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Journal articles on the topic 'Bifurcation'

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Author: Grafiati

Published: 4 June 2021

Last updated: 7 July 2024

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1

Liu, Ping, and Junping Shi. "A degenerate bifurcation from simple eigenvalue theorem." Electronic Research Archive 30, no. 1 (2021): 116–25. http://dx.doi.org/10.3934/era.2022006.

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<abstract><p>A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.</p></abstract>

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2

Aizawa, H., K. Ikeda, M. Osawa, and J. M. Gaspar. "Breaking and Sustaining Bifurcations in SN-Invariant Equidistant Economy." International Journal of Bifurcation and Chaos 30, no. 16 (December 28, 2020): 2050240. http://dx.doi.org/10.1142/s0218127420502405.

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This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group [Formula: see text] [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics.

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3

Chien, C. S., Z. Mei, and C. L. Shen. "Numerical Continuation at Double Bifurcation Points of a Reaction–Diffusion Problem." International Journal of Bifurcation and Chaos 08, no. 01 (January 1998): 117–39. http://dx.doi.org/10.1142/s0218127498000097.

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We study double bifurcations of a reaction–diffusion problem, and numerical methods for the continuation of bifurcating solution branches. To ensure a correct reflection of the bifurcation scenario in discretizations and to reduce imperfection of bifurcations, we consider a preservation of multiplicities of the bifurcation points in the discrete problems. A continuation-Arnoldi algorithm is exploited to trace the solution branches, and to detect secondary bifurcations. Numerical results on the Brusselator equations confirm our analysis.

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4

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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5

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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6

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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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

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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9

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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10

Zhuang, Xiaolan, Qi Wang, and Jiechang Wen. "Numerical Dynamics of Nonstandard Finite Difference Method for Nonlinear Delay Differential Equation." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1850133. http://dx.doi.org/10.1142/s021812741850133x.

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In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

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11

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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12

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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13

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 (June 30, 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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14

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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15

WANG, HUAILEI, and HAIYAN HU. "BIFURCATION ANALYSIS OF A DELAYED DYNAMIC SYSTEM VIA METHOD OF MULTIPLE SCALES AND SHOOTING TECHNIQUE." International Journal of Bifurcation and Chaos 15, no. 02 (February 2005): 425–50. http://dx.doi.org/10.1142/s0218127405012326.

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This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.

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16

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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17

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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18

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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19

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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20

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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21

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 degree theory. Finally, two neural network models with four or seven neurons are used to verify our theoretical results.

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22

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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23

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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24

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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25

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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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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Zhusubaliyev, Zhanybai T., Ulanbek A. Sopuev, Dmitry A. Bushuev, Andrey S. Kucherov, and Aitibek Z. Abdirasulov. "On bifurcations of chaotic attractors in a pulse width modulated control system." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 20, no. 1 (2024): 62–78. http://dx.doi.org/10.21638/11701/spbu10.2024.106.

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This paper discusses bifurcational phenomena in a control system with pulse-width modulation of the first kind. We show that the transition from a regular dynamics to chaos occurs in a sequence of classical supercritical period doubling and border collision bifurcations. As a parameter is varied, one can observe a cascade of doubling of the cyclic chaotic intervals, which are associated with homoclinic bifurcations of unstable periodic orbits. Such transition are also refereed as merging bifurcation (known also as merging crisis). At the bifurcation point, the unstable periodic orbit collides with some of the boundaries of a chaotic attractor and as a result, the periodic orbit becomes a homoclinic. This condition we use for obtain equations for bifurcation boundaries in the form of an explicit dependence on the parameters. This allow us to determine the regions of stability for periodic orbits and domains of the existence of four-, two- and one-band chaotic attractors in the parameter plane.

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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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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 verifies the theoretical results.

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Jiang, Chu-Yang, and Hsiao-Dong Chiang. "Pseudo-Pitchfork Bifurcation of Feasible Regions in Power Systems." International Journal of Bifurcation and Chaos 28, no. 01 (January 2018): 1830002. http://dx.doi.org/10.1142/s0218127418300021.

