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Journal articles on the topic 'Bifurcation from infinity'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Krasnosel'skii, Alexander M., and Dmitrii I. Rachinskii. "Subharmonic bifurcation from infinity." Journal of Differential Equations 226, no. 1 (July 2006): 30–53. http://dx.doi.org/10.1016/j.jde.2005.09.011.

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2

S. Kozyakin, Victor, Alexander M. Krasnosel’skii, and Dmitrii I. Rachinskii. "Arnold tongues for bifurcation from infinity." Discrete & Continuous Dynamical Systems - S 1, no. 1 (2008): 107–16. http://dx.doi.org/10.3934/dcdss.2008.1.107.

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3

Korman, Philip. "An oscillatory bifurcation from infinity, and from zero." Nonlinear Differential Equations and Applications NoDEA 15, no. 3 (October 2008): 335–46. http://dx.doi.org/10.1007/s00030-008-7024-1.

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4

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

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

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5

WANG, HUAILEI, HAIYAN HU, and ZAIHUA WANG. "GLOBAL DYNAMICS OF A DUFFING OSCILLATOR WITH DELAYED DISPLACEMENT FEEDBACK." International Journal of Bifurcation and Chaos 14, no. 08 (August 2004): 2753–75. http://dx.doi.org/10.1142/s0218127404010990.

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This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

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6

Gámez, José L., and Juan F. Ruiz-Hidalgo. "Bifurcation from Infinity and Resonance Results at High Eigenvalues in Dimension One." Journal of Function Spaces and Applications 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/284696.

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This paper is devoted to two different but related tags: firstly, the side of the bifurcation from infinity at every eigenvalue of the problem−u″(t)=λu(t)+g(t,u(t)),u∈H01(0,π), secondly, the solutions of the associated resonant problem at any eigenvalue. From the global shape of the nonlinearitygwe obtain computable integral values which will decide the behavior of the bifurcations and, consequently, the possibility of finding solutions of the resonant problems.

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7

Ward, James R. "A global continuation theorem and bifurcation from infinity for infinite-dimensional dynamical systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 4 (1996): 725–38. http://dx.doi.org/10.1017/s0308210500023039.

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A general continuation theorem for isolated sets in infinite-dimensional dynamical systems is proved for a class of semiflows. This result is then used to prove the existence of continua of full bounded solutions bifurcating from infinity for systems of reaction—diffusion equations.

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8

Xing, Siyuan, and Albert C. J. Luo. "On an origami structure of period-1 motions to homoclinic orbits in the Rössler system." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 12 (December 2022): 123121. http://dx.doi.org/10.1063/5.0131970.

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In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2 n motions [Formula: see text]. On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.

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9

Li, Chunqiu, Desheng Li, and Zhijun Zhang. "Dynamic Bifurcation from Infinity of Nonlinear Evolution Equations." SIAM Journal on Applied Dynamical Systems 16, no. 4 (January 2017): 1831–68. http://dx.doi.org/10.1137/16m1107358.

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10

Llibre, J., and E. Ponce. "Hopf bifurcation from infinity for planar control systems." Publicacions Matemàtiques 41 (January 1, 1997): 181–98. http://dx.doi.org/10.5565/publmat_41197_11.

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11

GENDELMAN, O. V. "DEGENERATE BIFURCATION SCENARIOS IN THE DYNAMICS OF COUPLED OSCILLATORS WITH SYMMETRIC NONLINEARITIES." International Journal of Bifurcation and Chaos 16, no. 01 (January 2006): 169–78. http://dx.doi.org/10.1142/s021812740601468x.

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We study the degenerate bifurcations of nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. The potentials of both the oscillator and the coupling spring are adopted to be even-power polynomials with non-negative coefficients. Coupling parameter ε is defined and the bifurcations of the nonlinear normal modes structure with change of this coupling parameter are revealed. The degeneracy in the dynamics is manifested by a "bifurcation from infinity" where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Other (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1 : 1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by the mechanism of successive cusp catastrophes with the growth of coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincaré sections). In the particular case of pure cubic nonlinearity of the oscillator and the coupling spring, an agreement between quantitative analytical predictions and numerical results is observed.

