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Journal articles on the topic 'Bifurcation problems'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

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 dimen

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2

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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3

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

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Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the even

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4

WU, ZHIQIANG, PEI YU, and KEQI WANG. "BIFURCATION ANALYSIS ON A SELF-EXCITED HYSTERETIC SYSTEM." International Journal of Bifurcation and Chaos 14, no. 08 (2004): 2825–42. http://dx.doi.org/10.1142/s0218127404010862.

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This paper investigates periodic bifurcation solutions of a mechanical system which involves a van der Pol type damping and a hysteretic damper representing restoring force. This system has recently been studied based on the singularity theory for bifurcations of smooth functions. However, the results do not actually take into account the property of nonsmoothness involved in the system. In particular, the transition varieties due to constraint boundaries were ignored, resulting in failure in finding some important bifurcation solutions. To reveal all possible bifurcation patterns for such sys

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5

Armbruster, D., and G. Dangelmayr. "Coupled stationary bifurcations in non-flux boundary value problems." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 1 (1987): 167–92. http://dx.doi.org/10.1017/s0305004100066500.

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AbstractCoupled stationary bifurcations in nonlinear operator equations for functions, which are defined on a real interval with non-flux boundary conditions at the ends, are analysed in the framework of imperfect bifurcation theory. The bifurcation equations resulting from a Lyapunov–Schmidt reduction possess a natural structure which can be obtained by taking real parts of a diagonal action in ℂ2 of the symmetry group 0(2). A complete unfolding theory is developed and bifurcation equations are classified up to codimension two. Structurally stable bifurcation diagrams are given and their depe

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6

Sridhar, Lakshmi N. "Elimination of oscillation causing Hopf bifurcations in engineering problems." Journal of AppliedMath 2, no. 5 (2024): 1826. http://dx.doi.org/10.59400/jam1826.

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Bifurcation analysis was performed on various engineering process problems that exhibit undesirable oscillation causing Hopf bifurcations. Hopf bifurcations result in oscillatory behavior which is problematic for optimization and control tasks. Additionally, the presence of oscillations causes a reduction in product quality and in some cases causes equipment damage. The hyperbolic tangent function activation factor is normally used in neural networks and optimal control problems to eliminate spikes in optimum profiles. Spikes are similar to oscillatory profiles and this is the motivation to in

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7

Cliffe, K. A., A. Spence, and S. J. Tavener. "The numerical analysis of bifurcation problems with application to fluid mechanics." Acta Numerica 9 (January 2000): 39–131. http://dx.doi.org/10.1017/s0962492900000398.

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In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the

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8

Aydin Akgun, Fatma. "Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution." International Journal of Differential Equations 2021 (November 26, 2021): 1–6. http://dx.doi.org/10.1155/2021/7516324.

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In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

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9

Neiman, Aleksander. "The cumulant approach for the investigation of bifurcations of dynamical systems driven by the external noise." Izvestiya VUZ. Applied Nonlinear Dynamics 3, no. 3 (1995): 8–21. https://doi.org/10.18500/0869-6632-1995-3-3-8-21.

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The problems of bifurcation analysis of noisy systems are considered. The technique of bifurcation analysis based on the cumulant expansion is proposed. The noise influence оn the mode-lockirg bifurcations in the circle map and оn the period-doubling bifurcations in the Feigenbaum map is considered as examples.

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10

Deng, Baoyang, Michael O'Connor, and Bill Goodwine. "Bifurcations and symmetry in two optimal formation control problems for mobile robotic systems." Robotica 35, no. 8 (2016): 1712–31. http://dx.doi.org/10.1017/s026357471600045x.

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SUMMARYThis paper studies bifurcations in the solution structure of an optimal control problem for mobile robotic formation control. In particular, this paper studies a group of mobile robots operating in a two-dimensional environment. Each robot has a predefined initial state and final state and we compute an optimal path between the two states for every robot. The path is optimized with respect to two factors, the control effort and the deviation from a desired “formation,” and a bifurcation parameter gives the relative weight given to each factor. Using an asymptotic analysis, we show that

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11

AMDJADI, FARIDON. "MULTIPLE HOPF BIFURCATION AND CHAOTIC REVERSING WAVES IN PROBLEMS WITH O(2) SYMMETRY." International Journal of Bifurcation and Chaos 14, no. 05 (2004): 1831–38. http://dx.doi.org/10.1142/s0218127404010199.

