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Journal articles on the topic 'Hopf’s bifurcation'

Author: Grafiati

Published: 28 May 2022

Last updated: 31 July 2025

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1

Martín, J. C., and L. Mora. "$C^2$-perturbations of Hopf’s bifurcation points and homoclinic tangencies." Proceedings of the American Mathematical Society 128, no. 4 (1999): 1241–45. http://dx.doi.org/10.1090/s0002-9939-99-05106-0.

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2

Mesa, Fernando, German Correa Velez, and Jose Jose Barba Ortega. "Hopf bifurcation in the study of synchronous motor stability." Ciencia en Desarrollo 13, no. 1 (2022): 1–7. http://dx.doi.org/10.19053/01217488.v13.n1.2022.12650.

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   In this document the dynamic model of the synchronous motor is presented, which has a typical structure of Lienard-type systems, the theory of dynamic systems is used, especially bifurcations, in this case, Hopf’s, which will be applied to the described model, to show the variations in the balance points of the system by taking the voltage of the bus to which it is connected as a variable parameter.  

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3

Ojeda Toro, Juan Carlos, Izabela Dobrosz-Gómez, and Miguel Ángel Gómez García. "Setting Safe Operation Conditions for Acetyl Chloride Hydrolysis through Dynamic Modelling and Bifurcation Analysis." Modelling and Simulation in Engineering 2023 (November 18, 2023): 1–19. http://dx.doi.org/10.1155/2023/9685811.

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Acetyl chloride hydrolysis is a highly sensitive exothermic reaction that has presented several industrial safety issues. In the present study, a multiparameter mathematical model, previously developed and applied to simulate the oscillatory thermal behavior of an experimental continuous stirred tank reactor, was used to determine the static/dynamic bifurcation behavior of this reactive system. The values predicted by the model showed good agreement with the experimental data reported in the literature. Full topological classification of its fixed points and iterative maps was obtained: unique

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4

Nikolov, Svetoslav, and Valentin Nedev. "Bifurcation Analysis and Dynamic Behaviour of an Inverted Pendulum with Bounded Control." Journal of Theoretical and Applied Mechanics 46, no. 1 (2016): 17–32. http://dx.doi.org/10.1515/jtam-2016-0002.

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Abstract This paper presents an investigation on the behaviour of con- ventional inverted pendulum with an inertia disk in its free extreme. The system is actuated by means of torques applied to the disk by a DC mo- tor, mounted on the pendulum’s arm. Thus, the system is underactuated since the pendulum can rotate freely around its pivot point. The dynam- ical model is given with three ordinary nonlinear differential equations. Using Poincare-Andronov-Hopf’s theory, we find a new analytical formula for the first Lyapunov’s value at the boundary of stability. It enables one to study in detail t

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5

Bouachir, Amel, Mahmoud Mamou, Redha Rebhi, and Smail Benissaad. "Linear and Nonlinear Stability Analyses of Double-Diffusive Convection in a Vertical Brinkman Porous Enclosure under Soret and Dufour Effects." Fluids 6, no. 8 (2021): 292. http://dx.doi.org/10.3390/fluids6080292.

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Analytical and numerical investigations were performed to study the influence of the Soret and Dufour effects on double-diffusive convection in a vertical porous layer filled with a binary mixture and subject to horizontal thermal and solute gradients. In particular, the study was focused on the effect of Soret and Dufour diffusion on bifurcation types from the rest state toward steady convective state, and then toward oscillatory convective state. The Brinkman-extended Darcy model and the Boussinesq approximation were employed to model the convective flow within the porous layer. Following pa

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6

Xu, Chaoqun, and Sanling Yuan. "Spatial Periodic Solutions in a Delayed Diffusive Predator–Prey Model with Herd Behavior." International Journal of Bifurcation and Chaos 25, no. 11 (2015): 1550155. http://dx.doi.org/10.1142/s0218127415501552.

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A delayed diffusive predator–prey model with herd behavior subject to Neumann boundary conditions is studied both theoretically and numerically. Applying Hopf bifurcation analysis, we obtain the critical conditions under which the model generates spatially nonhomogeneous bifurcating periodic solutions. It is shown that the spatially homogeneous Hopf bifurcations always exist and that the spatially nonhomogeneous Hopf bifurcations will arise when the diffusion coefficients are suitably small. The explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurc

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7

Li, Wei, Chunrui Zhang, and Mi Wang. "Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges." Discrete Dynamics in Nature and Society 2024 (May 11, 2024): 1–18. http://dx.doi.org/10.1155/2024/9185585.

