April  2015, 35(4): 1561-1588. doi: 10.3934/dcds.2015.35.1561

Spiraling bifurcation diagrams in superlinear indefinite problems

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040-Madrid

2. 

Departamento de Matemáticas, Universidad Carlos III de Madrid Campus de Leganés, Avda. Universidad 30, 28911 Leganés, Madrid

3. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid

Received  August 2013 Revised  April 2014 Published  November 2014

This paper computes and discusses a series of intricate global bifurcation diagrams for a class of one-dimensional superlinear indefinite boundary value problems arising in population dynamics, under non-homogeneous boundary conditions, measured by $M>0$; the main bifurcation parameter being the amplitude $b$ of the superlinear terms.
Citation: Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Spiraling bifurcation diagrams in superlinear indefinite problems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1561-1588. doi: 10.3934/dcds.2015.35.1561
References:
[1]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign,, J. Funct. Anal., 141 (1996), 159.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[2]

E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods,, SIAM Classics in Applied Mathematics 45, (2003).  doi: 10.1137/1.9780898719154.  Google Scholar

[3]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Diff. Eqns., 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[4]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems,, Top. Meth. Nonl. Anal., 4 (1994), 59.   Google Scholar

[5]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems,, Nonl. Diff. Eqns. Appns., 2 (1995), 553.  doi: 10.1007/BF01210623.  Google Scholar

[6]

M. D. Bertness and R. M. Callaway, Positive interactions in communities,, Trends in Ecology and Evolution, 9 (1994), 191.  doi: 10.1016/0169-5347(94)90088-4.  Google Scholar

[7]

F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part I: Branches of Nonsingular Solutions,, Numer. Math., 36 (1980), 1.  doi: 10.1007/BF01395985.  Google Scholar

[8]

F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points,, Numer. Math., 37 (1981), 1.  doi: 10.1007/BF01396184.  Google Scholar

[9]

F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points,, Numer. Math., 38 (1981), 1.  doi: 10.1007/BF01395805.  Google Scholar

[10]

R. M. Callaway and L. R. Walker, Competition and facilitation: A synthetic approach to interactions in plant communities,, Ecology, 78 (1997), 1958.  doi: 10.2307/2265936.  Google Scholar

[11]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods. Fundamentals in Single Domains,, Springer-Verlag, (2006).   Google Scholar

[12]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion,, Diff. Int. Eqns., 7 (1994), 411.   Google Scholar

[13]

J. H. Connell, On the prevalence and relative importance of interspecific competition: evidence from field experiments,, Amer. Natur., 122 (1983), 661.  doi: 10.1086/284165.  Google Scholar

[14]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[15]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation from simple eigenvalues and linearized stability,, Arch. Rat. Mech. Anal., 52 (1973), 161.   Google Scholar

[16]

M. Crouzeix and J. Rappaz, On Numerical Approximation in Bifurcation Theory,, Recherches en Mathématiques Appliquées 13. Masson, (1990).   Google Scholar

[17]

J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies,, SIAM J. of Sci. Stat. Comput., 7 (1986), 599.  doi: 10.1137/0907040.  Google Scholar

[18]

J. García-Melián, Multiplicity of positive solutions to boundary blow up elliptic problems with sign-changing weights,, J. Funct. Anal., 261 (2011), 1775.  doi: 10.1016/j.jfa.2011.05.018.  Google Scholar

[19]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).   Google Scholar

[20]

M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory,, Comm. Pure Appl. Math., 32 (1979), 21.  doi: 10.1002/cpa.3160320103.  Google Scholar

[21]

M. Golubitsky and D. G. Shaeffer, Singularity and Groups in Bifurcation Theory,, Springer, (1985).   Google Scholar

[22]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations,, J. Diff. Eqns., 167 (2000), 36.  doi: 10.1006/jdeq.2000.3772.  Google Scholar

[23]

R. Gómez-Reñasco and J. López-Gómez, The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations,, Diff. Int. Eqns., 14 (2001), 751.   Google Scholar

[24]

G. E. Hutchinson, The Ecological Theater and the Evolutionary Play,, Yale University Press, (1965).   Google Scholar

[25]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).   Google Scholar

[26]

