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Journal articles on the topic 'Hopf bifurcation'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 March 2025

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1

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 (October 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 bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorems for partial functional differential equations.

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2

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 (November 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 critical value of the delay τ. Finally, numerical simulation results are given to support the theoretical predictions.

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3

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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4

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 (March 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 for supporting the theoretical results are also included.

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5

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, the global existence of periodic solutions is established by using a global Hopf bifurcation result.

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6

Zhang, Huayong, Ju Kang, Tousheng Huang, Xuebing Cong, Shengnan Ma, and Hai Huang. "Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web." Mathematical Problems in Engineering 2018 (September 27, 2018): 1–21. http://dx.doi.org/10.1155/2018/8394651.

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Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

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7

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 (October 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 steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

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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

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 (October 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 solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

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10

GUO, SHANGJIANG, and YUAN YUAN. "PATTERN FORMATION IN A RING NETWORK WITH DELAY." Mathematical Models and Methods in Applied Sciences 19, no. 10 (October 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 unstable equilibria, periodic solutions, and quasi-periodic solutions.

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11

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 differential equations, we show the global existence of the periodic solutions.

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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 spatially homogeneous. Hopf bifurcations occurred in the model when the prey-taxis and time delay coexisted, and at the first critical value of $ \tau $, spatially homogeneous or nonhomogeneous periodic solutions might emerge. It was also discovered that the bifurcation curves will intersect, which implied that Hopf-Hopf bifurcations can occur. Finally, we did numerical simulations to validate our outcomes.</p>

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13

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 (August 22, 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 if the diffusive rates are suitably large, which has the same properties as Hopf bifurcation of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system generates the spatially nonhomogeneous Hopf bifurcation. Furthermore, we derive conditions for determining the direction of spatially nonhomogeneous Hopf bifurcations and the stability of bifurcating periodic solutions. These results indicate that the diffusion plays an important role for deriving the complex spatiotemporal dynamics.

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14

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 instability results of the spatially nonhomogeneous periodic solutions by Yi et al., 2009, indicates that, in this model, all the oscillatory patterns from Hopf bifurcations are unstable.

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15

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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16

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 (February 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 other conditions, except for the previous spatially homogeneous Hopf bifurcations, the system also undergoes a spatially inhomogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through another sequence of critical values. In particular, in order to determine the direction and stability of periodic solutions bifurcating from spatially homogeneous Hopf bifurcations, the explicit formulas are given by using the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, to verify our theoretical predictions, some numerical simulations are also included.

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17

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 some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.</p>

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18

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 (December 28, 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 bifurcation.

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19

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 (February 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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20

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 normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.

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21

Tigan, G., J. Llibre, and L. Ciurdariu. "Degenerate Fold–Hopf Bifurcations in a Rössler-Type System." International Journal of Bifurcation and Chaos 27, no. 05 (May 2017): 1750068. http://dx.doi.org/10.1142/s0218127417500687.

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We study the Hopf and the fold–Hopf bifurcations of the Rössler-type differential system [Formula: see text] with [Formula: see text]. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold–Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium [Formula: see text] when [Formula: see text]. This Hopf bifurcation becomes a fold–Hopf bifurcation when [Formula: see text].

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22

WEI, JUNJIE, and DEJUN FAN. "HOPF BIFURCATION ANALYSIS IN A MACKEY–GLASS SYSTEM." International Journal of Bifurcation and Chaos 17, no. 06 (June 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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23

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 evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model.

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24

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 results.

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25

ALGABA, A., M. MERINO, E. FREIRE, E. GAMERO, and A. J. RODRÍGUEZ-LUIS. "ON THE HOPF–PITCHFORK BIFURCATION IN THE CHUA'S EQUATION." International Journal of Bifurcation and Chaos 10, no. 02 (February 2000): 291–305. http://dx.doi.org/10.1142/s0218127400000190.

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We study some periodic and quasiperiodic behaviors exhibited by the Chua's equation with a cubic nonlinearity, near a Hopf–pitchfork bifurcation. We classify the types of this bifurcation in the nondegenerate cases, and point out the presence of a degenerate Hopf–pitchfork bifurcation. In this degenerate situation, analytical and numerical study shows a diversity of bifurcations of periodic orbits. We find a secondary Hopf bifurcation of periodic orbits, where invariant torus appears. This secondary Hopf bifurcation is bounded by a Takens–Bogdanov bifurcation of periodic orbits. Here, a sequence of period-doubling bifurcations of invariant tori is detected. Resonance phenomena are also analyzed. In the case of strong resonance 1:4, we show a new sequence of period-doubling bifurcations of 4T invariant tori.