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Local bifurcations occur in power systems, causing changes in power system dynamic behaviors. These local bifurcations include the saddle-node bifurcation, Hopf bifurcation, and structure-induced bifurcation. This paper presents a new type of bifurcation that can occur in the optimal power flow (OPF) problem. This new type of bifurcation, termed pseudo-pitchfork bifurcation, involves bifurcations of the feasible components of the OPF problem and the disappearance of local optimal power flow solutions. The main features of this special type of bifurcation are demonstrated on several power systems with different loading condition parameters and different constraint parameters. Then the computation consideration and physical meaning of the pseudo-pitchfork bifurcation are roughly discussed. It is also demonstrated that a pseudo-pitchfork bifurcation occurring between feasible components can help us interpret the loss or birth of optimal power flow solutions and can lead to powerful numerical methods for computing high-quality optimal power flow solutions.

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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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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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33

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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XUEJUN, GAO. "BIFURCATION BEHAVIORS OF THE TWO-STATE VARIABLE FRICTION LAW OF A ROCK MASS SYSTEM." International Journal of Bifurcation and Chaos 23, no. 11 (November 2013): 1350184. http://dx.doi.org/10.1142/s0218127413501848.

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Based on the stability and bifurcation theory of dynamical systems, the bifurcation behaviors and chaotic motions of the two-state variable friction law of a rock mass system are investigated by the bifurcation diagrams based on the continuation method and the Poincaré maps. The stick-slip of the rock mass is formulated as an initial values problem for an autonomous system of three coupled nonlinear ordinary differential equations (ODEs) of first order. The results of linear stability analysis indicate that there is an equilibrium position in the rock mass system. Furthermore, numerical results of nonlinear analysis indicate that the equilibrium position loses its stability from a sup-critical Hopf bifurcation point, and then the bifurcating periodic motion evolves into chaotic motion through a series of period-doubling bifurcations with the decreasing of the control parameter. The stick-slip and chaotic motions evolve into infinity in the end with some unstable periodic motions.

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35

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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Chang, Yu, Xiaoli Wang, and Dashun Xu. "Bifurcation Analysis of a Power System Model with Three Machines and Four Buses." International Journal of Bifurcation and Chaos 26, no. 05 (May 2016): 1650082. http://dx.doi.org/10.1142/s0218127416500826.

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The bifurcation phenomena in a power system with three machines and four buses are investigated by applying bifurcation theory and harmonic balance method. The existence of saddle-node bifurcation and Hopf bifurcation is analyzed in time domain and in frequency domain, respectively. The approach of the fourth-order harmonic balance is then applied to derive the approximate expressions of periodic solutions bifurcated from Hopf bifurcations and predict their frequencies and amplitudes. Since the approach is valid only in some neighborhood of a bifurcation point, numerical simulations and the software Auto2007 are utilized to verify the predictions and further study bifurcations of these periodic solutions. It is shown that the power system may have various types of bifurcations, including period-doubling bifurcation, torus bifurcation, cyclic fold bifurcation, and complex dynamical behaviors, including quasi-periodic oscillations and chaotic behavior. These findings help to better understand the dynamics of the power system and may provide insight into the instability of power systems.

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37

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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38

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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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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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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LIU, JUNLI, and TAILEI ZHANG. "HOPF BIFURCATION AND STABILITY ANALYSIS FOR A STAGE-STRUCTURED SYSTEM." International Journal of Biomathematics 03, no. 01 (March 2010): 21–41. http://dx.doi.org/10.1142/s1793524510000830.

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In this paper, we considered a time-delay predator–prey system, in which the prey has two life stages, juvenile and mature. Delay was regarded as the bifurcation parameter, we analyzed the characteristic equation of the system at the positive equilibrium, stability of the positive equilibrium and existence of Hopf bifurcation with delay τ in the term of degree are investigated. The explicit formulae which determine the direction of the bifurcations, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. To verify our theoretical results, a numerical example is also included.

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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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ROCŞOREANU, CARMEN, NICOLAIE GIURGIŢEANU, and ADELINA GEORGESCU. "CONNECTIONS BETWEEN SADDLES FOR THE FITZHUGH–NAGUMO SYSTEM." International Journal of Bifurcation and Chaos 11, no. 02 (February 2001): 533–40. http://dx.doi.org/10.1142/s0218127401002213.