H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems,, Tata Insitute of Fundamental Research, (1987).   Google Scholar

[27]

J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos,, Cuadernos de Matemática y Mecánica, (1988).   Google Scholar

[28]

J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems,, Comm. Part. Diff. Eqns., 22 (1997), 1787.  doi: 10.1080/03605309708821320.  Google Scholar

[29]

J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems,, Trans. Amer. Math. Soc., 352 (2000), 1825.  doi: 10.1090/S0002-9947-99-02352-1.  Google Scholar

[30]

J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems,, Sci. Math. Jpn., 61 (2005), 493.   Google Scholar

[31]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra. Stationary partial differential equations,, in Handbook of Differential Equations: Stationary partial differential equations. Vol. II (eds. M. Chipot and P. Quittner), II (2005), 211.  doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[32]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions,, J. Diff. Eqns., 224 (2006), 385.  doi: 10.1016/j.jde.2005.08.008.  Google Scholar

[33]

J. López-Gómez, J. C. Eilbeck, K. Duncan and M. Molina-Meyer, Structure of solution manifolds in a strongly coupled elliptic system, IMA Conference on Dynamics of Numerics and Numerics of Dynamics (Bristol, 1990),, IMA J. Numer. Anal., 12 (1992), 405.  doi: 10.1093/imanum/12.3.405.  Google Scholar

[34]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications,, Diff. Int. Eqns., 7 (1994), 383.   Google Scholar

[35]

J. López-Gómez and M. Molina-Meyer, Superlinear indefinite systems: Beyond Lotka-Volterra models,, J. Diff. Eqns., 221 (2006), 343.  doi: 10.1016/j.jde.2005.05.009.  Google Scholar

[36]

J. López-Gómez and M. Molina-Meyer, The competitive exclusion principle versus biodiversity through segregation and further adaptation to spatial heterogeneities,, Theoretical Population Biology, 69 (2006), 94.   Google Scholar

[37]

J. López-Gómez and M. Molina-Meyer, Biodiversity through co-opetition,, Discrete and Continuous Dynamical Systems B, 8 (2007), 187.  doi: 10.3934/dcdsb.2007.8.187.  Google Scholar

[38]

J. López-Gómez and M. Molina-Meyer, Modeling coopetition,, Mathematics and Computers in Simulation, 76 (2007), 132.  doi: 10.1016/j.matcom.2007.01.035.  Google Scholar

[39]

J. López-Gómez, M. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions,, J. Diff. Eqns., 255 (2013), 503.  doi: 10.1016/j.jde.2013.04.019.  Google Scholar

[40]

J. López-Gómez, M. Molina-Meyer and M. Villareal, Numerical computation of coexistence states,, SIAM J. Numer. Anal., 29 (1992), 1074.  doi: 10.1137/0729065.  Google Scholar

[41]

J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems,, Comm. Pure Appl. Anal., 13 (2014), 1.  doi: 10.3934/cpaa.2014.13.1.  Google Scholar

[42]

J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight,, J. Diff. Eqns., 188 (2003), 33.  doi: 10.1016/S0022-0396(02)00073-6.  Google Scholar

[43]

L. Ping, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573.  doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[44]

F. I. Pugnaire (Editor), Positive Plant Interactions and Community Dynamics,, Fundación BBVA, (2010).   Google Scholar

[45]

M. B. Saffo, Invertebrates in endosymbiotic associations,, Amer. Zool., 32 (1992), 557.  doi: 10.1093/icb/32.4.557.  Google Scholar

[46]

T. W. Shoener, Field experiments on interspecific competition,, Amer. Natur., 122 (1983), 240.  doi: 10.1086/284133.  Google Scholar

[47]

J. L. Wulff, Clonal organisms and the evolution of mutualism. In Jackson, J.B.C., Buss, L.W., Cook, R.E. (Eds.),, Population Biology and Evolution of Clonal Organisms, (1985), 437.   Google Scholar

show all references

References:
[1]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign,, J. Funct. Anal., 141 (1996), 159.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[2]

