September  2011, 31(3): 763-777. doi: 10.3934/dcds.2011.31.763

On the birth of minimal sets for perturbed reversible vector fields

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

2. 

Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, 13083–859 Campinas, SP, Brazil, Brazil

Received  June 2010 Revised  May 2011 Published  August 2011

The results in this paper fit into a program to study the existence of periodic orbits, invariant cylinders and tori filled with periodic orbits in perturbed reversible systems. Here we focus on bifurcations of one-parameter families of periodic orbits for reversible vector fields in $\mathbb{R}^4$. The main used tools are normal forms theory, Lyapunov-Schmidt method and averaging theory.
Citation: Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763
References:
[1]

Acta Appl. Math., 70 (2002), 23-41. doi: 10.1023/A:1013909812387.  Google Scholar

[2]

Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996.  Google Scholar

[3]

in "Real and Complex Singularities" (eds. M. Manoel, M. C. Romero Fuster and C. T. C. Wall), Cambridge University Press, (2010), 380, 46-70. Google Scholar

[4]

Transactions of the American Mathematical Society, 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

[5]

International J. of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar

[6]

Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[7]

Dover Publications, 2009. Google Scholar

[8]

Annali di Matematica Pura ed Applicata, 187 (2008), 105-117. doi: 10.1007/s10231-006-0036-8.  Google Scholar

[9]

J. Dynam. Differential Equations, 8 (1996), 71-102.  Google Scholar

[10]

Bull. Braz. Math. Soc. (N.S.), 40 (2009), 511-537.  Google Scholar

[11]

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 20 (2010), 3341-3344. doi: 10.1142/S0218127410027738.  Google Scholar

[12]

Communications on Pure and Applied Analysis, 10 (2011), 1257-1266. Google Scholar

[13]

in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995) (ed. C. Simó), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, (1999), 489-493.  Google Scholar

[14]

2nd edition, Applied Mathematical Sciences, 59, Springer, New York, 2007.  Google Scholar

[15]

Chaos, 1 (1991), 160-167. doi: 10.1063/1.165858.  Google Scholar

[16]

Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.  Google Scholar

[17]

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 7 (1997), 569-584. doi: 10.1142/S0218127497000406.  Google Scholar

[18]

Res. Notes in Math., 75, Pitman (Advanced Publishing Program), Boston, MA, 1982.  Google Scholar

[19]

Universitext, Springer-Verlag, Berlin, 1990.  Google Scholar

[20]

Ph.D. thesis, University Ilmenau, 1999. Google Scholar

show all references

References:
[1]

Acta Appl. Math., 70 (2002), 23-41. doi: 10.1023/A:1013909812387.  Google Scholar

[2]

Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996.  Google Scholar

[3]

in "Real and Complex Singularities" (eds. M. Manoel, M. C. Romero Fuster and C. T. C. Wall), Cambridge University Press, (2010), 380, 46-70. Google Scholar

[4]

Transactions of the American Mathematical Society, 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

[5]

International J. of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar

[6]

Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[7]

Dover Publications, 2009. Google Scholar

[8]

Annali di Matematica Pura ed Applicata, 187 (2008), 105-117. doi: 10.1007/s10231-006-0036-8.  Google Scholar

[9]

J. Dynam. Differential Equations, 8 (1996), 71-102.  Google Scholar

[10]

Bull. Braz. Math. Soc. (N.S.), 40 (2009), 511-537.  Google Scholar

[11]

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 20 (2010), 3341-3344. doi: 10.1142/S0218127410027738.  Google Scholar

[12]

Communications on Pure and Applied Analysis, 10 (2011), 1257-1266. Google Scholar

[13]

in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995) (ed. C. Simó), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, (1999), 489-493.  Google Scholar

[14]

2nd edition, Applied Mathematical Sciences, 59, Springer, New York, 2007.  Google Scholar

[15]

Chaos, 1 (1991), 160-167. doi: 10.1063/1.165858.  Google Scholar

[16]

Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.  Google Scholar

[17]

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 7 (1997), 569-584. doi: 10.1142/S0218127497000406.  Google Scholar

[18]

Res. Notes in Math., 75, Pitman (Advanced Publishing Program), Boston, MA, 1982.  Google Scholar

[19]

Universitext, Springer-Verlag, Berlin, 1990.  Google Scholar

[20]

Ph.D. thesis, University Ilmenau, 1999. Google Scholar

[1]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[2]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[3]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

[4]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[5]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037

[6]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026

[7]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015

[8]

Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021008

[9]

Zhenbing Gong, Yanping Chen, Wenyu Tao. Jump and variational inequalities for averaging operators with variable kernels. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021045

[10]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[11]

Demou Luo, Qiru Wang. Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3427-3453. doi: 10.3934/dcdsb.2020238

[12]

Wen Si. Response solutions for degenerate reversible harmonic oscillators. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3951-3972. doi: 10.3934/dcds.2021023

[13]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[14]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[15]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597

[16]

Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226

[17]

Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021053

[18]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[19]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023

[20]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (0)

[Back to Top]