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12

Rousseau, C. "The Bifurcation Diagram of Cubic Polynomial Vector Fields on CP1." Canadian Mathematical Bulletin 60, no. 2 (June 1, 2017): 381–401. http://dx.doi.org/10.4153/cmb-2016-095-3.

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AbstractIn this paper we give the bifurcation diagram of the family of cubic vector fields z=, depending on the values of . The bifurcation diagram is in ℝ 4, but its conic structure allows describing it for parameter values in . There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

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13

Aliyev, Ziyatkhan S., Nazim A. Neymatov, and Humay Sh Rzayeva. "Unilateral Global Bifurcation from Infinity in Nonlinearizable One-Dimensional Dirac Problems." International Journal of Bifurcation and Chaos 31, no. 01 (January 2021): 2150005. http://dx.doi.org/10.1142/s021812742150005x.

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In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

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14

Gavrilov, Lubomir, and Iliya D. Iliev. "Bifurcations of Limit Cycles From Infinity in Quadratic Systems." Canadian Journal of Mathematics 54, no. 5 (October 1, 2002): 1038–64. http://dx.doi.org/10.4153/cjm-2002-038-6.

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AbstractWe investigate the bifurcation of limit cycles in one-parameter unfoldings of quadractic differential systems in the plane having a degenerate critical point at infinity. It is shown that there are three types of quadratic systems possessing an elliptic critical point which bifurcates from infinity together with eventual limit cycles around it. We establish that these limit cycles can be studied by performing a degenerate transformation which brings the system to a small perturbation of certain well-known reversible systems having a center. The corresponding displacement function is then expanded in a Puiseux series with respect to the small parameter and its coefficients are expressed in terms of Abelian integrals. Finally, we investigate in more detail four of the cases, among them the elliptic case (Bogdanov-Takens system) and the isochronous center S3. We show that in each of these cases the corresponding vector space of bifurcation functions has the Chebishev property: the number of the zeros of each function is less than the dimension of the vector space. To prove this we construct the bifurcation diagram of zeros of certain Abelian integrals in a complex domain.

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15

Degla, Guy. "On Bifurcation from Infinity and Multipoint Boundary Value Problems." Advances in Pure Mathematics 04, no. 04 (2014): 108–17. http://dx.doi.org/10.4236/apm.2014.44018.

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16

Aida, Chihiro, Chao-Nien Chen, Kousuke Kuto, and Hirokazu Ninomiya. "Bifurcation from infinity with applications to reaction-diffusion systems." Discrete & Continuous Dynamical Systems - A 40, no. 6 (2020): 3031–55. http://dx.doi.org/10.3934/dcds.2020053.

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17

Rybicki, klawomir. "On bifurcation from infinity for S1-equivariant potential operators." Nonlinear Analysis: Theory, Methods & Applications 31, no. 3-4 (February 1998): 343–61. http://dx.doi.org/10.1016/s0362-546x(96)00314-8.

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18

Kryszewski, Wojciech, and Andrzej Szulkin. "Bifurcation from infinity for an asymptotically linear Schrödinger equation." Journal of Fixed Point Theory and Applications 16, no. 1-2 (December 2014): 411–35. http://dx.doi.org/10.1007/s11784-015-0221-8.

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19

Umezu, Kenichiro. "Bifurcation from Infinity for Asymptotically Linear Elliptic Eigenvalue Problems." Journal of Mathematical Analysis and Applications 267, no. 2 (March 2002): 651–64. http://dx.doi.org/10.1006/jmaa.2001.7799.

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20

Coyle, J., P. W. Eloe, and J. Henderson. "Bifurcation from Infinity and Higher Order Ordinary Differential Equations." Journal of Mathematical Analysis and Applications 195, no. 1 (October 1995): 32–43. http://dx.doi.org/10.1006/jmaa.1995.1340.

