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Journal articles on the topic 'Hopf Equation'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Tobisch, Elena, and Efim Pelinovsky. "Modular Hopf equation." Applied Mathematics Letters 97 (November 2019): 1–5. http://dx.doi.org/10.1016/j.aml.2019.05.009.

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2

N O, Onuoha. "Transformation of Parabolic Partial Differential Equations into Heat Equation Using Hopf Cole Transform." International Journal of Science and Research (IJSR) 12, no. 6 (2023): 1741–43. http://dx.doi.org/10.21275/sr23612082710.

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3

Wang, Chuncheng. "Normal Forms for Partial Neutral Functional Differential Equations with Applications to Diffusive Lossless Transmission Line." International Journal of Bifurcation and Chaos 30, no. 02 (2020): 2050028. http://dx.doi.org/10.1142/s0218127420500285.

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A class of partial neutral functional differential equations are considered. For the linearized equation, the semigroup properties and formal adjoint theory are established. Based on these results, we develop two algorithms of normal form computation for the nonlinear equation, and then use them to study Hopf bifurcation problems of such equations. In particular, it is shown that the normal forms, derived from these two different approaches, for the Hopf bifurcation are exactly the same. As an illustration, the diffusive lossless transmission line equation where a Hopf singularity occurs is st

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4

Ghosal, Amitava. "Wiener‐Hopf Equation Revisited." Kybernetes 23, no. 6/7 (1994): 128–32. http://dx.doi.org/10.1108/03684929410068415.

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5

Militaru, G. "The hopf modules category and the hopf equation." Communications in Algebra 26, no. 10 (1998): 3071–97. http://dx.doi.org/10.1080/00927879808826329.

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6

Boziev, Oleg L., and Mukhamed A. Abazokov. "APPROXIMATION OF THE HOPF EQUATION BY LOADED EQUATIONS." Bulletin of the Moscow State Regional University (Physics and Mathematics), no. 1 (2020): 28–36. http://dx.doi.org/10.18384/2310-7251-2020-1-28-36.

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7

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

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

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8

ZHENG, HUIHUI, FANGSHU LI, and TIANSHUI MA. "HOPF CO-BRACE, BRAID EQUATION AND BICROSSED." Mathematical Reports 25(75), no. 3 (2023): 481–93. http://dx.doi.org/10.59277/mrar.2023.25.75.3.481.

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In this paper, we mainly give some equivalent characterisations of Hopf cobraces, show that the full subcategory HCB(A) of Hopf co-braces is equivalent to the full subcategory C(A) of bijective 1-cocycles, and prove that the full subcategory HCB(A) is also equivalent to the category M(A) of Hopf matched pairs. Moreover, we construct many Hopf co-braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel’d doubles of finite dimensional Hopf algebras. And we also give a sufficient and necessary condition for a given bicrossed coproduct A ▷◁ H to be a Hopf co-brace if A or H is a

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9

Sgibnev, M. S. "Homogeneous conservative Wiener-Hopf equation." Sbornik: Mathematics 198, no. 9 (2007): 1341–50. http://dx.doi.org/10.1070/sm2007v198n09abeh003886.

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10

Mirsaburova, Gulbaxor. "DERIVATION OF THE WIENER-HOPF INTEGRAL EQUATION." Multidisciplinary Journal of Science and Technology 4, no. 10 (2024): 284–92. https://doi.org/10.5281/zenodo.14001950.

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The problem with Bitsadze-Samarskii conditions on the boundary of ellipticity and a segment of the degeneracy line and the displacement condition on pieces of the boundary characteristics of the Gelleristedt equation with a singular coefficient is investigated. The uniqueness of the solution to the problem is proved using the maximum principle, and the existence of the solution is proved using the method of integral equations.

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11

Dzhunushaliev, Vladimir, and Vladimir Folomeev. "Nonperturbative QED on the Hopf Bundle." Physical Sciences Forum 2, no. 1 (2021): 43. http://dx.doi.org/10.3390/ecu2021-09286.