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Multiple Hopf bifurcation in problems with O(2) symmetry leads to the standing and the traveling wave solutions. Swapping the branches at this point is considered by studying an O(2) symmetric problem on [Formula: see text]. The torus-doubling cascade bifurcations are investigated using canonical coordinate transformation. It is shown that direction reversing chaos can be obtained through a symmetry-increasing bifurcation.

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12

Pla, Francisco, and Henar Herrero. "Reduced Basis Method Applied to Eigenvalue Problems from Convection." International Journal of Bifurcation and Chaos 29, no. 03 (2019): 1950028. http://dx.doi.org/10.1142/s0218127419500287.

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The reduced basis method is a suitable technique for finding numerical solutions to partial differential equations that must be obtained for many values of parameters. This method is suitable when researching bifurcations and instabilities of stationary solutions for partial differential equations. It is necessary to solve numerically the partial differential equations along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions for a large number of bifurcation parameter values. In this paper, the reduced basis method has been used to solve eigenva

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13

FURTER, JACQUES-ELIE, and ANGELA MARIA SITTA. "NONDEGENERATE UMBILICS, THE PATH FORMULATION AND GRADIENT BIFURCATION PROBLEMS." International Journal of Bifurcation and Chaos 19, no. 09 (2009): 2965–77. http://dx.doi.org/10.1142/s021812740902458x.

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Parametrized contact-equivalence is a successful theory for the understanding and classification of the qualitative local behavior of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view making explicit the singular behavior due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter. We show how to use path formulation to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the nondegenerate umbilics singularities are the generic cores in four

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14

Parker, Albert E., and Alexander G. Dimitrov. "Symmetry-Breaking Bifurcations of the Information Bottleneck and Related Problems." Entropy 24, no. 9 (2022): 1231. http://dx.doi.org/10.3390/e24091231.

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In this paper, we investigate the bifurcations of solutions to a class of degenerate constrained optimization problems. This study was motivated by the Information Bottleneck and Information Distortion problems, which have been used to successfully cluster data in many different applications. In the problems we discuss in this paper, the distortion function is not a linear function of the quantizer. This leads to a challenging annealing optimization problem, which we recast as a fixed-point dynamics problem of a gradient flow of a related dynamical system. The gradient system possesses an SN s

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15

HURD, R. A., and E. MEISTER. "GENERALIZED WAVEGUIDE BIFURCATION PROBLEMS." Quarterly Journal of Mechanics and Applied Mathematics 41, no. 1 (1988): 127–39. http://dx.doi.org/10.1093/qjmam/41.1.127.

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16

Lüneburg, E., and R. A. Hurd. "Two Waveguide Bifurcation Problems." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 65, no. 11 (1985): 551–59. http://dx.doi.org/10.1002/zamm.19850651108.

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17

Du, Chaoxiong, and Wentao Huang. "Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model." AIMS Mathematics 8, no. 11 (2023): 26715–30. http://dx.doi.org/10.3934/math.20231367.

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<abstract><p>The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on

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18

MENASRI, ABDELLAH. "DYNAMIC ANALYSIS OF A CHAOTIC 3D QUADRATIC SYSTEM USING PLANAR PROJECTION." Journal of Mathematical Analysis 13, no. 3 (2022): 14–26. http://dx.doi.org/10.54379/jma-2022-3-2.

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The theory of dynamical systems is one of the most important theorems of scientific research because it relies heavily on most of the major fields of applied mathematics to give a sufficiently broad view of reality, but it still poses some problems, especially with regard to the modeling of certain physical phenomena. Since most of these systems are designed as continuous or discrete dynamic systems with large dimensions and multiple bifurcation parameters, researchers face major problems in qualitative study. In this paper, we propose a method to study bifurcations of continuous three-dimensi

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19

CHAMPNEYS, ALAN R., and MARK D. GROVES. "A global investigation of solitary-wave solutions to a two-parameter model for water waves." Journal of Fluid Mechanics 342 (July 10, 1997): 199–229. http://dx.doi.org/10.1017/s0022112097005193.

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The model equationformula herearises as the equation for solitary-wave solutions to a fifth-order long-wave equation for gravity–capillary water waves. Being Hamiltonian, reversible and depending upon two parameters, it shares the structure of the full steady water-wave problem. Moreover, all known analytical results for local bifurcations of solitary-wave solutions to the full water-wave problem have precise counterparts for the model equation.At the time of writing two major open problems for steady water waves are attracting particular attention. The first concerns the possible existence of

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20

MANOEL, MIRIAM, and IAN STEWART. "DEGENERATE BIFURCATIONS WITH Z2⊕Z2-SYMMETRY." International Journal of Bifurcation and Chaos 09, no. 08 (1999): 1653–67. http://dx.doi.org/10.1142/s0218127499001140.