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21

Mavinga, Nsoki, and Rosa Pardo. "Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 3 (March 20, 2017): 649–71. http://dx.doi.org/10.1017/s0308210516000251.

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We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

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22

Cano-Casanova, Santiago. "Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions." Advanced Nonlinear Studies 20, no. 1 (February 1, 2020): 31–51. http://dx.doi.org/10.1515/ans-2019-2051.

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AbstractThis article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.

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23

Avrutin, Viktor, Zhanybai T. Zhusubaliyev, Arindam Saha, Soumitro Banerjee, Irina Sushko, and Laura Gardini. "Dangerous Bifurcations Revisited." International Journal of Bifurcation and Chaos 26, no. 14 (December 30, 2016): 1630040. http://dx.doi.org/10.1142/s0218127416300408.

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A dangerous border collision bifurcation has been defined as the dynamical instability that occurs when the basins of attraction of stable fixed points shrink to a set of zero measure as the parameter approaches the bifurcation value from either side. This results in almost all trajectories diverging off to infinity at the bifurcation point, despite the eigenvalues of the fixed points before and after the bifurcation being within the unit circle. In this paper, we show that similar bifurcation phenomena also occur when the stable orbit in question is of a higher periodicity or is chaotic. Accordingly, we propose a generalized definition of dangerous bifurcation suitable for any kind of attracting sets. We report two types of dangerous border collision bifurcations and show that, in addition to the originally reported mechanism typically involving singleton saddle cycles, there exists one more situation where the basin boundary is formed by a repelling closed invariant curve.

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24

Makrooni, Roya, Laura Gardini, and Iryna Sushko. "Bifurcation Structures in a Family of 1D Discontinuous Linear-Hyperbolic Invertible Maps." International Journal of Bifurcation and Chaos 25, no. 13 (December 15, 2015): 1530039. http://dx.doi.org/10.1142/s0218127415300396.

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We consider a family of one-dimensional discontinuous invertible maps from an application in engineering. It is defined by a linear function and by a hyperbolic function with real exponent. The presence of vertical and horizontal asymptotes of the hyperbolic branch leads to particular codimension-two border collision bifurcation (BCB) such that if the parameter point approaches the bifurcation value from one side then the related cycle undergoes a regular BCB, while if the same bifurcation value is approached from the other side then a nonregular BCB occurs, involving periodic points at infinity, related to the asymptotes of the map. We investigate the bifurcation structure in the parameter space. Depending on the exponent of the hyperbolic branch, different period incrementing structures can be observed, where the boundaries of a periodicity region are related either to subcritical, or supercritical, or degenerate flip bifurcations of the related cycle, as well as to a regular or nonregular BCB. In particular, if the exponent is positive and smaller than one, then the period incrementing structure with bistability regions is observed and the corresponding flip bifurcations are subcritical, while if the exponent is larger than one, then the related flip bifurcations are supercritical and, thus, also the regions associated with cycles of double period are involved into the incrementing structure.

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25

Du, Chaoxiong, Yirong Liu, and Qi Zhang. "Limit Cycle Bifurcation of Infinity and Degenerate Singular Point in Three-Dimensional Vector Field." International Journal of Bifurcation and Chaos 26, no. 09 (August 2016): 1650152. http://dx.doi.org/10.1142/s0218127416501522.

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Our work focuses on investigating limit cycle bifurcation for infinity and a degenerate singular point of a fifth degree system in three-dimensional vector field. By using singular value method to compute focal values carefully, we give the expressions of the focal values (Lyapunov constants) at the origin and at infinity. Moreover, we obtain that four limit cycles at most can bifurcate from the origin and three limit cycles can bifurcate from infinity. At the same time, we show the structure of limit cycles from the origin and the infinity. It is interesting for this kind of nonlinear phenomenon that a string of large limit cycles encircle a string of small limit cycles by simultaneous Hopf bifurcation, which is hardly seen for similar published results in three-dimensional vector field, our result is new.

