May  2012, 32(5): 1627-1637. doi: 10.3934/dcds.2012.32.1627

The center--focus problem and small amplitude limit cycles in rigid systems

1. 

Instituto de Ciências Exatas, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500–903, Itajubá, MG, Brazil, Brazil

Received  December 2010 Revised  July 2011 Published  January 2012

In this paper we study the center--focus problem in families of rigid systems. We give explicit necessary and sufficient conditions to the unique equilibrium to be a center. We also study small amplitude limit cycles in these families of systems.
Citation: Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627
References:
[1]

A. Algaba and M. Reyes, Computing center conditions for vector fields with constant angular speed,, J. Comput. Appl. Math., 154 (2003), 143. doi: 10.1016/S0377-0427(02)00818-X. Google Scholar

[2]

T. R. Blows and N. G. Lloyd, The number of small-amplitude limit cycles of Liénard equations,, Math. Proc. Camb. Phil. Soc., 95 (1984), 359. doi: 10.1017/S0305004100061636. Google Scholar

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A. Cima, A. Gasull and F. Mañosas, Cyclicity of a family of vector fields,, J. Math. Anal. Appl., 196 (1995), 921. doi: 10.1006/jmaa.1995.1451. Google Scholar

[4]

C. Chicone, "Ordinary Differential Equations with Applications,", Texts in Applied Mathematics, 34 (1999). Google Scholar

[5]

C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree,, J. Math. Anal. Appl., 195 (1995), 719. doi: 10.1006/jmaa.1995.1385. Google Scholar

[6]

A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems,, Proc. Amer. Math. Soc., 133 (2005), 751. doi: 10.1090/S0002-9939-04-07542-2. Google Scholar

[7]

A. Gasull and J. Torregrosa, Limit cycles for rigid cubic systems,, J. Math. Anal. Appl., 303 (2005), 391. doi: 10.1016/j.jmaa.2004.07.030. Google Scholar

[8]

A. M. Ljapunov, "Stability of Motion,", With a contribution by V. A. Pliss and an introduction by V. P. Basov, (1966). Google Scholar

show all references

References:
[1]

A. Algaba and M. Reyes, Computing center conditions for vector fields with constant angular speed,, J. Comput. Appl. Math., 154 (2003), 143. doi: 10.1016/S0377-0427(02)00818-X. Google Scholar

[2]

T. R. Blows and N. G. Lloyd, The number of small-amplitude limit cycles of Liénard equations,, Math. Proc. Camb. Phil. Soc., 95 (1984), 359. doi: 10.1017/S0305004100061636. Google Scholar

[3]

A. Cima, A. Gasull and F. Mañosas, Cyclicity of a family of vector fields,, J. Math. Anal. Appl., 196 (1995), 921. doi: 10.1006/jmaa.1995.1451. Google Scholar

[4]

C. Chicone, "Ordinary Differential Equations with Applications,", Texts in Applied Mathematics, 34 (1999). Google Scholar

[5]

C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree,, J. Math. Anal. Appl., 195 (1995), 719. doi: 10.1006/jmaa.1995.1385. Google Scholar

[6]

A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems,, Proc. Amer. Math. Soc., 133 (2005), 751. doi: 10.1090/S0002-9939-04-07542-2. Google Scholar

[7]

A. Gasull and J. Torregrosa, Limit cycles for rigid cubic systems,, J. Math. Anal. Appl., 303 (2005), 391. doi: 10.1016/j.jmaa.2004.07.030. Google Scholar

[8]

A. M. Ljapunov, "Stability of Motion,", With a contribution by V. A. Pliss and an introduction by V. P. Basov, (1966). Google Scholar

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