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26

BUNGAY, SHARENE D., and SUE ANN CAMPBELL. "PATTERNS OF OSCILLATION IN A RING OF IDENTICAL CELLS WITH DELAYED COUPLING." International Journal of Bifurcation and Chaos 17, no. 09 (September 2007): 3109–25. http://dx.doi.org/10.1142/s0218127407018907.

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We investigate the behavior of a neural network model consisting of three neurons with delayed self and nearest-neighbor connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use numerical simulation and numerical bifurcation analysis to investigate the dynamics near the pitchfork–Hopf interaction points. Our numerical investigations reveal that multiple secondary Hopf bifurcations and pitchfork bifurcations of limit cycles may emanate from the pitchfork–Hopf points. Further, these secondary bifurcations give rise to ten different types of periodic solutions. In addition, the secondary bifurcations can lead to multistability between equilibrium points and periodic solutions in some regions of parameter space. We conclude by generalizing our results into conjectures about the secondary bifurcations that emanate from codimension two pitchfork–Hopf bifurcation points in systems with Dn symmetry.

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27

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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28

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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29

Guckenheimer, John, and Yuri Kuznetsov. "Hopf-Hopf bifurcation." Scholarpedia 3, no. 8 (2008): 1856. http://dx.doi.org/10.4249/scholarpedia.1856.

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REVEL, GUSTAVO, DIEGO M. ALONSO, and JORGE L. MOIOLA. "A GALLERY OF OSCILLATIONS IN A RESONANT ELECTRIC CIRCUIT: HOPF-HOPF AND FOLD-FLIP INTERACTIONS." International Journal of Bifurcation and Chaos 18, no. 02 (February 2008): 481–94. http://dx.doi.org/10.1142/s0218127408020409.

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In this work, the dynamics of a coupled electric circuit is studied. Several bifurcation diagrams associated with the truncated normal form of the Hopf-Hopf bifurcation are presented. The bifurcation curves are obtained by numerical continuation methods. The existence of quasi-periodic solutions with two (2D torus) and three (3D torus) frequency components is shown. These, in certain way, are close (or have a tendency to end up) to chaotic motion. Furthermore, two fold-flip bifurcations are detected in the vicinity of the Hopf-Hopf bifurcation, and are classified correspondingly. The analysis is completed with time simulations, the continuation of several limit cycle bifurcations and the indication of resonance points.

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31

Amdjadi, F., and P. J. Aston. "Detection of Tertiary Hopf Bifurcations Arising from Mode Interactions." International Journal of Bifurcation and Chaos 07, no. 07 (July 1997): 1691–98. http://dx.doi.org/10.1142/s0218127497001321.

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In the unfolding of a mode interaction, in addition to the primary bifurcations, there are also secondary bifurcations which occur on the primary branches giving rise to mixed mode solutions. A further tertiary Hopf bifurcation arises in some cases from the mixed mode solutions. The detection of Hopf bifurcation points is a numerically expensive procedure and so we consider whether it is possible to predict the existence of the tertiary Hopf bifurcation by considering only the geometric structure of the primary and secondary branches. We show that in some cases, it is possible to show that no Hopf bifurcation exists while in other cases, more information in the form of the stability of the trivial solution is required to determine whether or not the Hopf bifurcation exists. An algorithm for determining the existence of the Hopf bifurcation is given.

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32

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 manifold theory. Numerical simulations to support the analytical conclusions are carried out.

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Guo, Yuxiao, and Weihua Jiang. "Hopf Bifurcation in Two Groups of Delay-Coupled Kuramoto Oscillators." International Journal of Bifurcation and Chaos 25, no. 10 (September 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 the Kuramoto system, simulations are performed to support our analytic results.

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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 (July 8, 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 equations due to Li and Muldowney.

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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 (May 9, 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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van Kekem, Dirk L., and Alef E. Sterk. "Wave propagation in the Lorenz-96 model." Nonlinear Processes in Geophysics 25, no. 2 (April 27, 2018): 301–14. http://dx.doi.org/10.5194/npg-25-301-2018.

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Abstract. In this paper we study the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter n and the forcing parameter F. For F > 0 the first bifurcation is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at these bifurcations represents a traveling wave. Its spatial wave number increases linearly with n, but its period tends to a finite limit as n → ∞. For F < 0 and odd n, the first bifurcation is again a supercritical Hopf bifurcation, but in this case the period of the traveling wave also grows linearly with n. For F < 0 and even n, however, a Hopf bifurcation is preceded by either one or two pitchfork bifurcations, where the number of the latter bifurcations depends on whether n has remainder 2 or 0 upon division by 4. This bifurcation sequence leads to stationary waves and their spatiotemporal properties also depend on the remainder after dividing n by 4. Finally, we explain how the double-Hopf bifurcation can generate two or more stable waves with different spatiotemporal properties that coexist for the same parameter values n and F.

