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Journal articles on the topic 'Bifurcation problems'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

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

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

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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 (January 1998): 117–39. http://dx.doi.org/10.1142/s0218127498000097.

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

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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 (September 28, 2013): 20120467. http://dx.doi.org/10.1098/rsta.2012.0467.

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

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4

Armbruster, D., and G. Dangelmayr. "Coupled stationary bifurcations in non-flux boundary value problems." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 1 (January 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 dependence on the wave numbers of the unstable modes is clarified.

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5

WU, ZHIQIANG, PEI YU, and KEQI WANG. "BIFURCATION ANALYSIS ON A SELF-EXCITED HYSTERETIC SYSTEM." International Journal of Bifurcation and Chaos 14, no. 08 (August 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 systems, a new method is developed in this paper. With this method, a continuous, piecewise smooth bifurcation problem can be transformed into several subbifurcation problems with either single-sided or double-sided constraints. Further, the constrained bifurcation problems are converted to unconstrained problems and then singularity theory is employed to find transition varieties. Explicit formulas are applied to reconsider the mechanical system. Numerical simulations are carried out to verify analytical predictions. Moreover, symbolic notation for a sequence of bifurcations is introduced to easily show the characteristics of bifurcations, and also the comparison of different bifurcations. The method developed in this paper can be easily extended to study bifurcation problems with other types of nonsmoothness.

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6

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 performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

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7

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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8

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 (July 14, 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 for small values of the bifurcation parameter (corresponding to heavily weighting the control effort) a single unique solution is expected, and that as the bifurcation parameter becomes large (corresponding to heavily weighting maintaining the formation) a large number of solutions is expected. Between the asymptotic extremes, a numerical investigation indicates a solution bifurcation structure with a cascade of increasing numbers of solutions, reminiscent, but not the same as, period-doubling bifurcations leading to chaos in dynamical systems. Furthermore, we show that if the system is symmetric, the bifurcation structure possesses symmetries, and also present a symmetry-breaking example of a non-holonomic system. Knowledge and understanding of the existence and structure of bifurcations in the solutions of this type of formation control problem are important for robotics engineers because common optimization approaches based on gradient-descent are only likely to converge to the single nearest solution, and a more global study provides a deeper and more comprehensive understanding of the nature of this important problem in robotics.

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9

AMDJADI, FARIDON. "MULTIPLE HOPF BIFURCATION AND CHAOTIC REVERSING WAVES IN PROBLEMS WITH O(2) SYMMETRY." International Journal of Bifurcation and Chaos 14, no. 05 (May 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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10

Pla, Francisco, and Henar Herrero. "Reduced Basis Method Applied to Eigenvalue Problems from Convection." International Journal of Bifurcation and Chaos 29, no. 03 (March 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 eigenvalue problems derived from the linear stability analysis of stationary solutions in a two-dimensional Rayleigh–Bénard convection problem. The bifurcation parameter is the Rayleigh number, which measures buoyancy. The reduced basis considered belongs to the eigenfunction spaces derived from the eigenvalue problems for different types of solutions in the bifurcation diagram depending on the Rayleigh number. The eigenvalue with the largest real part and its corresponding eigenfunction are easily calculated and the bifurcation points are correctly captured. The resulting matrices are small, which enables a drastic reduction in the computational cost of solving the eigenvalue problems.

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11

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 (September 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 situations: the general or gradient problems, with or without ℤ2 symmetry where ℤ2 acts on the second component of ℝ2 via κ(x,y) = (x,-y). The universal unfolding of the umbilic singularities have an interesting "Russian doll" type of structure of miniversal unfoldings in all those categories. With the path formulation approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance, some internal hierarchy of parameters). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new. We discuss some applications to the bifurcation of a loaded cylindrical panel. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.

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12

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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13

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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14

Parker, Albert E., and Alexander G. Dimitrov. "Symmetry-Breaking Bifurcations of the Information Bottleneck and Related Problems." Entropy 24, no. 9 (September 2, 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 symmetry due to its invariance in relabeling representative classes. Its flow hence passes through a series of bifurcations with specific symmetry breaks. Here, we show that the dynamical system related to the Information Bottleneck problem has an additional spurious symmetry that requires more-challenging analysis of the symmetry-breaking bifurcation. For the Information Bottleneck, we determine that when bifurcations occur, they are only of pitchfork type, and we give conditions that determine the stability of the bifurcating branches. We relate the existence of subcritical bifurcations to the existence of first-order phase transitions in the corresponding distortion function as a function of the annealing parameter, and provide criteria with which to detect such transitions.

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15

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

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16

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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17

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

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18

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

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19

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

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20

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

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21

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

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22

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

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23

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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24

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

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25

MENASRI, ABDELLAH. "DYNAMIC ANALYSIS OF A CHAOTIC 3D QUADRATIC SYSTEM USING PLANAR PROJECTION." Journal of Mathematical Analysis 13, no. 3 (June 30, 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-dimensional dynamic systems in general and chaotic systems in particular, which contains many bifurcation parameters. This method is mainly based on the projection on the plane and on the appropriate bifurcation parameter.

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26

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 solitary waves of elevation as local bifurcation phenomena in a particular parameter regime; the second, larger, issue is the determination of the global bifurcation picture for solitary waves. Given that the above equation is a good model for solitary waves of depression, it seems natural to study the above issues for this equation; they are comprehensively treated in this article.The equation is found to have branches of solitary waves of elevation bifurcating from the trivial solution in the appropriate parameter regime, one of which is described by an explicit solution. Numerical and analytical investigations reveal a rich global bifurcation picture including multi-modal solitary waves of elevation and depression together with interactions between the two types of wave. There are also new orbit-flip bifurcations and associated multi-crested solitary waves with non-oscillatory tails.

