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A note on a generalization of Diliberto’s Theorem for certain differential equations of higher dimension. (English) Zbl 1099.37032
Summary: In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by S. P. Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.
37E99 Low-dimensional dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
34D10 Perturbations of ordinary differential equations
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[1] L. Adamec: Kinetical systems. Appl. Math. 42 (1997), 293–309. · Zbl 0903.34043
[2] L. Adamec: A note on the transition mapping for n-dimensional systems. Submitted.
[3] I. Agricola, T. Friedrich: Global Analysis. American Mathematical Society, Rhode Island, 2002. · Zbl 1005.58001
[4] J. Andres: On the multivalued Poincaré operators. Topol. Meth. Nonlin. Anal. 10 (1997), 171–182. · Zbl 0909.47038
[5] J. Andres: Poincarés translation multioperator revisted. In: Proceedings of the 3rd Polish Symposium of Nonlinear Analalysis, Łódź, January 29–31, 2001. Lecture Notes Nonlinear Anal. 3 (2002), 7–22.
[6] A. A. Andronov, E. A. Leontovich, I. I. Gordon, and I. I. Mayer: Theory of Bifurcation of Dynamical System on the Plane. John Wiley & Sons, New York-London-Sydney, 1973.
[7] C. Chicone: Ordinary Differential Equations with Applications. Springer-Verlag, New York, 1999. · Zbl 0937.34001
[8] S. P. Diliberto: On systems of ordinary differential equations. In: Contributions to the Theory of Nonlinear Oscillations. Ann. Math. Stud. 20 (1950), 1–38. · Zbl 0039.09402
[9] P. Hartman: Ordinary Differential Equations. John Wiley & Sons, New York-London-Sydney, 1964. · Zbl 0125.32102
[10] J. Kurzweil: Ordinary Differential Equations. Elsevier, Amsterdam-Oxford-New York-Tokyo, 1986. · Zbl 0667.34002
[11] M. Medved’: A construction of realizations of perturbations of Poincaré maps. Math. Slovaca 36 (1986), 179–190.
[12] H. Poincaré: Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris, 1892.
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