We present geometric conditions for the existence of a closed trajectory with symmetry in three-dimensional nonlinear systems. A generator with quadratic nonlinearity and a Chua circuit are considered as examples.
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V. E. Puzyrev, “Asymptotic stability of uniform rotations of a symmetric gyroscope with elastically fastened rotor,” Mekh. Tverd. Tela, Issue 34, 100–107 (2004).
V. S. Anishchenko, Complex Vibrations in Simple Systems [in Russian], Nauka, Moscow (1990).
V. A. Pliss, Nonlocal Problems of the Theory of Oscillations [in Russian], Nauka, Moscow (1964).
M. Belhaq and F. Lakrad, “Analytics of homoclinic bifurcations in three-dimensional systems,” Int. J. Bifurcation Chaos, 12, No. 11, 2479–2486 (2002).
G. A. Leonov, Strange Attractors and Classical Stability Theory, St. Petersburg University Press (2008).
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics. Part II, World Scientific, Singapore (2001).
N. V. Nikitina, Nonlinear Systems with Complex and Chaotic Behaviors of the Trajectories [in Russian], Feniks, Kiev (2012).
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhteorizdat, Leningrad (1949).
A. A. Martynyuk and A. Yu. Obolenskii, Investigation of the Stability of Autonomous Systems of Comparison [in Russian], Preprint No. 78.28, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1978).
A. A. Martynyuk and A. Yu. Obolenskii, “On the stability of solutions of autonomous Wa˙zewski systems,” Differents. Uravn., 16, No. 8, 1392–1407 (1980).
A. A. Martynyuk, “Asymptotic stability criterion for nonlinear monotonic systems and its applications (Review),” Int. Appl. Mech., 47, No. 5, 475–534 (2011).
N. V. Nikitina, “On the conditions of closure of trajectories in the three-dimensional space,” Dop. Nats. Akad. Nauk. Ukr., No. 7, 72–77 (2013).
V. S. Anishchenko, V.V. Astakhov, T. E. Vadivasova, O.V. Sosnovtseva, C.W.Wu, and L. O. Chua, “Dynamics of two coupled Chua’s circuits,” Int. J. Bifurcation Chaos, 5, No. 6, 1677–1699 (1995).
V. V. Astakhov, A. V. Shabunin, A. N. Sil’chenko, G. I. Strelkova, and V. S. Anishchenko, “Nonlinear dynamics of two Chua generators coupled via the capacitor,” Radiotekhn. E´lektron., 42, No. 3, 320–327 (1997)
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Translated from Neliniini Kolyvannya, Vol. 17, No. 2, pp. 268–280, April–June, 2014.
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Martynyuk, A.A., Nikitina, N.V. On Periodic Motion and Bifurcations in Three-Dimensional Nonlinear Systems. J Math Sci 208, 593–606 (2015). https://doi.org/10.1007/s10958-015-2469-3
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DOI: https://doi.org/10.1007/s10958-015-2469-3