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This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidence

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8

Xu, Changjin. "Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters." Abstract and Applied Analysis 2012 (2012): 1–20. http://dx.doi.org/10.1155/2012/264870.

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A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.

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9

Yan, Xiang-Ping, and Wan-Tong Li. "Global existence of periodic solutions in a simplified four-neuron BAM neural network model with multiple delays." Discrete Dynamics in Nature and Society 2006 (2006): 1–18. http://dx.doi.org/10.1155/ddns/2006/57254.

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We consider a simplified bidirectional associated memory (BAM) neural network model with four neurons and multiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to Wu and Bendixson's criterion for high-dimensional ordinary differential equations due to Li and Muldowney. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the sum of two delays. Numerical simulations supporting the theoretical analysis are also included.

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10

Xu, Changjin, Maoxin Liao, and Xiaofei He. "Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays." International Journal of Applied Mathematics and Computer Science 21, no. 1 (2011): 97–107. http://dx.doi.org/10.2478/v10006-011-0007-0.

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Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations

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11

SONG, YONGLI, JUNJIE WEI, and MAOAN HAN. "LOCAL AND GLOBAL HOPF BIFURCATION IN A DELAYED HEMATOPOIESIS MODEL." International Journal of Bifurcation and Chaos 14, no. 11 (2004): 3909–19. http://dx.doi.org/10.1142/s0218127404011697.

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In this paper, we consider the following nonlinear differential equation [Formula: see text] We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second

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12

Zhang, Fengrong, and Ruining Chen. "Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis." Electronic Research Archive 32, no. 7 (2024): 4723–40. http://dx.doi.org/10.3934/era.2024215.

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<p>This paper explored a delayed diffusive prey-predator model with prey-taxis involving the volume-filling mechanism subject to homogeneous Neumann boundary condition. To figure out the impact on the dynamic of the prey-predator model due to prey-taxis and time delay, we treated the prey-tactic coefficient $ \chi $ and time delay $ \tau $ as the bifurcating parameters and did stability and bifurcation analysis. It showed that the time delay will induce Hopf bifurcations in the absence of prey-taxis, and the bifurcation periodic solution at the first critical value of $ \tau $ was spatia

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13

Zhai, Yanhui, Ying Xiong, Xiaona Ma, and Haiyun Bai. "Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/242410.

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The paper investigated an avian influenza virus propagation model with nonlinear incidence rate and delay based on SIR epidemic model. We regard delay as bifurcating parameter to study the dynamical behaviors. At first, local asymptotical stability and existence of Hopf bifurcation are studied; Hopf bifurcation occurs when time delay passes through a sequence of critical values. An explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions is derived by applying the normal form theory and center manifold theorem. What is more,

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14

Liu, Ping, Junping Shi, Rui Wang, and Yuwen Wang. "Bifurcation Analysis of a Generic Reaction–Diffusion Turing Model." International Journal of Bifurcation and Chaos 24, no. 04 (2014): 1450042. http://dx.doi.org/10.1142/s0218127414500424.

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A generic Turing type reaction–diffusion system derived from the Taylor expansion near a constant equilibrium is analyzed. The existence of Hopf bifurcations and steady state bifurcations is obtained. The bifurcation direction and the stability of the bifurcating periodic obits are calculated. Numerical simulations are included to show the rich spatiotemporal dynamics.

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15

Zhang, Yan, and Zhenhua Bao. "Studies on the Existence of Unstable Oscillatory Patterns Bifurcating from Hopf Bifurcations in a Turing Model." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/574921.

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We revisit a homogeneous reaction-diffusion Turing model subject to the Neumann boundary conditions in the one-dimensional spatial domain. With the help of the Hopf bifurcation theory applicable to the reaction-diffusion equations, we are capable of proving the existence of Hopf bifurcations, which suggests the existence of spatially homogeneous and nonhomogeneous periodic solutions of this particular system. In particular, we also prove that the spatial homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point of the system are always unstable. This together with the

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16

LIU, JIANXIN, and JUNJIE WEI. "ON HOPF BIFURCATION OF A DELAYED PREDATOR–PREY SYSTEM WITH DIFFUSION." International Journal of Bifurcation and Chaos 23, no. 02 (2013): 1350023. http://dx.doi.org/10.1142/s0218127413500235.