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Bifurcation problems with the symmetry group Z2⊕Z2 of the rectangle are common in applied science, for example, whenever a Euclidean invariant PDE is posed on a rectangular domain. In this work we derive normal forms for one-parameter bifurcations of steady states with symmetry of the group Z2⊕Z2. We study degeneracies of Z2⊕Z2-codimension 3 and modality 1. We also deduce persistent bifurcation diagrams when the system is subject to symmetry-preserving perturbations.

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21

Yumagulov, Marat G., and Natalia A. Vasenina. "Spectral Properties of the "Reaction-Diffusion" System Operators and Bifurcations Signs." Вестник Пермского университета. Математика. Механика. Информатика, no. 2(65) (2024): 17–25. http://dx.doi.org/10.17072/1993-0550-2024-2-17-25.

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The article discusses differential equations that arise when modeling reaction-diffusion systems. Questions about the stability of equilibrium points in critical cases, as well as about bifurcations in the vicinity of such points, are studied. The main attention is paid to the linearized problem operators spectral properties study. The spectrum discreteness was established, the root properties and invariant subspaces were studied, and formulas for eigenfunctions were proposed. As an application, questions about the multiple equilibrium bifurcation signs and Andronov–Hopf bifurcation in the vic

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22

Damon, J. "Topological equivalence of bifurcation problems." Nonlinearity 1, no. 2 (1988): 311–31. http://dx.doi.org/10.1088/0951-7715/1/2/002.

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23

Demirska, Ewa. "External approximation of bifurcation problems." ESAIM: Mathematical Modelling and Numerical Analysis 20, no. 1 (1986): 25–46. http://dx.doi.org/10.1051/m2an/1986200100251.

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24

Goltser, Ya M. "Some bifurcation problems of stability." Nonlinear Analysis: Theory, Methods & Applications 30, no. 3 (1997): 1461–67. http://dx.doi.org/10.1016/s0362-546x(97)00044-8.

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25

Kim, In-Sook. "A solvability of bifurcation problems." Nonlinear Analysis: Theory, Methods & Applications 63, no. 5-7 (2005): e1809-e1816. http://dx.doi.org/10.1016/j.na.2004.12.025.

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26

Kertész, V. "Bifurcation problems with high codimensions." Mathematical and Computer Modelling 31, no. 4-5 (2000): 99–108. http://dx.doi.org/10.1016/s0895-7177(00)00027-3.

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27

López-Gómez, J., and C. Mora-Corral. "Counting solutions in bifurcation problems." Journal of Mathematical Sciences 150, no. 5 (2008): 2395–407. http://dx.doi.org/10.1007/s10958-008-0138-5.

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28

Zou, Jiancheng. "FINITE DETERMINATION OF BIFURCATION PROBLEMS." Acta Mathematica Scientia 18, no. 4 (1998): 399–403. http://dx.doi.org/10.1016/s0252-9602(17)30594-5.

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29

Rheinboldt, W. C., Hans D. Mittelmann, and Dirk Roose. "Continuation Techniques and Bifurcation Problems." Mathematics of Computation 56, no. 193 (1991): 383. http://dx.doi.org/10.2307/2008554.

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30

Salahifard, H., and S. Mansour Vaezpour. "Bifurcation problems for noncompact operators." Miskolc Mathematical Notes 17, no. 1 (2016): 571. http://dx.doi.org/10.18514/mmn.2016.1290.

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31

Percell, Peter B., and Peter N. Brown. "Finite Determination of Bifurcation Problems." SIAM Journal on Mathematical Analysis 16, no. 1 (1985): 28–46. http://dx.doi.org/10.1137/0516003.

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32

Evstigneev, Nikolay M., and Nikolai A. Magnitskii. "Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition." Mathematics 11, no. 18 (2023): 3875. http://dx.doi.org/10.3390/math11183875.

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The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific

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KOWALCZYK, P., M. DI BERNARDO, A. R. CHAMPNEYS, et al. "TWO-PARAMETER DISCONTINUITY-INDUCED BIFURCATIONS OF LIMIT CYCLES: CLASSIFICATION AND OPEN PROBLEMS." International Journal of Bifurcation and Chaos 16, no. 03 (2006): 601–29. http://dx.doi.org/10.1142/s0218127406015015.

