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Journal articles on the topic 'Bifurcations'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 June 2025

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1

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 (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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2

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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3

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

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

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4

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

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

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6

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 (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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7

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

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

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8

Mutakabbir Khan, Md, and Md Jasim Uddin. "Complexity analysis with chaos control: A discretized ratio-dependent Holling-Tanner predator-prey model with Fear effect in prey population." PLOS One 20, no. 6 (2025): e0324299. https://doi.org/10.1371/journal.pone.0324299.

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This study explores a novel two-dimensional discrete-time ratio-dependent Holling-Tanner predator-prey model, incorporating the impact of the Fear effect on the prey population. The study focuses on identifying stationary points and analyzing bifurcations around the positive fixed point, with an emphasis on their biological significance. Our examination of bifurcations at the interior fixed point uncovers a variety of generic bifurcations, including one-parameter bifurcations, period-doubling, and Neimark-Sacker bifurcations. To further understand NS bifurcation, we establish non-degeneracy condition. The system’s bifurcating and fluctuating behavior is managed using Ott—Grebogi—Yorke (OGY) control technique. From an ecological perspective, these findings underscore the substantial role of the Fear effect in shaping predator-prey dynamics. The research is extended to a networked context, where interconnected prey-predator populations demonstrate the influence of coupling strength and network structure on the system’s dynamics. The theoretical results are validated through numerical simulations, which encompass local dynamical classifications, calculations of maximum Lyapunov exponents, phase portrait analyses, and bifurcation diagrams.

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9

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 (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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10

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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11

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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12

GOLUBITSKY, M., and M. KRUPA. "STABILITY COMPUTATIONS FOR NILPOTENT HOPF BIFURCATIONS IN COUPLED CELL SYSTEMS." International Journal of Bifurcation and Chaos 17, no. 08 (2007): 2595–603. http://dx.doi.org/10.1142/s0218127407018658.

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Vanderbauwhede and van Gils, Krupa, and Langford studied unfoldings of bifurcations with purely imaginary eigenvalues and a nonsemisimple linearization, which generically occurs in codimension three. In networks of identical coupled ODE these nilpotent Hopf bifurcations can occur in codimension one. Elmhirst and Golubitsky showed that these bifurcations can lead to surprising branching patterns of periodic solutions, where the type of bifurcation depends in part on the existence of an invariant subspace corresponding to partial synchrony. We study the stability of some of these bifurcating solutions. In the absence of partial synchrony the problem is similar to the generic codimension three problem. In this case we show that the bifurcating branches are generically unstable. When a synchrony subspace is present we obtain partial stability results by using only those near identity transformations that leave this subspace invariant.

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13

TURAEV, D. "ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS." International Journal of Bifurcation and Chaos 06, no. 05 (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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14

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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15

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

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

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16

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

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

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17

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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18

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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19

JENSEN, CARSTEN NORDSTRØM, MARTIN GOLUBITSKY, and HANS TRUE. "SYMMETRY, GENERIC BIFURCATIONS, AND MODE INTERACTION IN NONLINEAR RAILWAY DYNAMICS." International Journal of Bifurcation and Chaos 09, no. 07 (1999): 1321–31. http://dx.doi.org/10.1142/s0218127499000924.

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We investigate Cooperrider's complex bogie, a mathematical model of a railway bogie running on an ideal straight track. The speed of the bogie v is the control parameter. Taking symmetry into account, we find that the generic bifurcations from a symmetric periodic solution of the model are Hopf bifurcations for maps (or Neimark bifurcations), saddle-node bifurcations, and pitchfork bifurcations. The last ones are symmetry-breaking bifurcations. By variation of an additional parameter, bifurcations of higher degeneracy are possible. In particular, we consider mode interactions near a degenerate bifurcation. The bifurcation analysis and path-finding are done numerically.

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20

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

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

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21

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 (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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22

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

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

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23

ABED, E. H., H. O. WANG, J. C. ALEXANDER, A. M. A. HAMDAN, and H. C. LEE. "DYNAMIC BIFURCATIONS IN A POWER SYSTEM MODEL EXHIBITING VOLTAGE COLLAPSE." International Journal of Bifurcation and Chaos 03, no. 05 (1993): 1169–76. http://dx.doi.org/10.1142/s0218127493000969.