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26

Li, Chunqiu, and Jintao Wang. "Bifurcation from infinity of the Schrödinger equation via invariant manifolds." Nonlinear Analysis 213 (December 2021): 112490. http://dx.doi.org/10.1016/j.na.2021.112490.

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27

Kwon, Ohsang, and Youngae Lee. "Bifurcation from infinity for a quasilinear equation with general nonlinearity." Kodai Mathematical Journal 42, no. 3 (October 2019): 611–32. http://dx.doi.org/10.2996/kmj/1572487235.

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28

Byeon, Jaeyoung, and Youngae Lee. "Variational approach to bifurcation from infinity for nonlinear elliptic problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 143, no. 2 (March 18, 2013): 269–301. http://dx.doi.org/10.1017/s0308210511000801.

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For any N ⩾ 1 and sufficiently small ε > 0, we find a positive solution of a nonlinear elliptic equationwhen lim|x| → ∞V(x) = m > 0 and some optimal conditions on f are satisfied. Furthermore, we investigate the asymptotic behaviour of the solution as ε → 0.

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29

Esteban, Marina, Enrique Ponce, and Francisco Torres. "Periodic Orbit Bifurcations in Planar Hysteretic Systems without Equilibria." International Journal of Bifurcation and Chaos 30, no. 07 (June 15, 2020): 2030016. http://dx.doi.org/10.1142/s0218127420300165.

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This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from 3D systems with slow–fast dynamics. We focus our attention on the symmetric case without equilibria, determining the existence of periodic orbits as well as their stability, and possible bifurcations. New analytical characterizations of bifurcations in these hysteretic systems are obtained. In particular, bifurcations of periodic orbits from infinity, grazing and saddle-node bifurcations of periodic orbits are studied in detail and the corresponding bifurcation sets are provided. Finally, the study of the hysteretic systems is shown to be useful in detecting periodic orbits for some [Formula: see text]D piecewise linear systems.

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30

Shen, Wenguo. "Global Bifurcation from Intervals for the Monge-Ampère Equations and Its Applications." Journal of Function Spaces 2018 (2018): 1–7. http://dx.doi.org/10.1155/2018/9269458.

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We shall establish the global bifurcation results from the trivial solutions axis or from infinity for the Monge-Ampère equations: det(D2u)=λm(x)-uN+m(x)f1(x,-u,-u′,λ)+f2(x,-u,-u′,λ), in B, u(x)=0, on ∂B, where D2u=(∂2u/∂xi∂xj) is the Hessian matrix of u, where B is the unit open ball of RN, m∈C(B¯,[0,+∞)) is a radially symmetric weighted function and m(r):=m(x)≢0 on any subinterval of [0,1], λ is a positive parameter, and the nonlinear term f1,f2∈C(B¯×R+3,R+), but f1 is not necessarily differentiable at the origin and infinity with respect to u, where R+=[0,+∞). Some applications are given to the Monge-Ampère equations and we use global bifurcation techniques to prove our main results.

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31

Ma, Ruyun. "Bifurcation from infinity and multiple solutions for periodic boundary value problems." Nonlinear Analysis: Theory, Methods & Applications 42, no. 1 (September 2000): 27–39. http://dx.doi.org/10.1016/s0362-546x(98)00327-7.

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32

Cowley, S. J., and F. T. Smith. "On the stability of Poiseuille-Couette flow: a bifurcation from infinity." Journal of Fluid Mechanics 156, no. -1 (July 1985): 83. http://dx.doi.org/10.1017/s0022112085002002.

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33

Arrieta, José M., Rosa Pardo, and Anibal Rodríguez-Bernal. "Equilibria and global dynamics of a problem with bifurcation from infinity." Journal of Differential Equations 246, no. 5 (March 2009): 2055–80. http://dx.doi.org/10.1016/j.jde.2008.09.002.

