river for the current and previous year. Practical verification of the developed algorithm was carried out on the basis of the new set of data on the water level in the river and the number of holes. The developed algorithm can be used in predicting the spread (leaching from the soil) of pollutants.

Vibration mitigation of a pipe conveying fluid with a passive geometrically nonlinear absorber: a tuning optimal design

The objective of the presented article is to investigate the nonlinear dynamics and vibration of a simply supported fluid conveying pipe coupled with a geometrically nonlinear absorber, as well as the occurrence of the targeted energy transfer phenomena in the system. In addition, given the sensitivity of passive nonlinear absorbers performance to uncertainties in design (e.g., damping coefficient) and aleatory parameters such as the initial excitation of the primary system, the optimal design of the absorber in presence of uncertainties is also examined in this paper. The absorber consists of two linear springs, a lightweight mass, and one linear damping configured in a particular formation and is coupled to the pipe in grounded and ungrounded configurations. It was indicated that the absorber can perform as a tuned mass damper (TMD), nonlinear energy sink (NES), or a modified NES based on the bifurcation parameter value, and finding the optimal type of the absorber is left to the optimization solver without any preceding knowledge of the physics of the system. It is illustrated that the stability of the pipe and the absorber effectiveness is reduced by increasing the inside flow velocity. Moreover, the efficiency of the absorber slightly dependents on the density of the inside fluid. The results show that the stochastic optimally designed absorber can nonlinearly resonate with the pipe over extensive frequency ranges which leads to a synchronized motion of the pipe and absorber. In this situation a significant irreversible energy pumping from the pipe to the absorber may occur that make the motion localizes to the absorber and reduce the amplitude of the pipe for a wide range of initial excitation and fluid velocity. Also, comparing the present work to other similar works yields a great result.

Stochastic stability and Hopf bifurcation analysis of a singular bio-economic model with stochastic fluctuations

In this paper, we study a class of singular stochastic bio-economic models described by differential-algebraic equations due to the influence of economic factors. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. Stochastic stability and Hopf bifurcations can be analytically determined based on the singular boundary theory of diffusion process, the Maximal Lyapunov exponent and the invariant measure theory. The critical value of the stochastic Hopf bifurcation parameter is obtained and the position of Hopf bifurcation drifting with the parameter increase is presented as a result. Practical example is presented to verify the effectiveness of the results.

Regimes of two-dimensional energy channeling in the inertially coupled unit-cell model subjected to an asymmetric potential

We analyze the stationary and non-stationary states emerging in a two-dimensional (2D) model consisting of an outer element that incorporates an internal rotator and is subjected to a 2D anharmonic and non-symmetric local potential. We focus on the effect of the asymmetry of the potential on the mechanism of formation and bifurcations of various stationary and non-stationary regimes in the system. The non-stationary regimes are characterized by an intense bi-directional energy transfer between the axial and lateral vibrations of the main element as well as the regimes of unidirectional energy locking. We show that the transitions between the different states of the system are controlled by the motion of the internal rotator. Our analysis is based on the complexification-averaging procedure and the singular multiscale asymptotic method. Investigation of the system's dynamics in the vicinity of the 1:1:1 resonance enables us to reproduce the bifurcation structure of the stationary regimes and describe the formation mechanisms of highly non-stationary states of intense energy transfer. Our analysis shows that the most important effect of the asymmetric potential is the breakdown of a unique complete bi-directional energy channeling mechanism into two separate mechanisms that correspond to incomplete horizontal-to-vertical and vertical-to-horizontal energy transfer, respectively, and are enabled at two different critical values of the main bifurcation parameter. Results of the analysis are in good agreement with the numerical simulations of the full model.

Reduced Basis Method Applied to Eigenvalue Problems from Convection

The reduced basis method is a suitable technique for finding numerical solutions to partial differential equations that must be obtained for many values of parameters. This method is suitable when researching bifurcations and instabilities of stationary solutions for partial differential equations. It is necessary to solve numerically the partial differential equations along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions for a large number of bifurcation parameter values. In this paper, the reduced basis method has been used to solve eigenvalue problems derived from the linear stability analysis of stationary solutions in a two-dimensional Rayleigh–Bénard convection problem. The bifurcation parameter is the Rayleigh number, which measures buoyancy. The reduced basis considered belongs to the eigenfunction spaces derived from the eigenvalue problems for different types of solutions in the bifurcation diagram depending on the Rayleigh number. The eigenvalue with the largest real part and its corresponding eigenfunction are easily calculated and the bifurcation points are correctly captured. The resulting matrices are small, which enables a drastic reduction in the computational cost of solving the eigenvalue problems.

Analysis of a predator–prey model with specific time scales: a geometrical approach proving the occurrence of canard solutions

We study a predator-prey model with different characteristic time scales for the prey and predator populations, assuming that the predator dynamics is much slower than the prey one. Geometrical Singular Perturbation theory provides the mathematical framework for analyzing the dynamical properties of the model. This model exhibits a Hopf bifurcation and we prove that when this bifurcation occurs, a canard phenomenon arises. We provide an analytic expression to get an approximation of the bifurcation parameter value for which a maximal canard solution occurs. The model is the well-known Rosenzweig-MacArthur predator-prey differential system. An invariant manifold with a stable and an unstable branches occurs and a geometrical approach is used to explicitly determine a solution at the intersection of these branches. The method used to perform this analysis is based on Blow-up techniques. The analysis of the vector field on the blown-up object at an equilibrium point where a Hopf bifurcation occurs with zero perturbation parameter representing the time scales ratio, allows to prove the result. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon.

Interaction-induced time-symmetry breaking in driven quantum oscillators

We study parametrically driven quantum oscillators and show that, even for weak coupling between the oscillators, they can exhibit various many-body states with broken time-translation symmetry. In the quantum-coherent regime, the symmetry breaking occurs via a nonequilibrium quantum phase transition. For dissipative oscillators, the main effect of the weak coupling is to make the switching rate of an oscillator between its period-2 states dependent on the states of other oscillators. This allows mapping the oscillators onto a system of coupled spins. Away from the bifurcation parameter values where the period-2 states emerge, the stationary state corresponds to having a microscopic current in the spin system, in the presence of disorder. In the vicinity of the bifurcation point or for identical oscillators, the stationary state can be mapped on that of the Ising model with an effective temperature $\propto \hbar$, for low temperature. Closer to the bifurcation point the coupling can not be considered weak and the system maps onto coupled overdamped Brownian particles performing quantum diffusion in a potential landscape.

A Fixed-Frequency Optimization of PWM Current Controller—Modeling and Design of Control Parameters

Design and modeling of the control parameters are important to guarantee the operation in a desired regime and to have the optimal dynamic behavior. To achieve these objectives, a proposed continuous-time averaging model is developed for a pulsewidth modulation current controller based on a modulated hysteresis controller while ensuring a fixed frequency operation. Stability analysis is performed using the eigenvalues of the system and estimating the bifurcation point for different cases to investigate parameter tuning. Moreover, the dynamic and static behaviors of the current for different values of bifurcation parameter are presented. The current controller in a two-loop control scheme is investigated. The simulation and experimental results are given to confirm the proposed method.