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27

Shibata, Tetsutaro. "Inverse bifurcation problems for nonlinear Sturm–Liouville problems." Inverse Problems 27, no. 5 (March 29, 2011): 055003. http://dx.doi.org/10.1088/0266-5611/27/5/055003.

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28

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

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29

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. Furthermore, we establish the asymptotic shape of the spike layer solutionu2(λ,t), which corresponds toλ=λ2(ξ), asλ→∞.

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30

KOWALCZYK, P., M. DI BERNARDO, A. R. CHAMPNEYS, S. J. HOGAN, M. HOMER, P. T. PIIROINEN, YU A. KUZNETSOV, and A. NORDMARK. "TWO-PARAMETER DISCONTINUITY-INDUCED BIFURCATIONS OF LIMIT CYCLES: CLASSIFICATION AND OPEN PROBLEMS." International Journal of Bifurcation and Chaos 16, no. 03 (March 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 fully described by an appropriate Poincaré map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

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31

MANOEL, MIRIAM, and IAN STEWART. "DEGENERATE BIFURCATIONS WITH Z2⊕Z2-SYMMETRY." International Journal of Bifurcation and Chaos 09, no. 08 (August 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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32

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 (November 2001): 1260–67. http://dx.doi.org/10.1007/bf02437849.

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33

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 (January 2016): 1669–75. http://dx.doi.org/10.1016/j.apm.2015.07.022.

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34

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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35

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

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36

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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37

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

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38

Schmitt, Klaus. "Bifurcation problems for second order systems." Nonlinear Analysis 201 (December 2020): 112042. http://dx.doi.org/10.1016/j.na.2020.112042.

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39

Guo-bin, Zhang, and Zhang Dun-mu. "Graded stable unfoldings in bifurcation problems." Wuhan University Journal of Natural Sciences 5, no. 3 (September 2000): 257–64. http://dx.doi.org/10.1007/bf02830131.

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40

Guckenheimer, John. "Multiple bifurcation problems for chemical reactors." Physica D: Nonlinear Phenomena 20, no. 1 (May 1986): 1–20. http://dx.doi.org/10.1016/0167-2789(86)90093-x.

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41

Brindley, J., C. Kaas-Petersen, and A. Spence. "Path-following methods in bifurcation problems." Physica D: Nonlinear Phenomena 34, no. 3 (March 1989): 456–61. http://dx.doi.org/10.1016/0167-2789(89)90269-8.

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42

Mittelmann, H. D., and H. Weber. "Multi-Grid Solution of Bifurcation Problems." SIAM Journal on Scientific and Statistical Computing 6, no. 1 (January 1985): 49–60. http://dx.doi.org/10.1137/0906005.

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43

Fujisaka, Hisato, and Chikara Sato. "An algebraic analysis for bifurcation problems." Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 74, no. 9 (1991): 22–32. http://dx.doi.org/10.1002/ecjc.4430740903.

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44

Poore, A. B., and C. A. Tiahrt. "Bifurcation problems in nonlinear parametric programming." Mathematical Programming 39, no. 2 (June 1987): 189–205. http://dx.doi.org/10.1007/bf02592952.

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45

UETA, Tetsushi, and Seiya AMOH. "To Tackle Bifurcation Problems with Python." IEICE ESS Fundamentals Review 16, no. 3 (January 1, 2023): 139–46. http://dx.doi.org/10.1587/essfr.16.3_139.

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46

Surovyatkina, E. "Prebifurcation noise amplification and noise-dependent hysteresis as indicators of bifurcations in nonlinear geophysical systems." Nonlinear Processes in Geophysics 12, no. 1 (January 13, 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 condition that the contribution of the non-linear term becomes comparable (in the sense of mean squares) with that of the linear term. The method has proved to be efficient for two simple bifurcation models (period doubling bifurcation and pitchfork bifurcation) and might be helpful in application to geophysics problems. The transition of a nonlinear system through the bifurcation point offers a new opportunity for estimating the internal noise using the magnitude of the noise-dependent hysteretic loop, which occurs when the control parameter is changed in the forward and backward direction.

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47

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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Gazor, Majid, and Mahsa Kazemi. "Normal Form Analysis of ℤ2-Equivariant Singularities." International Journal of Bifurcation and Chaos 29, no. 02 (February 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 procedures for computing normal forms, universal unfoldings, local transition varieties and persistent bifurcation diagram classifications are presented. Finally, we consider several Chua circuit type systems to demonstrate the applicability of our Maple library. We show how Singularity can be used for local equilibrium bifurcation analysis of such systems and their possible small perturbations. A brief user interface of [Formula: see text]-equivariant extension of Singularity is also presented.

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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 perturbed bifurcation problems, Newton's and chord methods can be used to follow the solution curves in a neighborhood of the perturbed bifurcation point while the perturbation parameter is kept fixed.

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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 physical problem implies invariance with respect to translations and reflections in the horizontal direction normal to the applied wind stress (so-called O(2) symmetry). A multiple bifurcation at a double-zero point serves as an organizing centre for dynamics over a wide range of parameter values. This double zero, or Takens–Bogdanov, bifurcation leads to doubly periodic motions manifested as modulated travelling waves. Other multiple bifurcation points appear as double-Hopf bifurcations. It is believed that this paper gives the first quantitative comparison of dynamics of double-diffusive type predicted by rationally derived amplitude equations and by full nonlinear partial differential equations. The implications for physically observable natural phenomena are discussed. This problem has been treated previously, but the earlier numerical treatment is in error, and is corrected here. When the Stokes drift gradient due to surface waves is not constant, the analogy with the common formulations of double-diffusion problems is compromised. Our bifurcation analyses are extended here to include the case of exponentially decaying Stokes drift gradient.

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