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This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimension-one events have recently been identified, causing for example, period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be full

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34

Shibata, Tetsutaro. "Asymptotic length of bifurcation curves related to inverse bifurcation problems." Journal of Mathematical Analysis and Applications 438, no. 2 (2016): 629–42. http://dx.doi.org/10.1016/j.jmaa.2016.02.014.

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Shibata, Tetsutaro. "Inverse bifurcation problems for nonlinear Sturm–Liouville problems." Inverse Problems 27, no. 5 (2011): 055003. http://dx.doi.org/10.1088/0266-5611/27/5/055003.

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36

Surovyatkina, E. "Prebifurcation noise amplification and noise-dependent hysteresis as indicators of bifurcations in nonlinear geophysical systems." Nonlinear Processes in Geophysics 12, no. 1 (2005): 25–29. http://dx.doi.org/10.5194/npg-12-25-2005.

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Abstract. The phenomena of prebifurcation noise amplification and noise-dependent hysteresis are studied as prospective indicators of bifurcations ("noisy precursor") in nonlinear Geophysical systems. The phenomenon of prebifurcation noise amplification arises due to decreasing of damping coefficients just before bifurcation. A simple method for the estimation of the forced fluctuation variance is suggested which is based on results of linear theory up to the boundary of its validity. The upper level for the fluctuation variance before the onset of the bifurcation is estimated from the conditi

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Gazor, Majid, та Mahsa Kazemi. "Normal Form Analysis of ℤ2-Equivariant Singularities". International Journal of Bifurcation and Chaos 29, № 02 (2019): 1950015. http://dx.doi.org/10.1142/s0218127419500159.

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Singular parametric systems usually experience bifurcations when their parameters slightly vary around certain critical values, that is, surprising changes occur in their dynamics. The bifurcation analysis is important due to their applications in real world problems. Here, we provide a brief review of the mathematical concepts in the extension of our developed Maple library, Singularity, for the study of [Formula: see text]-equivariant local bifurcations. We explain how the process of this analysis is involved with algebraic objects and tools from computational algebraic geometry. Our procedu

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Shibata, Tetsutaro. "Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems." International Journal of Differential Equations 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/138629.

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We consider the nonlinear eigenvalue problemu″(t)+λf(u(t))=0, u(t)>0, t∈I=:(-1,1), u(1)=u(-1)=0, wheref(u)is a cubic-like nonlinear term andλ>0is a parameter. It is known by Korman et al. (2005) that, under the suitable conditions onf(u), there exist exactly three bifurcation branchesλ=λj(ξ)(j=1,2,3), and these curves are parameterized by the maximum normξof the solutionuλcorresponding toλ. In this paper, we establish the precise global structures forλj(ξ)(j=1,2,3), which can be applied to the inverse bifurcation problems. The precise local structures forλj(ξ)(j=1,2,3) are also discussed

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Cox, Stephen M., Sidney Leibovich, Irene M. Moroz, and Amit Tandon. "Nonlinear dynamics in Langmuir circulations with O(2) symmetry." Journal of Fluid Mechanics 241 (August 1992): 669–704. http://dx.doi.org/10.1017/s0022112092002192.

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A direct comparison is made between the dynamics obtained by weakly nonlinear theory and full numerical simulations for Langmuir circulations in a density-stratified layer having finite depth and infinite horizontal extent. In one limit, the mathematical formulation employed is analogous to that of double-diffusion phenonema with the flux of one diffusing quantity fixed at the boundaries of the layer. These problems have multiple bifurcation points, but their amplitude equations have no intrinsic (nonlinear) degeneracies, in contrast to ‘standard’ double-diffusion problems. The symmetry of the

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40

Gazzola, Filippo. "Critical growth problems for polyharmonic operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 2 (1998): 251–63. http://dx.doi.org/10.1017/s0308210500012774.

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We prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower-order perturbations of the critical term. The results highlight the phenomenon of bifurcation of the critical dimensions discovered by Pucci and Serrin; moreover, we show that another bifurcation seems to appear for ‘nonresonant’ dimensions.

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Elgindi, M. B. M., and R. W. Langer. "On the numerical solution of perturbed bifurcation problems." International Journal of Mathematics and Mathematical Sciences 18, no. 3 (1995): 561–70. http://dx.doi.org/10.1155/s0161171295000718.