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A delayed predator–prey system with diffusion and Dirichlet boundary conditions is considered. By regarding the growth rate a of prey as a main bifurcation parameter, we show that Hopf bifurcation occurs when the parameter a is varied. Then, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived.

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17

Xu, Changjin, and Xiaofei He. "Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms." Abstract and Applied Analysis 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/697630.

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A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the n

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18

Xu, Changjin, and Peiluan Li. "Dynamical Analysis in a Delayed Predator-Prey Model with Two Delays." Discrete Dynamics in Nature and Society 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/652947.

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A class of Beddington-DeAngelis functional response predator-prey model is considered. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are given.

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19

Liu, Qingsong, Yiping Lin, and Jingnan Cao. "Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge." Computational and Mathematical Methods in Medicine 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/619132.

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A modified Leslie-Gower predator-prey system with two delays is investigated. By choosingτ1andτ2as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional diffe

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20

Wang, Shaoli, and Zhihao Ge. "The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge." Abstract and Applied Analysis 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/168340.

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The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main

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21

Yang, Ting. "Multistability and Hidden Attractors in a Three-Dimensional Chaotic System." International Journal of Bifurcation and Chaos 30, no. 06 (2020): 2050087. http://dx.doi.org/10.1142/s021812742050087x.

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This paper proposes a novel three-dimensional autonomous chaotic system. Interestingly, when the system has infinitely many stable equilibria, it is found that the system also has infinitely many hidden chaotic attractors. We show that the period-doubling bifurcations are the routes to chaos. Moreover, the Hopf bifurcations at all equilibria are investigated and it is also found that all the Hopf bifurcations simultaneously occur. Furthermore, the approximate expressions and stabilities of bifurcating limit cycles are obtained by using normal form theory and bifurcation theory.

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22

Liu, Yi Jing, Zhi Shu Li, Xiao Mei Cai, and Ya Lan Ye. "Local Stability and Hopf Bifurcation Analysis of the Arneodo’s System." Applied Mechanics and Materials 130-134 (October 2011): 2550–57. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.2550.

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The chaotic behaviors of the Arneodo’s system are investigated in this paper. Based on the Arneodo's system characteristic equation, the equilibria of the system and the conditions of Hopf bifurcations are obtained, which shows that Hopf bifurcations occur in this system. Then using the normal form theory, we give the explicit formulas which determine the stability of bifurcating periodic solutions and the direction of the Hopf bifurcation. Finally, some numerical examples are employed to demonstrate the effectiveness of the theoretical analysis.

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23

ZHANG, JIA-FANG, WAN-TONG LI, and XIANG-PING YAN. "BIFURCATION AND SPATIOTEMPORAL PATTERNS IN A HOMOGENEOUS DIFFUSION-COMPETITION SYSTEM WITH DELAYS." International Journal of Biomathematics 05, no. 06 (2012): 1250049. http://dx.doi.org/10.1142/s1793524512500490.

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A competitive Lotka–Volterra reaction-diffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive constant steady state of the system is globally asymptotically stable if the interspecies competition is weaker than the intraspecies one and is unstable if the interspecies competition dominates over the intraspecies one. If the latter holds, then we show that Hopf bifurcation can occur as the parameters (delays) in the system cross some critical values. In particular, we prove that these Hopf bifurcations are all spatially homogeneous

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24

Niu, Ben, Yuxiao Guo, and Yanfei Du. "Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System." International Journal of Bifurcation and Chaos 28, no. 11 (2018): 1850136. http://dx.doi.org/10.1142/s0218127418501365.

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Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic

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25

Astakhov, Sergey, Oleg Astakhov, Vladimir Astakhov, and Jürgen Kurths. "Bifurcational Mechanism of Multistability Formation and Frequency Entrainment in a van der Pol Oscillator with an Additional Oscillatory Circuit." International Journal of Bifurcation and Chaos 26, no. 07 (2016): 1650124. http://dx.doi.org/10.1142/s0218127416501248.

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In this paper, the bifurcational mechanism of frequency entrainment in a van der Pol oscillator coupled with an additional oscillatory circuit is studied. It is shown that bistability observed in the system is based on two bifurcations: a supercritical Andronov–Hopf bifurcation and a sub-critical Neimark–Sacker bifurcation. The attracting basin boundaries are determined by stable and unstable invariant manifolds of a saddle two-dimensional torus.

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26

Zhou, Xiaojian, Xin Chen, and Yongzhong Song. "Hopf Bifurcation of a Differential-Algebraic Bioeconomic Model with Time Delay." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/768364.