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Dynamic bifurcations, including Hopf and period-doubling bifurcations, are found to occur in a power system dynamic model recently employed in voltage collapse studies. The occurrence of dynamic bifurcations is ascertained in a region of state and parameter space linked with the onset of voltage collapse. The work focuses on a power system model studied by Dobson & Chiang [1989]. The presence of the dynamic bifurcations, and the resulting implications for dynamic behavior, necessitate a re-examination of the role of saddle node bifurcations in the voltage collapse phenomenon. The bifurcation analysis is performed using the reactive power demand at a load bus as the bifurcation parameter. It is determined that the power system model under consideration exhibits two Hopf bifurcations in the vicinity of the saddle node bifurcation. Between the Hopf bifurcations, i.e., in the "Hopf window," period-doubling bifurcations are found to occur. Simulations are given to illustrate the various types of dynamic behaviors associated with voltage collapse for the model. In particular, it is seen that an oscillatory transient may play a role in the collapse.

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24

Tikjha, Wirot, and Laura Gardini. "Bifurcation Sequences and Multistability in a Two-Dimensional Piecewise Linear Map." International Journal of Bifurcation and Chaos 30, no. 06 (2020): 2030014. http://dx.doi.org/10.1142/s0218127420300141.

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Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifurcations, (B) degenerate flip bifurcations, supercritical and subcritical, (C) degenerate transcritical bifurcations and (D) supercritical center bifurcations. Each of these is characterized by a particular dynamic behavior, and may be related to attracting or repelling cycles. We consider different bifurcation routes, showing the interplay between all these kinds of bifurcations, and their role in the phase plane in determining attracting sets and basins of attraction.

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Ferrer, Josep, Marta Peña, and Antoni Susin. "Codimension-3 Bifurcation for Continuous Saddle Bimodal Linear Dynamical Systems." International Journal of Bifurcation and Chaos 28, no. 02 (2018): 1850025. http://dx.doi.org/10.1142/s0218127418500256.

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We continue the study of the structural stability and the bifurcations of planar continuous bimodal linear dynamical systems (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we complete the study when one of the subsystems is a saddle, leading to a 3D bifurcation diagram where a large catalogue of bifurcations appears: four surfaces of codimension-1 bifurcations; two sequences of surfaces of additional codimension-1 bifurcations; two lines of codimension-2 bifurcations; and one codimension-3 bifurcation.

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26

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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Li, Yingguo. "Dynamics of a Discrete Internet Congestion Control Model." Discrete Dynamics in Nature and Society 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/628369.

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We consider a discrete Internet model with a single link accessed by a single source, which responds to congestion signals from the network. Firstly, the stability of the equilibria of the system is investigated by analyzing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Neimark-Sacker bifurcations occur when the delay passes a sequence of critical values. Then, the explicit algorithm for determining the direction of the Neimark-Sacker bifurcations and the stability of the bifurcating periodic solutions is derived. Finally, some numerical simulations are given to verify the 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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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 (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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Saputra, Kie Van Ivanky. "Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations." International Journal of Bifurcation and Chaos 25, no. 06 (2015): 1550091. http://dx.doi.org/10.1142/s0218127415500911.

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We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

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

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

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32

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 (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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33

Liu, Yongjie, Yu Jiang, and Hengnian Li. "Bifurcations of Periodic Orbits in the Gravitational Field of Irregular Bodies: Applications to Bennu and Steins." Aerospace 9, no. 3 (2022): 151. http://dx.doi.org/10.3390/aerospace9030151.

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We investigate the topological types and bifurcations of periodic orbits in the gravitational field of irregular bodies by the well-known two parameter analysis method. Results show that the topological types of periodic orbits are determined by the locations of these two parameters and that the bifurcation types correspond to their variation paths in the plane. Several new paths corresponding to doubling period bifurcations, tangent bifurcations and Neimark–Sacker bifurcations are discovered. Then, applications in detecting bifurcations of periodic orbits near asteroids 101955 Bennu and 2867 Steins are presented. It is found that tangent bifurcations may occur three times when continuing the vertical orbits near the equilibrium points of 101955 Bennu. The continuation stops as the Jacobi energy reaches a local maximum. However, while continuing the vertical orbits near the equilibrium points of 2867 Steins, the tangent bifurcation and pseudo period-doubling bifurcation occur. The continuation can always go on, and the orbit ultimately becomes nearly circular.

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34

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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35

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

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The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present 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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36

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 (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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37

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

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A competitive Lotka–Volterra reaction-diffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive constant steady state of the system is globally asymptotically stable if the interspecies competition is weaker than the intraspecies one and is unstable if the interspecies competition dominates over the intraspecies one. If the latter holds, then we show that Hopf bifurcation can occur as the parameters (delays) in the system cross some critical values. In particular, we prove that these Hopf bifurcations are all spatially homogeneous 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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38

WEI, HSIU-CHUAN. "NUMERICAL REVISIT TO A CLASS OF ONE-PREDATOR, TWO-PREY MODELS." International Journal of Bifurcation and Chaos 20, no. 08 (2010): 2521–36. http://dx.doi.org/10.1142/s0218127410027143.