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34

Przybycin, Jolanta. "Bifurcation from infinity for the special class of nonlinear differential equations." Journal of Differential Equations 65, no. 2 (November 1986): 235–39. http://dx.doi.org/10.1016/0022-0396(86)90035-5.

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35

Nagata, M. "Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity." Journal of Fluid Mechanics 217 (August 1990): 519–27. http://dx.doi.org/10.1017/s0022112090000829.

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Finite-amplitude solutions of plane Couette flow are discovered. They take a steady three-dimensional form. The solutions are obtained numerically by extending the bifurcation problem of a circular Couette system between co-rotating cylinders with a narrow gap to the case with zero average rotation rate.

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36

SHUKLA, PRIYANKA, and MEHEBOOB ALAM. "Weakly nonlinear theory of shear-banding instability in a granular plane Couette flow: analytical solution, comparison with numerics and bifurcation." Journal of Fluid Mechanics 666 (November 16, 2010): 204–53. http://dx.doi.org/10.1017/s0022112010004143.

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A weakly nonlinear theory, in terms of the well-known Landau equation, has been developed to describe the nonlinear saturation of the shear-banding instability in a rapid granular plane Couette flow using the amplitude expansion method. The nonlinear modes are found to follow certain symmetries of the base flow and the fundamental mode, which helped to identify analytical solutions for the base-flow distortion and the second harmonic, leading to an exact calculation of the first Landau coefficient. The present analytical solutions are used to validate a spectral-based numerical method for the nonlinear stability calculation. The regimes of supercritical and subcritical bifurcations for the shear-banding instability have been identified, leading to the prediction that the lower branch of the neutral stability contour in the (H, φ0)-plane, where H is the scaled Couette gap (the ratio between the Couette gap and the particle diameter) and φ0 is the mean density or the volume fraction of particles, is subcritically unstable. The predicted finite-amplitude solutions represent shear localization and density segregation along the gradient direction. Our analysis suggests that there is a sequence of transitions among three types of pitchfork bifurcations with increasing mean density: from (i) the bifurcation from infinity in the Boltzmann limit to (ii) subcritical bifurcation at moderate densities to (iii) supercritical bifurcation at larger densities to (iv) subcritical bifurcation in the dense limit and finally again to (v) supercritical bifurcation near the close packing density. It has been shown that the appearance of subcritical bifurcation in the dense limit depends on the choice of the contact radial distribution function and the constitutive relations. The scalings of the first Landau coefficient, the equilibrium amplitude and the phase diagram, in terms of mode number and inelasticity, have been demonstrated. The granular plane Couette flow serves as a paradigm that supports all three possible types of pitchfork bifurcations, with the mean density (φ0) being the single control parameter that dictates the nature of the bifurcation. The predicted bifurcation scenario for the shear-band formation is in qualitative agreement with particle dynamics simulations and the experiment in the rapid shear regime of the granular plane Couette flow.

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Genoud, François. "Bifurcation from infinity for an asymptotically linear problem on the half-line." Nonlinear Analysis: Theory, Methods & Applications 74, no. 13 (September 2011): 4533–43. http://dx.doi.org/10.1016/j.na.2011.04.019.

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38

Ma, Ruyun. "Bifurcation from infinity and multiple solutions for some discrete Sturm–Liouville problems." Computers & Mathematics with Applications 54, no. 4 (August 2007): 535–43. http://dx.doi.org/10.1016/j.camwa.2007.03.001.

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39

Gámez, José L., and Juan F. Ruiz-Hidalgo. "A detailed analysis on local bifurcation from infinity for nonlinear elliptic problems." Journal of Mathematical Analysis and Applications 338, no. 2 (February 2008): 1458–68. http://dx.doi.org/10.1016/j.jmaa.2007.06.019.

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40

Cosner, Chris. "Bifurcation from higher eigenvalues in nonlinear elliptic equations: Continua that meet infinity." Nonlinear Analysis: Theory, Methods & Applications 12, no. 3 (January 1988): 271–77. http://dx.doi.org/10.1016/0362-546x(88)90113-7.