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We investigate the dynamics of a differential-algebraic bioeconomic model with two time delays. Regarding time delay as a bifurcation parameter, we show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Using the theories of normal form and center manifold, we also give the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical tests are provided to verify our theoretical analysis.

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27

Bılazeroğlu, Şeyma, Huseyin Merdan, and Luca Guerrini. "Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays." Discrete & Continuous Dynamical Systems - S 15, no. 3 (2022): 535. http://dx.doi.org/10.3934/dcdss.2021150.

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<p style='text-indent:20px;'>Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine so

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28

Liu, Ming, and Xiaofeng Xu. "Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/367589.

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The dynamics of a 2-dimensional neural network model in neutral form are investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results.

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29

Cai, Yongli, Zhanji Gui, Xuebing Zhang, Hongbo Shi, and Weiming Wang. "Bifurcations and Pattern Formation in a Predator–Prey Model." International Journal of Bifurcation and Chaos 28, no. 11 (2018): 1850140. http://dx.doi.org/10.1142/s0218127418501407.

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In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the s

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30

Liu, Qingsong, Yiping Lin, Jingnan Cao, and Jinde Cao. "Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System." Discrete Dynamics in Nature and Society 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/139375.

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The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

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31

YAN, XIANG-PING, and WAN-TONG LI. "STABILITY AND HOPF BIFURCATION FOR A DELAYED COOPERATIVE SYSTEM WITH DIFFUSION EFFECTS." International Journal of Bifurcation and Chaos 18, no. 02 (2008): 441–53. http://dx.doi.org/10.1142/s0218127408020434.

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The main purpose of this paper is to investigate the stability and Hopf bifurcation for a delayed two-species cooperative diffusion system with Neumann boundary conditions. By linearizing the system at the positive equilibrium and analyzing the corresponding characteristic equation, the asymptotic stability of positive equilibrium and the existence of Hopf oscillations are demonstrated. It is shown that, under certain conditions, the system undergoes only a spatially homogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through a sequence of critical values; under the

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32

WEI, JUNJIE, and DEJUN FAN. "HOPF BIFURCATION ANALYSIS IN A MACKEY–GLASS SYSTEM." International Journal of Bifurcation and Chaos 17, no. 06 (2007): 2149–57. http://dx.doi.org/10.1142/s0218127407018282.

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The dynamics of a Mackey–Glass equation with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [1994].

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33

Zeng, Bing, and Pei Yu. "Analysis of Zero-Hopf Bifurcation in Two Rössler Systems Using Normal Form Theory." International Journal of Bifurcation and Chaos 30, no. 16 (2020): 2030050. http://dx.doi.org/10.1142/s0218127420300505.

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In recent publications [Llibre, 2014; Llibre & Makhlouf, 2020], time-averaging method was applied to studying periodic orbits bifurcating from zero-Hopf critical points of two Rössler systems. It was shown that the averaging method is successful for a certain type of zero-Hopf critical points, but fails for some type of such critical points. In this paper, we apply normal form theory to reinvestigate the bifurcation and show that the method of normal forms is applicable for all types of zero-Hopf bifurcations, revealing why the time-averaging method fails for some type of zero-Hopf bifurca

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34

Yao, Yong, Zuxiong Li, Huili Xiang, Hailing Wang, and Zhijun Liu. "Hopf bifurcation analysis in a turbidostat model with Beddington–DeAngelis functional response and discrete delay." International Journal of Biomathematics 10, no. 05 (2017): 1750061. http://dx.doi.org/10.1142/s1793524517500619.

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In this paper, regarding the time delay as a bifurcation parameter, the stability and Hopf bifurcation of the model of competition between two species in a turbidostat with Beddington–DeAngelis functional response and discrete delay are studied. The Hopf bifurcations can be shown when the delay crosses the critical value. Furthermore, based on the normal form and the center manifold theorem, the type, stability and other properties of the bifurcating periodic solutions are determined. Finally, some numerical simulations are given to illustrate the results.

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35

Wang, Hong Yan, and Hong Mei Wang. "Stability and Bifurcation Analysis in a Stage-Structured Predator-Prey Model with Delay." Applied Mechanics and Materials 513-517 (February 2014): 3723–27. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.3723.

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Hopf bifurcation occurs in most of dynamics systems when the influence from the past state varies. In modeling population dynamics, it is more reasonable taking into account the time delays. In this paper, a stage-structured predator-prey system with delay is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifo

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36

Guo, Yuxiao, and Weihua Jiang. "Hopf Bifurcation in Two Groups of Delay-Coupled Kuramoto Oscillators." International Journal of Bifurcation and Chaos 25, no. 10 (2015): 1550129. http://dx.doi.org/10.1142/s0218127415501291.