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Some observations are made on a class of one-predator, two-prey models via numerical analysis. The simulations are performed with the aid of an adaptive grid method for constructing bifurcation diagrams and cell-to-cell mapping for global analysis. A two-dimensional bifurcation diagram is constructed to show that regions of coexistence of all three species, which imply the balance of competitive and predatory forces, are surrounded by regions of extinction of one or two species. Two or three coexisting attractors which may have a chaotic member are found in some regions of the bifurcation diagram. Their separatrices are computed to show the domains of attraction. The bifurcation diagram also contains codimension-two bifurcation points including Bogdanov–Takens, Gavrilov–Guckenheimer, and Bautin bifurcations. The dynamics in the vicinity of these codimension-two bifurcation points are discussed. Some global bifurcations including homoclinic and heteroclinic bifurcations are investigated. They can account for the disappearance of chaotic attractors and limit cycles. Bifurcations of limit cycles such as transcritical and saddle-node bifurcations are also studied in this work. Finally, some relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.

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39

Liu, Wei, and Yaolin Jiang. "Analysis of a Delayed Predator–Prey System with Harvesting." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 3-4 (2018): 335–49. http://dx.doi.org/10.1515/ijnsns-2017-0094.

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AbstractThis article is concerned with a Leslie–Gower predator–prey system with the predator being harvested and the prey having a delay due to the gestation of prey species. By regarding the gestation delay as a bifurcation parameter, we first derive some sufficient conditions on the stability of positive equilibrium point and the existence of Hopf bifurcations basing on the local parametrization method for differential-algebra system. In succession, we also investigate the direction of Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold by employing the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, several numerical simulations are given.

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40

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

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

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41

Liu, Yongjian, Chunbiao Li, and Aimin Liu. "Analysis of Geometric Invariants for Three Types of Bifurcations in 2D Differential Systems." International Journal of Bifurcation and Chaos 31, no. 07 (2021): 2150105. http://dx.doi.org/10.1142/s0218127421501054.

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Little is known about bifurcations in two-dimensional (2D) differential systems from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. Based on the KCC geometric invariants, three types of static bifurcations in 2D differential systems, i.e. saddle-node bifurcation, transcritical bifurcation, and pitchfork bifurcation, are discussed in this paper. The dynamics far from fixed points of the systems generating bifurcations are characterized by the deviation curvature and nonlinear connection. In the nonequilibrium region, the nonlinear stability of systems is not simple but involves alternation between stability and instability, even though systems are invariably Jacobi-unstable. The results also indicate that the dynamics in the nonequilibrium region are node-like for three typical static bifurcations.

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42

NIKOLAEV, E. V., V. N. BIKTASHEV, and A. V. HOLDEN. "ON BIFURCATIONS OF SPIRAL WAVES IN THE PLANE." International Journal of Bifurcation and Chaos 09, no. 08 (1999): 1501–16. http://dx.doi.org/10.1142/s021812749900105x.

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We describe the simplest bifurcations of spiral waves in reaction–diffusion systems in the plane and present the list of model systems. One-parameter bifurcations of one-armed spiral waves are fold and Hopf bifurcations. Multiarmed spiral waves may additionally undergo a period-doubling pitchfork bifurcation, when two congruent spiral wave solutions, having the "double" period, branch from the original spiral wave at the bifurcation point.

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43

Akmanova, S. V., and M. G. Yumagulov. "On local bifurcations in nonlinear continuous-discrete dynamical systems." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 2 (February 25, 2025): 3–14. https://doi.org/10.26907/0021-3446-2025-2-3-14.

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The dynamics of nonlinear continuous-discrete (hybrid) systems and its dependence on the sampling step h are studied. Such systems contain phase variables and equations with both continuous and discrete time. The main focus of the work is the issue of local bifurcations during loss of stability of equilibrium points of hybrid systems. Sufficient signs of bifurcations are given, the properties of bifurcations are studied, and possible bifurcation scenarios are determined. The concept of transversal bifurcation is introduced, meaning that the corresponding eigenvalue of the matrix of the linearized problem passes through the unit circle when the parameter h passes through the bifurcation point h0. It is shown that in a one-parameter formulation, two main scenarios are typical: transversal bifurcation of period doubling and transversal Andronov-Hopf bifurcation, while the scenario of transversal bifurcation of multiple equilibrium, as a rule, is not realized. Examples are given to illustrate the effectiveness of the proposed approaches in the problem of studying bifurcations in hybrid systems.