E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods,, SIAM Classics in Applied Mathematics 45, (2003).  doi: 10.1137/1.9780898719154.  Google Scholar

[3]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Diff. Eqns., 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[4]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems,, Top. Meth. Nonl. Anal., 4 (1994), 59.   Google Scholar

[5]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems,, Nonl. Diff. Eqns. Appns., 2 (1995), 553.  doi: 10.1007/BF01210623.  Google Scholar

[6]

M. D. Bertness and R. M. Callaway, Positive interactions in communities,, Trends in Ecology and Evolution, 9 (1994), 191.  doi: 10.1016/0169-5347(94)90088-4.  Google Scholar

[7]

F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part I: Branches of Nonsingular Solutions,, Numer. Math., 36 (1980), 1.  doi: 10.1007/BF01395985.  Google Scholar

[8]

F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part II: Limit Points,, Numer. Math., 37 (1981), 1.  doi: 10.1007/BF01396184.  Google Scholar

[9]

F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points,, Numer. Math., 38 (1981), 1.  doi: 10.1007/BF01395805.  Google Scholar

[10]

R. M. Callaway and L. R. Walker, Competition and facilitation: A synthetic approach to interactions in plant communities,, Ecology, 78 (1997), 1958.  doi: 10.2307/2265936.  Google Scholar

[11]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods. Fundamentals in Single Domains,, Springer-Verlag, (2006).   Google Scholar

[12]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion,, Diff. Int. Eqns., 7 (1994), 411.   Google Scholar

[13]

J. H. Connell, On the prevalence and relative importance of interspecific competition: evidence from field experiments,, Amer. Natur., 122 (1983), 661.  doi: 10.1086/284165.  Google Scholar

[14]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[15]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation from simple eigenvalues and linearized stability,, Arch. Rat. Mech. Anal., 52 (1973), 161.   Google Scholar

[16]

M. Crouzeix and J. Rappaz, On Numerical Approximation in Bifurcation Theory,, Recherches en Mathématiques Appliquées 13. Masson, (1990).   Google Scholar

[17]

J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies,, SIAM J. of Sci. Stat. Comput., 7 (1986), 599.  doi: 10.1137/0907040.  Google Scholar

[18]

J. García-Melián, Multiplicity of positive solutions to boundary blow up elliptic problems with sign-changing weights,, J. Funct. Anal., 261 (2011), 1775.  doi: 10.1016/j.jfa.2011.05.018.  Google Scholar

[19]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).   Google Scholar

[20]

M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory,, Comm. Pure Appl. Math., 32 (1979), 21.  doi: 10.1002/cpa.3160320103.  Google Scholar

[21]

M. Golubitsky and D. G. Shaeffer, Singularity and Groups in Bifurcation Theory,, Springer, (1985).   Google Scholar

[22]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations,, J. Diff. Eqns., 167 (2000), 36.  doi: 10.1006/jdeq.2000.3772.  Google Scholar

[23]

R. Gómez-Reñasco and J. López-Gómez, The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations,, Diff. Int. Eqns., 14 (2001), 751.   Google Scholar

[24]

G. E. Hutchinson, The Ecological Theater and the Evolutionary Play,, Yale University Press, (1965).   Google Scholar

[25]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).   Google Scholar

[26]

H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems,, Tata Insitute of Fundamental Research, (1987).   Google Scholar

[27]

J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos,, Cuadernos de Matemática y Mecánica, (1988).   Google Scholar

[28]

J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems,, Comm. Part. Diff. Eqns., 22 (1997), 1787.  doi: 10.1080/03605309708821320.  Google Scholar

[29]

J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems,, Trans. Amer. Math. Soc., 352 (2000), 1825.  doi: 10.1090/S0002-9947-99-02352-1.  Google Scholar

[30]

J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems,, Sci. Math. Jpn., 61 (2005), 493.   Google Scholar

[31]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra. Stationary partial differential equations,, in Handbook of Differential Equations: Stationary partial differential equations. Vol. II (eds. M. Chipot and P. Quittner), II (2005), 211.  doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[32]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions,, J. Diff. Eqns., 224 (2006), 385.  doi: 10.1016/j.jde.2005.08.008.  Google Scholar