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By studying the two-dimensional FitzHugh–Nagumo (F–N) dynamical system, points of Bogdanov–Takens bifurcation were detected (Sec. 1). Two of the curves of homoclinic bifurcation emerging from these points intersect each other at a point of double breaking saddle connection bifurcation (Sec. 2). Numerical investigations of the bifurcation curves emerging from this point, in the parameter plane, allowed us to find other types of codimension-one and -two bifurcations concerning the connections between saddles and saddle-nodes, referred to as saddle-node–saddle connection bifurcation and saddle-node–saddle with separatrix connection bifurcation, respectively. The local bifurcation diagrams corresponding to these bifurcations are presented in Sec. 3. An analogy between the bifurcation corresponding to the point of double homoclinic bifurcation and the point of double breaking saddle connection bifurcation is also presented in Sec. 3.

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45

TURAEV, D. "ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS." International Journal of Bifurcation and Chaos 06, no. 05 (May 1996): 919–48. http://dx.doi.org/10.1142/s0218127496000515.

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An analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension dc which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong-unstable foliations, and the Lyapunov dimension dL which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values dc and dL. A connection with the problem of hyperchaos is discussed.

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46

Zhang, Huayong, Shengnan Ma, Tousheng Huang, Xuebing Cong, Zichun Gao, and Feifan Zhang. "Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response." Discrete Dynamics in Nature and Society 2018 (2018): 1–18. http://dx.doi.org/10.1155/2018/2386954.

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We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.

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Li, Changzhao, and Hui Fang. "Stochastic Bifurcations of Group-Invariant Solutions for a Generalized Stochastic Zakharov–Kuznetsov Equation." International Journal of Bifurcation and Chaos 31, no. 03 (March 15, 2021): 2150040. http://dx.doi.org/10.1142/s0218127421500401.

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In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic bifurcations of group-invariant solutions can be considered as an indirect display of bifurcation phenomena of stochastic nonlinear wave equations. As a constructive example, we study stochastic bifurcations of group-invariant solutions for a generalized stochastic Zakharov–Kuznetsov equation.

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XIAO, DONGMEI, and SHIGUI RUAN. "CODIMENSION TWO BIFURCATIONS IN A PREDATOR–PREY SYSTEM WITH GROUP DEFENSE." International Journal of Bifurcation and Chaos 11, no. 08 (August 2001): 2123–31. http://dx.doi.org/10.1142/s021812740100336x.

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In this paper we study the qualitative behavior of a predator–prey system with nonmonotonic functional response. The system undergoes a series of bifurcations including the saddle-node bifurcation, the supercritical Hopf bifurcation, and the homoclinic bifurcation. For different parameter values the system could have a limit cycle or a homoclinic loop, or exhibit the so-called "paradox of enrichment" phenomenon. In the generic case, the model has the bifurcation of cusp-type codimension two (i.e. the Bogdanov–Takens bifurcation) but no bifurcations of codimension three.

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AFRAIMOVICH, V. S., and M. A. SHERESHEVSKY. "THE HAUSDORFF DIMENSION OF ATTRACTORS APPEARING BY SADDLE-NODE BIFURCATIONS." International Journal of Bifurcation and Chaos 01, no. 02 (June 1991): 309–25. http://dx.doi.org/10.1142/s0218127491000233.

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We consider the strange attractors which appear as a result of saddle-node vanishing bifurcations in two-dimensional, smooth dynamical systems. Some estimates and asymptotic formulas for the Hausdorff dimension of such attractors are obtained. The estimates demonstrate a dependence of the dimension growth rate after the bifurcation upon the "pre-bifurcational" picture.

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Fan, Li, and Sanyi Tang. "Global Bifurcation Analysis of a Population Model with Stage Structure and Beverton–Holt Saturation Function." International Journal of Bifurcation and Chaos 25, no. 12 (November 2015): 1550170. http://dx.doi.org/10.1142/s0218127415501709.

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In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton–Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov–Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically.

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