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We consider the Dirac equation and Maxwell’s electrodynamics in ℝ×S3 spacetime, where a three-dimensional sphere is the Hopf bundle S3→S2. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger–Dyson equations for Green’s functions is written down. To illustrate the suggested scheme of nonperturbative quantization, we write a simplified set of equations describing some physical situation. Additionally, we discuss the properties of quantum states and operators of interacting fiel

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12

ARNOLD, LUDWIG, N. SRI NAMACHCHIVAYA, and KLAUS R. SCHENK-HOPPÉ. "TOWARD AN UNDERSTANDING OF STOCHASTIC HOPF BIFURCATION." International Journal of Bifurcation and Chaos 06, no. 11 (1996): 1947–75. http://dx.doi.org/10.1142/s0218127496001272.

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In this paper, asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through the study of a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied. In the first part of this paper, we present an introduction to the theory of random dynamical systems (in particular, their generation, their invariant measures, the multiplicative ergodic theorem, and Lyapunov exponents). We then present the two concepts of stochastic bifurcation theory: Phenomenologica

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13

Wu, Hui, and Xiaohui Zhang. "On the BiHom-Type Nonlinear Equations." Mathematics 10, no. 22 (2022): 4360. http://dx.doi.org/10.3390/math10224360.

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In this paper, the Heisenberg doubles and Long dimodules of a BiHom-Hopf algebra are introduced. Then, we discussed the relation between BiHom-Hopf equation and BiHom-pentagon equation, and we obtain the solutions of BiHom-Hopf equation from Heisenberg doubles. We also showed that the parametric generalized Long dimodules can provide the solutions of BiHom-Yang-Baxter equation and generalized D-equation.

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14

Iwata, Yoritaka. "Abstract Formulation of the Miura Transform." Mathematics 8, no. 5 (2020): 747. http://dx.doi.org/10.3390/math8050747.

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Miura transform is known as the transformation between Korweg de-Vries equation and modified Korweg de-Vries equation. Its formal similarity to the Cole-Hopf transform has been noticed. This fact sheds light on the logarithmic type transformations as an origin of a certain kind of nonlinearity in the soliton equations. In this article, based on the logarithmic representation of operators in infinite-dimensional Banach spaces, a structure common to both Miura and Cole-Hopf transforms is discussed. In conclusion, the Miura transform is generalized as the transform in abstract Banach spaces, and

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15

McGregor, M. T. "On a Wiener-Hopf Integral Equation." Journal of Integral Equations and Applications 7, no. 4 (1995): 479–83. http://dx.doi.org/10.1216/jiea/1181075899.

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16

Li, Junyu. "Hopf bifurcation of the sunflower equation." Nonlinear Analysis: Real World Applications 10, no. 4 (2009): 2574–80. http://dx.doi.org/10.1016/j.nonrwa.2008.03.002.

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17

Liu, Hai-liang. "On discreteness of the Hopf equation." Acta Mathematicae Applicatae Sinica, English Series 24, no. 3 (2008): 423–40. http://dx.doi.org/10.1007/s10255-008-8021-1.

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18

Shabat, A. B. "Periodic solutions of the Hopf equation." Theoretical and Mathematical Physics 177, no. 2 (2013): 1471–78. http://dx.doi.org/10.1007/s11232-013-0116-z.

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19

Feldkord, Sven, Marco Reit, and Wolfgang Mathis. "Discretization analysis of bifurcation based nonlinear amplifiers." Advances in Radio Science 15 (September 21, 2017): 43–47. http://dx.doi.org/10.5194/ars-15-43-2017.

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Abstract. Recently, for modeling biological amplification processes, nonlinear amplifiers based on the supercritical Andronov–Hopf bifurcation have been widely analyzed analytically. For technical realizations, digital systems have become the most relevant systems in signal processing applications. The underlying continuous-time systems are transferred to the discrete-time domain using numerical integration methods. Within this contribution, effects on the qualitative behavior of the Andronov–Hopf bifurcation based systems concerning numerical integration methods are analyzed. It is shown exem

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20

Leung, A. Y. T., and T. Ge. "An Algorithm for Higher Order Hopf Normal Forms." Shock and Vibration 2, no. 4 (1995): 307–19. http://dx.doi.org/10.1155/1995/581272.

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Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the

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21

ALGABA, ANTONIO, MANUEL MERINO, FERNANDO FERNÁNDEZ-SÁNCHEZ, and ALEJANDRO J. RODRÍGUEZ-LUIS. "HOPF BIFURCATIONS AND THEIR DEGENERACIES IN CHUA'S EQUATION." International Journal of Bifurcation and Chaos 21, no. 09 (2011): 2749–63. http://dx.doi.org/10.1142/s0218127411030106.