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44

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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Kuznetsov, Yu A., S. Rinaldi, and A. Gragnani. "One-Parameter Bifurcations in Planar Filippov Systems." International Journal of Bifurcation and Chaos 13, no. 08 (2003): 2157–88. http://dx.doi.org/10.1142/s0218127403007874.

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We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

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46

KOSTOVA, TANYA, RENUKA RAVINDRAN, and MARIA SCHONBEK. "FITZHUGH–NAGUMO REVISITED: TYPES OF BIFURCATIONS, PERIODICAL FORCING AND STABILITY REGIONS BY A LYAPUNOV FUNCTIONAL." International Journal of Bifurcation and Chaos 14, no. 03 (2004): 913–25. http://dx.doi.org/10.1142/s0218127404009685.

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We study several aspects of FitzHugh–Nagumo's (FH–N) equations without diffusion. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined Lyapunov functional. We show the existence of both supercritical and subcritical Hopf bifurcations. We demonstrate that the number of all bifurcation diagrams is 8 but that the possible sequential occurrences of bifurcation events is much richer. We present a numerical study of an example exhibiting a series of various bifurcations, including subcritical Hopf bifurcations, homoclinic bifurcations and saddle-node bifurcations of equilibria and of periodic solutions. Finally, we study periodically forced FH–N equations. We prove that phase-locking occurs independently of the magnitude of the periodic forcing.

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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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48

Delgado, Joaquín, Lucía Ivonne Hernández-Martínez, and Javier Pérez-López. "Global Bifurcation Map of the Homogeneous States in the Gray–Scott Model." International Journal of Bifurcation and Chaos 27, no. 07 (2017): 1730024. http://dx.doi.org/10.1142/s0218127417300245.

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We study the spatially homogeneous time-dependent solutions and their bifurcations in the Gray–Scott model. We find the global map of bifurcations by a combination of rigorous verification of the existence of Takens–Bogdanov and a Bautin bifurcation, in the space of two parameters [Formula: see text]–[Formula: see text]. With the aid of numerical continuation of local bifurcation curves we give a global description of all the possible bifurcations.

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BASHKIRTSEVA, IRINA, LEV RYASHKO, and PAVEL STIKHIN. "NOISE-INDUCED CHAOS AND BACKWARD STOCHASTIC BIFURCATIONS IN THE LORENZ MODEL." International Journal of Bifurcation and Chaos 23, no. 05 (2013): 1350092. http://dx.doi.org/10.1142/s0218127413500922.

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We study the phenomena of stochastic D- and P-bifurcations of randomly forced limit cycles for the Lorenz model. As noise intensity increases, regular multiple limit cycles of this model in a period-doubling bifurcations zone are deformed to be stochastic attractors that look chaotic (D-bifurcation) and their multiplicity is reduced (P-bifurcation). In this paper for the comparative investigation of these bifurcations, the analysis of Lyapunov exponents and stochastic sensitivity function technique are used. A probabilistic mechanism of backward stochastic bifurcations for cycles of high multiplicity is analyzed in detail. We show that for a limit cycle with multiplicity two and higher, a threshold value of the noise intensity which marks the onset of chaos agrees with the first backward stochastic bifurcation.

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Sheng, Hao, Hsiao-Dong Chiang, and Yan-Feng Jiang. "Local Bifurcations of Electric Distribution Networks with Renewable Energy." International Journal of Bifurcation and Chaos 24, no. 07 (2014): 1450102. http://dx.doi.org/10.1142/s0218127414501028.

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Recent years have witnessed a growing trend towards the development and deployment of distributed generation (DG). It is shown that electric distribution networks with DGs can encounter two types of local bifurcations: saddle-node bifurcation and structure-induced bifurcation. The structure-induced bifurcation occurs when a transition between two structures of the distribution network takes place due to limited amount of reactive power supports from renewable energies. The saddle-node bifurcation occurs when the underlying distribution network reaches the limit of its delivery capability. The consequence of structure-induced bifurcation is an immediate instability induced by reactive power limits of renewable energy. It is numerically shown that both types of local bifurcations can occur at both small distribution networks and large-scale distribution networks with DGs. Physical explanations of these two local bifurcations are provided. Studies of local bifurcations in distribution networks provide insights regarding how to design controls to enhance distribution networks with DGs.

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