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41

Umezu, Kenichiro. "MULTIPLICITY OF POSITIVE SOLUTIONS UNDER NONLINEAR BOUNDARY CONDITIONS FOR DIFFUSIVE LOGISTIC EQUATIONS." Proceedings of the Edinburgh Mathematical Society 47, no. 2 (June 2004): 495–512. http://dx.doi.org/10.1017/s0013091503000294.

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AbstractIn this paper we consider the existence and multiplicity of positive solutions of a nonlinear elliptic boundary-value problem with nonlinear boundary conditions which arises in population dynamics. While bifurcation problems from lines of trivial solutions are studied, the existence of bifurcation positive solutions from infinity is discussed. The former will be caught by the reduction to a bifurcation equation following the Lyapunov and Schmidt procedure. The latter will be based on a variational argument depending on the corresponding constrained minimization problem.AMS 2000 Mathematics subject classification: Primary 35J65; 35J20; 35P30; 35B32; 92D25

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42

Arrieta, José M., Rosa Pardo, and Anibal Rodríguez-Bernal. "Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 137, no. 2 (2007): 225–52. http://dx.doi.org/10.1017/s0308210505000363.

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We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle.

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43

Aliyev, Ziyatkhan, and Leyla Nasirova. "Bifurcation from zero or infinity in nonlinearizable Sturm–Liouville problems with indefinite weight." Electronic Journal of Qualitative Theory of Differential Equations, no. 55 (2021): 1–16. http://dx.doi.org/10.14232/ejqtde.2021.1.55.

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44

Drabek, P., P. Girg, P. Takac, and M. Ulm. "The Fredholm alternative for the p-Laplacian: Bifurcation from infinity, existence and multiplicity." Indiana University Mathematics Journal 53, no. 2 (2004): 433–82. http://dx.doi.org/10.1512/iumj.2004.53.2396.

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Rynne, Bryan P. "Bifurcation from infinity in nonlinear sturm liouville problems with different linearizations at ‘u = ±∞’." Applicable Analysis 67, no. 3-4 (December 1997): 233–44. http://dx.doi.org/10.1080/00036819708840608.

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46

Llibre, Jaume, and Enrique Ponce. "Bifurcation of a periodic orbit from infinity in planar piecewise linear vector fields." Nonlinear Analysis: Theory, Methods & Applications 36, no. 5 (June 1999): 623–53. http://dx.doi.org/10.1016/s0362-546x(98)00175-8.

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47

Chiappinelli, R., J. Mawhin, and R. Nugari. "Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities." Nonlinear Analysis: Theory, Methods & Applications 18, no. 12 (June 1992): 1099–112. http://dx.doi.org/10.1016/0362-546x(92)90155-8.

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48

Rynne, Bryan P. "Bifurcation from Zero or Infinity in Sturm–Liouville Problems Which Are Not Linearizable." Journal of Mathematical Analysis and Applications 228, no. 1 (December 1998): 141–56. http://dx.doi.org/10.1006/jmaa.1998.6122.

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Dai, Guowei, Ruyun Ma, and Yanqiong Lu. "Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition." Journal of Mathematical Analysis and Applications 397, no. 1 (January 2013): 119–23. http://dx.doi.org/10.1016/j.jmaa.2012.07.056.

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50

LÓPEZ-RUIZ, R., and J. L. LÓPEZ. "BIFURCATION CURVES OF LIMIT CYCLES IN SOME LIÉNARD SYSTEMS." International Journal of Bifurcation and Chaos 10, no. 05 (May 2000): 971–80. http://dx.doi.org/10.1142/s0218127400000694.

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Abstract:

Liénard systems of the form [Formula: see text], with f(x) an even continuous function, are considered. The bifurcation curves of limit cycles are calculated exactly in the weak (ε → 0) and in the strongly (ε → ∞) nonlinear regime in some examples. The number of limit cycles does not increase when ε increases from zero to infinity in all the cases analyzed.

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