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Hopf bifurcation in two groups of Kuramoto's phase oscillators with delay-coupled interactions is investigated on the Ott–Antonsen's manifold. We find that the reduced delay differential system undergoes Hopf bifurcations when the coupling strength between two groups exceeds some critical values. With the increasing of time delay, stability switches are observed which leads to the synchrony switches for the Kuramoto system. The direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by deriving the normal forms on the center manifold. With respect to

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37

Wei, Junjie, and Chunbo Yu. "Hopf bifurcation analysis in a model of oscillatory gene expression with delay." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 4 (2009): 879–95. http://dx.doi.org/10.1017/s0308210507000091.

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The dynamics of a gene expression model with time delay are investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions has been derived by using the theory of the centre manifold and the normal forms method. The global existence of periodic solutions has been established using a global Hopf bifurcation result by Wu and a Bendixson criterion for higher-dimensional ordinary differential

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38

Qu, Ying, and Junjie Wei. "Global Hopf Bifurcation Analysis for a Time-Delayed Model of Asset Prices." Discrete Dynamics in Nature and Society 2010 (2010): 1–17. http://dx.doi.org/10.1155/2010/432821.

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A time-delayed model of speculative asset markets is investigated to discuss the effect of time delay and market fraction of the fundamentalists on the dynamics of asset prices. It proves that a sequence of Hopf bifurcations occurs at the positive equilibriumv, the fundamental price of the asset, as the parameters vary. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined using normal form method and center manifold theory. Global existence of periodic solutions is established combining a global Hopf bifurcation theorem with a Bendixson'

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Li, Li, and Jian Xu. "Bifurcation Analysis and Spatiotemporal Patterns in Unidirectionally Delay-Coupled Vibratory Gyroscopes." International Journal of Bifurcation and Chaos 28, no. 02 (2018): 1850029. http://dx.doi.org/10.1142/s0218127418500293.

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Time delay is inevitable in unidirectionally coupled drive-free vibratory gyroscope system. The effect of time delay on the gyroscope system is studied in this paper. To this end, amplitude death and Hopf bifurcation induced by small time delay are first investigated by analyzing the related characteristic equation. Then, the direction of Hopf bifurcations and stability of Hopf-bifurcating periodic oscillations are determined by calculating the normal form on the center manifold. Next, spatiotemporal patterns of these Hopf-bifurcating periodic oscillations are analyzed by using the symmetric b

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GUO, SHANGJIANG, and YUAN YUAN. "PATTERN FORMATION IN A RING NETWORK WITH DELAY." Mathematical Models and Methods in Applied Sciences 19, no. 10 (2009): 1797–852. http://dx.doi.org/10.1142/s0218202509004005.

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We consider a ring network of three identical neurons with delayed feedback. Regarding the coupling coefficients as bifurcation parameters, we obtain codimension one bifurcation (including a Fold bifurcation and Hopf bifurcation) and codimension two bifurcations (including Fold–Fold bifurcations, Fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulas for the normal form coefficients derived via the center manifold reduction that provide detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstab

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Wei, Fengying, Lanqi Wu, and Yuzhi Fang. "Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/624162.

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A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτpasses through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

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JIANG, ZHICHAO, ZHAOZHUANG GUO, and YUEFANG SUN. "STABILITY ANALYSIS OF A PREDATOR-PREY MODEL." International Journal of Biomathematics 05, no. 01 (2012): 1250007. http://dx.doi.org/10.1142/s1793524511001477.

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In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

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PENG, JIAN, LIANHUA WANG, YUEYU ZHAO, and SHANGJIANG GUO. "SYNCHRONIZATION AND BIFURCATION IN LIMIT CYCLE OSCILLATORS WITH DELAYED COUPLINGS." International Journal of Bifurcation and Chaos 21, no. 11 (2011): 3157–69. http://dx.doi.org/10.1142/s0218127411030428.

