On the limit cycles of the Floquet differential equation

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June 2014, 19(4): 1129-1136. doi: 10.3934/dcdsb.2014.19.1129

On the limit cycles of the Floquet differential equation

Jaume Llibre ^1, and Ana Rodrigues ^2,

1.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.	College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, United Kingdom

Received April 2012 Revised February 2014 Published April 2014

Abstract
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We provide sufficient conditions for the existence of limit cycles for the Floquet differential equations $\dot {\bf x}(t) = A{\bf x}(t)+ε(B(t){\bf x}(t)+b(t))$, where ${\bf x}(t)$ and $b(t)$ are column vectors of length $n$, $A$ and $B(t)$ are $n\times n$ matrices, the components of $b(t)$ and $B(t)$ are $T$--periodic functions, the differential equation $\dot {\bf x}(t)= A{\bf x}(t)$ has a plane filled with $T$--periodic orbits, and $ε$ is a small parameter. The proof of this result is based on averaging theory but only uses linear algebra.

Keywords: Floquet differential equation, averaging theory., periodic solution, limit cycle.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129

References:

[1]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Second Printing, Springer-Verlag, Berlin, 1997.
[2]	A. Buică, J.P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Communication on Pure and Applied Analysis, 6 (2007), 103-111. doi: 10.1016/j.physd.2011.11.007.
[3]	C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, New York, 1999.
[4]	D. G. de Figueiredo, Análise de Fourier e Equaçoes Diferenciais Parciais, Projeto Euclides 5, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1977 (in Portuguese).
[5]	M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Pure and Applied Mathematics 60, Academic Press, New York, 1974.
[6]	J. Llibre, M.A. Teixeira and J. Torregrosa, Limit cycles bifurcating from a $k$-dimensional isochronous set center contained in $R^n$ with $k \leq n$, Math. Phys. Anal. Geom., 10 (2007), 237-249. doi: 10.1007/s11040-007-9030-7.
[7]	P. Lochak and C. Meunier, Multiphase averaging for classical systems, Appl. Math. Sciences 72, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1044-3.
[8]	I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.
[9]	M. Roseau, Vibrations non Linéaires et Théorie de la Stabilité, (French) Springer Tracts in Natural Philosophy, Vol.8 Springer-Verlag, Berlin-New York, 1966.
[10]	J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Second edition, Applied Mathematical Sci. 59, Springer-Verlag, New York, 2007.
[11]	W. F. Trench, On nonautonomous linear systems of differential and difference equations with R-symmetric coefficient matrices, Linear Algebra Appl., 431 (2009), 2109-2117. doi: 10.1016/j.laa.2009.07.004.
[12]	W. F. Trench, Asymptotic preconditioning of linear homogeneous systems of differential equations, Linear Algebra Appl., 434, (2011), 1631-1637. doi: 10.1016/j.laa.2010.03.026.
[13]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.

show all references

References:

[1]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Second Printing, Springer-Verlag, Berlin, 1997.
[2]	A. Buică, J.P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Communication on Pure and Applied Analysis, 6 (2007), 103-111. doi: 10.1016/j.physd.2011.11.007.
[3]	C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, New York, 1999.
[4]	D. G. de Figueiredo, Análise de Fourier e Equaçoes Diferenciais Parciais, Projeto Euclides 5, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1977 (in Portuguese).
[5]	M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Pure and Applied Mathematics 60, Academic Press, New York, 1974.
[6]	J. Llibre, M.A. Teixeira and J. Torregrosa, Limit cycles bifurcating from a $k$-dimensional isochronous set center contained in $R^n$ with $k \leq n$, Math. Phys. Anal. Geom., 10 (2007), 237-249. doi: 10.1007/s11040-007-9030-7.
[7]	P. Lochak and C. Meunier, Multiphase averaging for classical systems, Appl. Math. Sciences 72, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1044-3.
[8]	I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.
[9]	M. Roseau, Vibrations non Linéaires et Théorie de la Stabilité, (French) Springer Tracts in Natural Philosophy, Vol.8 Springer-Verlag, Berlin-New York, 1966.
[10]	J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Second edition, Applied Mathematical Sci. 59, Springer-Verlag, New York, 2007.
[11]	W. F. Trench, On nonautonomous linear systems of differential and difference equations with R-symmetric coefficient matrices, Linear Algebra Appl., 431 (2009), 2109-2117. doi: 10.1016/j.laa.2009.07.004.
[12]	W. F. Trench, Asymptotic preconditioning of linear homogeneous systems of differential equations, Linear Algebra Appl., 434, (2011), 1631-1637. doi: 10.1016/j.laa.2010.03.026.
[13]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.

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