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Some numerical schemes, based upon Newton's and chord methods, for the computations of the perturbed bifurcation points as well as the solution curves through them, are presented. The initial guesses for Newton's and chord methods are obtained using the local analysis techniques and proved to fall into the neighborhoods of contraction for these methods. In applications the perturbation parameter represents a physical quantity and it is desirable to use it to parameterize the solution curves near the perturbed bifurcation point. In this regard, it is shown that, for certain classes of the p

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Elgindi, M. B. M. "On the application of Newton's and Chord methods to bifurcation problems." International Journal of Mathematics and Mathematical Sciences 17, no. 1 (1994): 147–54. http://dx.doi.org/10.1155/s0161171294000207.

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This paper is concerned with the applications of Newton's and chord methods in the computations of the bifurcation solutions in a neighborhood of a simple bifurcation point for prescribed values of the bifurcation parameter.

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Zhi-qiang, Wu, and Chen Yu-shu. "New bifurcation patterns in elementary bifurcation problems with single-side constraint." Applied Mathematics and Mechanics 22, no. 11 (2001): 1260–67. http://dx.doi.org/10.1007/bf02437849.

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44

Wangari, Isaac Mwangi, Stephen Davis, and Lewi Stone. "Backward bifurcation in epidemic models: Problems arising with aggregated bifurcation parameters." Applied Mathematical Modelling 40, no. 2 (2016): 1669–75. http://dx.doi.org/10.1016/j.apm.2015.07.022.

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45

Wang, Fosheng. "Bifurcation Problems for Generalized Beam Equations." Advances in Mathematical Physics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/635731.

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We investigate a class of bifurcation problems for generalized beam equations and prove that the one-parameter family of problems have exactly two bifurcation points via a unified, elementary approach. The proof of the main results relies heavily on calculus facts rather than such complicated arguments as Lyapunov-Schmidt reduction technique or Morse index theory from nonlinear functional analysis.

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Chen, Henry Y., Issam D. Moussa, Charles Davidson, and Ghassan S. Kassab. "Impact of main branch stenting on endothelial shear stress: role of side branch diameter, angle and lesion." Journal of The Royal Society Interface 9, no. 71 (2011): 1187–93. http://dx.doi.org/10.1098/rsif.2011.0675.

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In-stent restenosis and stent thrombosis remain clinically significant problems for bifurcation lesions. The objective of this study is to determine the haemodynamic effect of the side branch (SB) on main branch (MB) stenting. We hypothesize that the presence of a SB has a negative effect on MB wall shear stress (WSS), wall shear stress gradient (WSSG) and oscillatory shear index (OSI); and that the bifurcation diameter ratio (SB diameter/MB diameter) and angle are important contributors. We further hypothesized that stent undersizing exaggerates the negative effects on WSS, WSSG and OSI. To t

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Zhang, Yue, Kuanquan Wang, Yongfeng Yuan, Dong Sui, and Henggui Zhang. "Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley Model." Computational and Mathematical Methods in Medicine 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/761907.

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Hodgkin-Huxley (HH) equation is the first cell computing model in the world and pioneered the use of model to study electrophysiological problems. The model consists of four differential equations which are based on the experimental data of ion channels. Maximal conductance is an important characteristic of different channels. In this study, mathematical method is used to investigate the importance of maximal sodium conductanceg-Naand maximal potassium conductanceg-K. Applying stability theory, and takingg-Naandg-Kas variables, we analyze the stability and bifurcations of the model. Bifurcatio

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Field, M. J., and R. W. Richardson. "Symmetry breaking in equivariant bifurcation problems." Bulletin of the American Mathematical Society 22, no. 1 (1990): 79–85. http://dx.doi.org/10.1090/s0273-0979-1990-15846-x.

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Aliyev, Ziyatkhan, and Humay Rzayeva. "Global bifurcation for nonlinear Dirac problems." Electronic Journal of Qualitative Theory of Differential Equations, no. 46 (2016): 1–14. http://dx.doi.org/10.14232/ejqtde.2016.1.46.

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Mortici, Cristinel, and Silviu Sburlan. "A coincidence degree for bifurcation problems." Nonlinear Analysis: Theory, Methods & Applications 53, no. 5 (2003): 715–21. http://dx.doi.org/10.1016/s0362-546x(02)00308-5.

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