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We perform an analytical study of the Hopf bifurcations and their degeneracies in Chua's equation. In the case of the equilibrium at the origin only codimension-two Hopf bifurcations appear. However, for the nontrivial equilibria we prove the existence of codimension-three Hopf bifurcations. Numerical results are in strong agreement with the analytical ones.

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22

Mitsioulis, G. "A Wiener–Hopf theory for a semi-infinite dielectric slab." Canadian Journal of Physics 68, no. 11 (1990): 1348–51. http://dx.doi.org/10.1139/p90-192.

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We investigate the field distribution around a semi-infinite dielectric slab. The wave equation is transformed, through a Fourier transform, to the wave number domain. The boundary conditions are imposed and we end up with an equation of the Wiener–Hopf type. Certain functions are factorized and we find a system of equations for the unknown coefficients of the Fourier series expansion for the field at the mouth of the slab.

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23

Tang, Diandian, Shirui Zhang, and Jingli Ren. "Dynamics of a general jerky equation." Journal of Vibration and Control 25, no. 4 (2018): 922–32. http://dx.doi.org/10.1177/1077546318805583.

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Some classic nonlinear dynamical systems, such as Rössler's toroidal model, the Genesio model, and 19 Sprott's models, can be classified into seven distinct basic classes of jerky dynamics, labeled by [Formula: see text]. This paper is devoted to the dynamics of a general jerky equation which contains [Formula: see text] as parameters vary. It is shown that the system undergoes fold, Hopf, zero-Hopf, and Bogdanov–Takens bifurcations based on the center manifold theorem and normal form theory. Numerical simulations are also given to make the theoretical results visible and to detect more compli

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24

Bluman, G. W., and S. Kumei. "Symmetry-based algorithms to relate partial differential equations: II. Linearization by nonlocal symmetries." European Journal of Applied Mathematics 1, no. 3 (1990): 217–23. http://dx.doi.org/10.1017/s0956792500000188.

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An algorithm is presented to linearize nonlinear partial differential equations by non-invertible mappings. The algorithm depends on finding nonlocal symmetries of the given equations which are realized as appropriate local symmetries of a related auxiliary system. Examples include the Hopf-Cole transformation and the linearizations of a nonlinear heat conduction equation, a nonlinear telegraph equation, and the Thomas equations.

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25

Losson, J�r�me, and Michael C. Mackey. "A Hopf-like equation and perturbation theory for differential delay equations." Journal of Statistical Physics 69, no. 5-6 (1992): 1025–46. http://dx.doi.org/10.1007/bf01058760.

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26

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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27

MATSKEVICH, S. E. "BURGERS EQUATION AND KOLMOGOROV–PETROVSKY–PISKUNOV EQUATION ON MANIFOLDS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 14, no. 02 (2011): 199–208. http://dx.doi.org/10.1142/s0219025711004341.

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In this paper the analogue of the Hopf–Cole transformation is introduced for manifolds. It has been shown that under certain conditions this transformation connects the Burgers equation and Kolmogorov–Petrovsky–Piskunov equation on pseudo-Riemannian manifolds.

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28

MOUS, ILHEM, and ABDELHAMID LAOUAR. "A Numerical Solution of a Coupling System of Conformable Time-Derivative Two-Dimensional Burgers’ Equations." Kragujevac Journal of Mathematics 48, no. 1 (2024): 7–23. http://dx.doi.org/10.46793/kgjmat2401.007m.

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In this paper, we deal with a numerical solution of a coupling system of fractional conformable time-derivative two-dimensional (2D) Burgers’ equations. The presence of both the fractional time derivative and the nonlinear terms in this system of equations makes solving it more diﬃcult. Firstly, we use the Cole-Hopf transformation in order to reduce the coupling system of equations to a conformable time-derivative 2D heat equation for which the numerical solution is calculated by the explicit and implicit schemes. Secondly, we calculate the numerical solution of the proposed system by using bo

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29

Agore, A. L. "Constructing Hopf braces." International Journal of Mathematics 30, no. 02 (2019): 1850089. http://dx.doi.org/10.1142/s0129167x18500891.

