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The research of periodic solutions of time-varying differential models. (English) Zbl 1295.34052

Summary: We have studied the periodicity of solutions of some nonlinear time-varying differential models by using the theory of reflecting functions. We have established a new relationship between the linear differential system and the Riccati equations and applied the obtained results to discuss the behavior of periodic solutions of the Riccati equations.

MSC:

34C25 Periodic solutions to ordinary differential equations

References:

[1] J. D. Murray, Lectures on Nonlinear Differential Equations Models in Biology, World Book Publishing, Moscow, Russia, 1983. · Zbl 0525.92001
[2] V. I. Mironenko, Reflective Function and Periodic Solutions of Differential Equations, University Press, Minsk, Belarus, 1986, (Russian). · Zbl 0607.34038
[3] V. I. Mironenko, Analysis of Reflective Function and Multivariate Differential System, University Press, Gomel, Belarus, 2004. · Zbl 1079.34518
[4] V. I. Mironenko, “A reflecting function of a family of functions,” Differential Equations, vol. 36, no. 12, pp. 1636-1641, 2000. · Zbl 1001.34027 · doi:10.1023/A:1017548511544
[5] V. V. Mironenko, “Time-symmetry-preserving perturbations of differential systems,” Differential Equations, vol. 40, no. 10, pp. 1325-1332, 2004. · Zbl 1087.34521 · doi:10.1007/s10625-005-0064-y
[6] V. I. Mironenko and V. V. Mironenko, “Time symmetries and in-period transformations,” Applied Mathematics Letters, vol. 24, no. 10, pp. 1721-1723, 2011. · Zbl 1221.34089 · doi:10.1016/j.aml.2011.04.027
[7] L. A. Alisevich, “On linear system with triangular reflective function,” Differential Equations, vol. 25, no. 3, pp. 1446-1449, 1989.
[8] E. V. Musafirov, “Differential systems, the mapping over period for which is represented by a product of three exponential matrixes,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 647-654, 2007. · Zbl 1136.34003 · doi:10.1016/j.jmaa.2006.06.085
[9] P. P. Veresovich, “Nonautonomous two-dimensional quadratic systems that are equivalent to linear systems,” Differential Equations, vol. 34, no. 12, pp. 2257-2259, 1998. · Zbl 0411.34010
[10] S. V. Maĭorovskaya, “Quadratic systems with a linear reflecting function,” Differential Equations, vol. 45, no. 2, pp. 271-273, 2009. · Zbl 1180.34046 · doi:10.1134/S0012266109020153
[11] Z. Zhou, “On the qualitative behavior of periodic solutions of differential systems,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 600-611, 2009. · Zbl 1188.34049 · doi:10.1016/j.cam.2009.06.035
[12] Z. Zhou, “The structure of reflective function of polynomial differential systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 391-398, 2009. · Zbl 1179.34039 · doi:10.1016/j.na.2008.10.121
[13] Z. Zhou, “On the reflective function of polynomial differential system,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 18-26, 2003. · Zbl 1034.34008 · doi:10.1016/S0022-247X(02)00441-9
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