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In this paper, a system of three globally coupled limit cycle oscillators with a linear time-delayed coupling are investigated. Considering the delay as a parameter, we also study the effect of time delay on the dynamics. Next, Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction are obtained. Based on the symmetric Hopf bifurcation theorem, we investigate stable phase-locking and unstable waves. Then later, the directions of Hopf bifurcations are determined in some region, where stability switches may occur. The results show that the bifurcating

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WANG, LINGSHU, RUI XU, and GUANGHUI FENG. "STABILITY AND HOPF BIFURCATION OF A PREDATOR–PREY SYSTEM WITH TIME DELAY AND HOLLING TYPE-II FUNCTIONAL RESPONSE." International Journal of Biomathematics 02, no. 02 (2009): 139–49. http://dx.doi.org/10.1142/s1793524509000595.

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A predator–prey model with time delay and Holling type-II functional response is investigated. By choosing time delay as the bifurcation parameter and analyzing the associated characteristic equation of the linearized system, the local stability of the system is investigated and Hopf bifurcations are established. The formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.

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JI, J. C., X. Y. LI, Z. LUO, and N. ZHANG. "TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY." International Journal of Bifurcation and Chaos 22, no. 03 (2012): 1250060. http://dx.doi.org/10.1142/s0218127412500605.

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The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present pape

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FREIRE, EMILIO, ENRIQUE PONCE, and JAVIER ROS. "LIMIT CYCLE BIFURCATION FROM CENTER IN SYMMETRIC PIECEWISE-LINEAR SYSTEMS." International Journal of Bifurcation and Chaos 09, no. 05 (1999): 895–907. http://dx.doi.org/10.1142/s0218127499000638.

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The rapid bifurcation described by Kriegsmann [1987] is shown to be a generic bifurcation for planar symmetric piecewise-linear systems. The bifurcation can be responsible for the abrupt appearance of stable periodic oscillations. Although it has some similarities with the Hopf bifurcation for smooth systems, since the stability change of an equilibrium involves the appearance of one limit cycle, the dependence of the limit cycle amplitude on the bifurcation parameter is different from the Hopf's case. To characterize this bifurcation, accurate estimates for the amplitude and period of the bif

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Wang, Beibei, Min Zhao, Chuanjun Dai, Hengguo Yu, Nan Wang, and Pengfei Wang. "Dynamics Analysis of a Nutrient-Plankton Model with a Time Delay." Discrete Dynamics in Nature and Society 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/9797624.

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We analyze a nutrient-plankton system with a time delay. We choose the time delay as a bifurcation parameter and investigate the stability of a positive equilibrium and the existence of Hopf bifurcations. By using the center manifold theorem and the normal form theory, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are researched. The theoretical results indicate that the time delay can induce a positive equilibrium to switch from a stable to an unstable to a stable state and so on. Numerical simulations show that the theoretical results are corre

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Yang, Xinchao, Xiju Zong, Xingong Cheng, and Zhenlai Han. "Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection." Journal of Applied Mathematics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/875783.

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The stability and bifurcation analysis for a delay differential equation of hepatitis B virus infection is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. The existence of the Hopf bifurcation with delayτat the endemic equilibria is established by analyzing the distribution of the characteristic values. The explicit formulae which determine the direction of the bifurcations, stability, and the other properties of the bifurcating periodic solutions are given by using the normal form theory and the center manifold theorem. Numerical simulation ve

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Jiang, Jiao, and Yongli Song. "Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling." Abstract and Applied Analysis 2014 (2014): 1–18. http://dx.doi.org/10.1155/2014/368652.

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We investigate the dynamics of a delayed neural network model consisting ofnidentical neurons. We first analyze stability of the zero solution and then study the effect of time delay on the dynamics of the system. We also investigate the steady state bifurcations and their stability. The direction and stability of the Hopf bifurcation and the pitchfork bifurcation are analyzed by using the derived normal forms on center manifolds. Then, the spatiotemporal patterns of bifurcating periodic solutions are investigated by using the symmetric bifurcation theory, Lie group theory andS1-equivariant de

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Jiang, Jiao, and Pei Yu. "Multistable Phenomena Involving Equilibria and Periodic Motions in Predator–Prey Systems." International Journal of Bifurcation and Chaos 27, no. 03 (2017): 1750043. http://dx.doi.org/10.1142/s0218127417500432.

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In this paper, we consider a number of predator–prey systems with various types of functional responses. Detailed analysis on the dynamics and bifurcations of the systems are given. Particular attention is focused on the complex dynamics due to bifurcation of limit cycles, which may generate bistable or tristable phenomena involving equilibria and oscillating motions. It is shown that predator–prey systems can exhibit such bistable or tristable phenomena due to Hopf bifurcation, giving rise to the coexistence of stable equilibria and stable periodic solutions. Explicit conditions on the system

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