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We investigate Hopf braces, a concept recently introduced by Angiono, Galindo and Vendramin [I. Angiono, C. Galindo and L. Vendramin, Hopf braces and Yang–Baxter operators, Proc. Amer. Math. Soc. 145 (2017) 1981–1995] in connection to the quantum Yang–Baxter equation. More precisely, we propose two methods for constructing Hopf braces. The first one uses matched pairs of Hopf algebras while the second one relies on category-theoretic tools.

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30

Anikonov, Yu E., M. V. Neshchadim, and A. P. Chupakhin. "Multidimensional Hopf equation and some exact solutions." Sibirskii zhurnal industrial'noi matematiki 25, no. 1 (2022): 5–13. http://dx.doi.org/10.33048/sibjim.2022.25.101.

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31

Sgibnev, M. S. "On the Inhomogeneous Conservative Wiener–Hopf Equation." Siberian Mathematical Journal 58, no. 6 (2017): 1090–103. http://dx.doi.org/10.1134/s0037446617060180.

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32

GUO, QIAN, and CHANGPIN LI. "HOPF BIFURCATION OF A DELAYED DIFFERENTIAL EQUATION." International Journal of Bifurcation and Chaos 17, no. 04 (2007): 1367–74. http://dx.doi.org/10.1142/s0218127407017860.

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In this paper, we study Hopf bifurcation of a second-order nonlinear differential equation with time delay by using the Lyapunov–Schmidt reduction. The approximate analytical expressions of the periodic solutions bifurcated from the trivial solution are given. We also discuss the stability of these periodic solutions. The numerical simulations line up with the theoretical results.

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33

Lorenz, Martin. "On the class equation for Hopf algebras." Proceedings of the American Mathematical Society 126, no. 10 (1998): 2841–44. http://dx.doi.org/10.1090/s0002-9939-98-04392-5.

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34

Dybin, V. B. "THE WIENER-HOPF EQUATION AND BLASCHKE PRODUCTS." Mathematics of the USSR-Sbornik 70, no. 1 (1991): 205–30. http://dx.doi.org/10.1070/sm1991v070n01abeh001938.

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35

Afanasyev, Nikolaevich, and Vitalievich Titov. "Modified Wiener-Hopf equation in identification problems." Journal of Applied Engineering Science 16, no. 4 (2018): 592–98. http://dx.doi.org/10.5937/jaes16-14637.

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36

Buchstaber, V. M., and E. V. Koritskaya. "Quasilinear Burgers-Hopf equation and Stasheff polytopes." Functional Analysis and Its Applications 41, no. 3 (2007): 196–207. http://dx.doi.org/10.1007/s10688-007-0017-8.

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37

Valverde, J. C. "Simplest Normal Forms of Hopf–Neimark–Sacker Bifurcations." International Journal of Bifurcation and Chaos 13, no. 07 (2003): 1831–39. http://dx.doi.org/10.1142/s0218127403007667.

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According to [Yu, 1999], at most two terms remain in the amplitude equation of the normal form of a continuous system, expressed in polar coordinates, with a Hopf or Generalized Hopf singularity, if we (only) apply specific nonlinear transformation to the conventional normal form; but, at least one remains in the phase equation. In this paper we show that, using a particular nonlinear scaling, these terms in the phase equation can be eliminated, which simplifies the (diffeomorphic) normal form given by [Yu, 1999]. Besides, we have also treated the Neimark–Sacker and Generalized Neimark–Sacker

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38

Feng, Cheng, Yun-dong Li, and Gui-yu Ou. "Hopf Bifurcation of a Standing Cantilever Pipe Conveying Fluid with Time Delay." Mathematical Problems in Engineering 2022 (December 24, 2022): 1–15. http://dx.doi.org/10.1155/2022/3588068.

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In this article, the dynamical behavior of a standing cantilever pipe conveying fluid is investigated with time − delay. By applying piezoelectric materials and considering the time delay of voltage, the motion equation is built with motion-limiting constraints and elastic support. The motion equation is discretized into ordinary differential equations by the Galerkin method. A stability analysis of the equilibrium point with three parameters is obtained. The system will lose stability at the equilibrium point and generate Hopf branches. The central manifold theorem and the canonical type theo

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39

Zenyuk, Dmitry Alexeevich, and Georgii Gennadyevich Malinetskii. "Amplitude equation formalism for reaction—subdiffusion systems." Keldysh Institute Preprints, no. 93 (2021): 1–15. http://dx.doi.org/10.20948/prepr-2021-93.

