Abstract
The complex behavior of trajectories is investigated. A new approach is proposed to estimate the applicability limits of some models. Examples of real phenomena are considered
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REFERENCES
V. S. Anishchenko, Complex Oscillations in Simple Systems: Mechanisms, Structure, and Properties of Chaos in Radiophysical Systems [in Russian], Nauka, Moscow (1990).
V. I. Arnol’d, Ordinary Differential Equations [in Russian], Glavn. Red. Fiz.-Mat. Lit., Moscow (1984).
N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).
V. I. Zubov, Oscillations and Waves [in Russian], Izd. Leningr. Univ., Leningrad (1989).
A. A. Martynyuk and N. V. Nikitina, “Complex oscillations revisited,” Int. Appl. Mech., 41, No.2, 179–186 (2005).
Yu. A. Mitropol’skii, Nonlinear Mechanics: Single-Frequency Oscillations [in Russian], Inst. Mat. NAN Ukrainy, Kiev (1997).
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], GITTL, Moscow (1949).
M. I. Rabinovich and D. I. Trubetskov, An Introduction to the Theory of Oscillations and Waves [in Russian], NITs Regul. Khaotich. Dinam., Moscow (2000).
L. P. Shil’nikov, A. L. Shil’nikov, D. V. Turaev, and L. Chua, Qualitative Methods in Nonlinear Dynamics [in Russian], Pt. 1, Inst. Komp. Issled., Moscow (2004).
B. Aulbach and B. M. Kieninger, “On three definitions of chaos,” Nonlin. Dynam. Syst. Theor., 1, No.1, 23–38 (2001).
M-P. Boer, B. W. Kooi, and S. A. L. M. Kooijman, “Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain,” J. Vath. Biol., 39, No.1, 19–38 (1999).
A. A. Martynyuk and N. V. Nikitina, “Bistable oscillator theory revisited,” Int. Appl. Mech., 38, No.4, 489–497 (2002).
A. A. Martynyuk and N. V. Nikitina, “On an approximate solution of the Van der Pol equations with a large parameter,” Int. Appl. Mech., 38, No.8, 1017–1023 (2002).
A. A. Martynyuk and N. V. Nikitina, “A note on bifurcations of motions in the Lorenz system,” Int. Appl. Mech., 39, No.2, 224–231 (2003).
A. A. Martynyuk and N. V. Nikitina, “Saddle limit cycles in the Lorenz system,” Int. Appl. Mech., 39, No.5, 613–620 (2003).
A. A. Martynyuk and N. V. Nikitina, “Studying the complex oscillations of a star in the field of a galaxy,” Int. Appl. Mech., 40, No.4, 453–461 (2004).
V. Reitman, Regulare und chaotische Dynamik, Teubner, Leipzig (1996).
Y. Shinohara, “A geometric method for the numerical solution of non-linear equations and its application to non-linear oscillations,” Publ. Res. Inst. Math. Sci., Kyoto Univ., 8, No.1, 13–42 (1972).
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Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 108–118, March 2005.
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Martynyuk, A.A., Nikitina, N.V. Complex Behavior of a Trajectory in Single- and Double-Frequency Systems. Int Appl Mech 41, 315–323 (2005). https://doi.org/10.1007/s10778-005-0090-8
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DOI: https://doi.org/10.1007/s10778-005-0090-8