[33]

J. López-Gómez, J. C. Eilbeck, K. Duncan and M. Molina-Meyer, Structure of solution manifolds in a strongly coupled elliptic system, IMA Conference on Dynamics of Numerics and Numerics of Dynamics (Bristol, 1990),, IMA J. Numer. Anal., 12 (1992), 405.  doi: 10.1093/imanum/12.3.405.  Google Scholar

[34]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications,, Diff. Int. Eqns., 7 (1994), 383.   Google Scholar

[35]

J. López-Gómez and M. Molina-Meyer, Superlinear indefinite systems: Beyond Lotka-Volterra models,, J. Diff. Eqns., 221 (2006), 343.  doi: 10.1016/j.jde.2005.05.009.  Google Scholar

[36]

J. López-Gómez and M. Molina-Meyer, The competitive exclusion principle versus biodiversity through segregation and further adaptation to spatial heterogeneities,, Theoretical Population Biology, 69 (2006), 94.   Google Scholar

[37]

J. López-Gómez and M. Molina-Meyer, Biodiversity through co-opetition,, Discrete and Continuous Dynamical Systems B, 8 (2007), 187.  doi: 10.3934/dcdsb.2007.8.187.  Google Scholar

[38]

J. López-Gómez and M. Molina-Meyer, Modeling coopetition,, Mathematics and Computers in Simulation, 76 (2007), 132.  doi: 10.1016/j.matcom.2007.01.035.  Google Scholar

[39]

J. López-Gómez, M. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions,, J. Diff. Eqns., 255 (2013), 503.  doi: 10.1016/j.jde.2013.04.019.  Google Scholar

[40]

J. López-Gómez, M. Molina-Meyer and M. Villareal, Numerical computation of coexistence states,, SIAM J. Numer. Anal., 29 (1992), 1074.  doi: 10.1137/0729065.  Google Scholar

[41]

J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems,, Comm. Pure Appl. Anal., 13 (2014), 1.  doi: 10.3934/cpaa.2014.13.1.  Google Scholar

[42]

J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight,, J. Diff. Eqns., 188 (2003), 33.  doi: 10.1016/S0022-0396(02)00073-6.  Google Scholar

[43]

L. Ping, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573.  doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[44]

F. I. Pugnaire (Editor), Positive Plant Interactions and Community Dynamics,, Fundación BBVA, (2010).   Google Scholar

[45]

M. B. Saffo, Invertebrates in endosymbiotic associations,, Amer. Zool., 32 (1992), 557.  doi: 10.1093/icb/32.4.557.  Google Scholar

[46]

T. W. Shoener, Field experiments on interspecific competition,, Amer. Natur., 122 (1983), 240.  doi: 10.1086/284133.  Google Scholar

[47]

J. L. Wulff, Clonal organisms and the evolution of mutualism. In Jackson, J.B.C., Buss, L.W., Cook, R.E. (Eds.),, Population Biology and Evolution of Clonal Organisms, (1985), 437.   Google Scholar

[1]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515

[2]

Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1

[3]

Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112

[4]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561

[5]

Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047

[6]

Andrea Tellini. Imperfect bifurcations via topological methods in superlinear indefinite problems. Conference Publications, 2015, 2015 (special) : 1050-1059. doi: 10.3934/proc.2015.1050

[7]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505

[8]

Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861

[9]

Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21

[10]

Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103

[11]

E. Kapsza, Gy. Károlyi, S. Kovács, G. Domokos. Regular and random patterns in complex bifurcation diagrams. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 519-540. doi: 10.3934/dcdsb.2003.3.519

[12]

Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297

[13]

Behrouz Kheirfam. A weighted-path-following method for symmetric cone linear complementarity problems. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 141-150. doi: 10.3934/naco.2014.4.141

[14]

Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219

[15]

Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61

[16]

Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102

[17]

Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure & Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002

[18]

David Rojas, Pedro J. Torres. Bifurcation of relative equilibria generated by a circular vortex path in a circular domain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 749-760. doi: 10.3934/dcdsb.2019265

[19]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71

[20]

Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (4)

[Back to Top]