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The paper presents derivation of the amplitude equation for the Hopf bifurcation in the two-component system with nonlinear chemical kinetics and subdiffusion. Anomalous diffusion transport is described via Caputo fractional derivatives. The obtained amplitude equation is much more complex compared to the case of normal diffusion because solutions of fractional order linear differential equations have inconvenient behavior.

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40

ARINO, OVIDE, and EVA SÁNCHEZ. "AN INTEGRAL EQUATION OF CELL POPULATION DYNAMICS FORMULATED AS AN ABSTRACT DELAY EQUATION — SOME CONSEQUENCES." Mathematical Models and Methods in Applied Sciences 08, no. 04 (1998): 713–35. http://dx.doi.org/10.1142/s0218202598000329.

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From a nonlinear integral equation modelling cell proliferation, two other formulations are first proposed. One of them leads to a delay differential equation with values in a Banach space to which the theory of abstract delay differential equations is applied. Specific features of the equation are shown to entail interesting formulae for the dual product and the fundamental solution. Finally, preliminary steps are made in the study of what seems to be a Hopf bifurcation.

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41

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 (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 sequen

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42

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

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

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43

Zhao, Huitao, Yiping Lin, and Yunxian Dai. "Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays." Abstract and Applied Analysis 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/321930.

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A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obt

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44

Babeshko, V. A., O. V. Evdokimova, O. M. Babeshko, M. V. Zaretskaya, and V. S. Evdokimov. "THE EXACT SOLUTION OF THE WIENER–HOPF EQUATION ON THE SEGMENT FOR CONTACT PROBLEMS AND PROBLEMS OF THE THEORY OF CRACKS IN A LAYERED MEDIUM." Доклады Российской академии наук. Физика, технические науки 509, no. 1 (2023): 39–44. http://dx.doi.org/10.31857/s2686740023020025.

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This paper presents an approach that allows for the first time to construct an exact solution of the Wiener–Hopf integral equations on a finite segment for the case of meromorphic functions in Fourier transforms of the kernel. The Wiener–Hopf integral equation is traditionally considered set on a semi-infinite segment. However, in applications, there are often cases of their application specified on a finite segment. For these purposes, approximate methods of applying these integral equations have been developed. However, when considering the Wiener–Hopf integral equations generated by mixed p

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45

Vitanov, Nikolay K. "On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations." Axioms 12, no. 12 (2023): 1106. http://dx.doi.org/10.3390/axioms12121106.

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Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the

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46

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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47

NIU, BEN, and JUNJIE WEI. "BIFURCATION ANALYSIS OF A NFDE ARISING FROM MULTIPLE-DELAY FEEDBACK CONTROL." International Journal of Bifurcation and Chaos 21, no. 03 (2011): 759–74. http://dx.doi.org/10.1142/s0218127411028775.

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We study the stability and Hopf bifurcation of a neutral functional differential equation (NFDE) which is transformed from an amplitude equation with multiple-delay feedback control. By analyzing the distribution of the eigenvalues, the stability and existence of Hopf bifurcation are obtained. Furthermore, the direction and stability of the Hopf bifurcation are determined by using the center manifold and normal form theories for NFDEs. Finally, we carry out some numerical simulations to illustrate the results.

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48

ALGABA, ANTONIO, FERNANDO FERNÁNDEZ-SÁNCHEZ, MANUEL MERINO та ALEJANDRO J. RODRÍGUEZ-LUIS. "ANALYSIS OF THE T-POINT–HOPF BIFURCATION WITH ℤ2-SYMMETRY: APPLICATION TO CHUA'S EQUATION". International Journal of Bifurcation and Chaos 20, № 04 (2010): 979–93. http://dx.doi.org/10.1142/s0218127410026265.

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The aim of this work is twofold — on the one hand, to perform a theoretical analysis of the global behavior organized by a T-point–Hopf in ℤ2-symmetric systems; on the other hand, to apply the obtained results for a numerical study of Chua's equation, where for the first time this bifurcation is considered.In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-two heteroclinic cycle occurs. A more degenerate scenario appears when one of the equilibria involved in such a cycle undergoes a

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Zhou, Xiaobing, Murong Jiang, and Xiaomei Cai. "Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays." Discrete Dynamics in Nature and Society 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/569141.

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Abstract:

We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis.

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50

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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Abstract:

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

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