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Postlethwaite, C. M., G. Brown, and M. Silber. "Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1999 (September 28, 2013): 20120467. http://dx.doi.org/10.1098/rsta.2012.0467.

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Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

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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 (February 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 bifurcation theory of delay differential equations. Finally, it is found that numerical simulations agree with the associated analytic results. These phenomena could be induced although time delay is very small. Therefore, it is shown that time delay is an important factor which influences the sensitivity and accuracy of the gyroscope system and cannot be neglected during the design and manufacture.

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WANG, JINGNAN, and WEIHUA JIANG. "HOPF BIFURCATION ANALYSIS OF TWO SUNFLOWER EQUATIONS." International Journal of Biomathematics 05, no. 01 (January 2012): 1250001. http://dx.doi.org/10.1142/s1793524511001349.

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In this paper, two sunflower equations are considered. Using delay τ as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem's increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

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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 (March 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 paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.

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ITOVICH, GRISELDA R., and JORGE L. MOIOLA. "CHARACTERIZATION OF DYNAMIC BIFURCATIONS IN THE FREQUENCY DOMAIN." International Journal of Bifurcation and Chaos 12, no. 01 (January 2002): 87–101. http://dx.doi.org/10.1142/s0218127402004280.

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In this paper dynamical systems with certain degenerate Hopf bifurcations are considered. An analysis of the bifurcation behavior is proposed using several tools from the frequency domain approach. The analyzed bifurcations are the building blocks to understand the multiplicity of Hopf bifurcation points and to propose certain strategies in the future for controlling the bifurcation behavior in nonlinear systems.

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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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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's criterion for higher-dimensional ordinary differential equations.

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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 (November 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 periodic solutions are orbitally asymptotically stable. Numerical simulations are applied to verify the theoretical predictions.

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XU, XU. "LOCAL AND GLOBAL HOPF BIFURCATION IN A TWO-NEURON NETWORK WITH MULTIPLE DELAYS." International Journal of Bifurcation and Chaos 18, no. 04 (April 2008): 1015–28. http://dx.doi.org/10.1142/s0218127408020811.

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The paper presents a detailed analysis on the dynamics of a two-neuron network with time-delayed connections between the neurons and time-delayed feedback from each neuron to itself. On the basis of characteristic roots method and Hopf bifurcation theorems for functional differential equations, we investigate the existence of local Hopf bifurcation. In addition, the direction of Hopf bifurcation and stability of the periodic solutions bifurcating from the trivial equilibrium are determined based on the normal form theory and center manifold theorem. Moreover, employing the global Hopf bifurcation theory due to [Wu, 1998], we study the global existence of periodic solutions. It is shown that the local Hopf bifurcation indicates the global Hopf bifurcation after the second group critical value of the delay.

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Zhang, Wenjing, and Pei Yu. "Hopf and Generalized Hopf Bifurcations in a Recurrent Autoimmune Disease Model." International Journal of Bifurcation and Chaos 26, no. 05 (May 2016): 1650079. http://dx.doi.org/10.1142/s0218127416500796.

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This paper is concerned with bifurcation and stability in an autoimmune model, which was established to study an important phenomenon — blips arising from such models. This model has two equilibrium solutions, disease-free equilibrium and disease equilibrium. The positivity of the solutions of the model and the global stability of the disease-free equilibrium have been proved. In this paper, we particularly focus on Hopf bifurcation which occurs from the disease equilibrium. We present a detailed study on the use of center manifold theory and normal form theory, and derive the normal form associated with Hopf bifurcation, from which the approximate amplitude of the bifurcating limit cycles and their stability conditions are obtained. Particular attention is also paid to the bifurcation of multiple limit cycles arising from generalized Hopf bifurcation, which may yield bistable phenomenon involving equilibrium and oscillating motion. This result may explain some complex dynamical behavior in real biological systems. Numerical simulations are compared with the analytical predictions to show a very good agreement.

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Du, Wenju, Yandong Chu, Jiangang Zhang, Yingxiang Chang, Jianning Yu, and Xinlei An. "Bifurcation Analysis and Sliding Mode Control of Chaotic Vibrations in an Autonomous System." Journal of Applied Mathematics 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/726491.

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We study the bifurcations and sliding mode control of chaotic vibrations in an autonomous system. More precisely, a Hopf bifurcation controller is designed so as to control the unstable subcritical Hopf bifurcation to the stable supercritical Hopf bifurcation. Research result shows that the control method can work very well in Hopf bifurcation control. Besides, we controlled the system to any fixed point and any periodic orbit to eliminate the chaotic vibration by means of sliding mode method. And the numerical simulations were presented to confirm the effectiveness of the controller.

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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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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 (April 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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Yang, Ting. "Multistability and Hidden Attractors in a Three-Dimensional Chaotic System." International Journal of Bifurcation and Chaos 30, no. 06 (May 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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