References
E. F. Mishchenko and L. S. Pontryagin, “Periodic solutions of systems of differential equations that are close to discontinuous,”Dokl. Akad. Nauk SSSR,102, No. 5, 889–891 (1955).
E. F. Mishchenko and N. Kh. Rozov,Differential Equations with a Small Parameter and Relaxation Oscillations [in Russian], Nauka, Moscow (1975).
A. Yu. Kolesov, Yu. S. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “Asymptotic integration of a system in variations of a multidimensional relaxation cycle,”Diff. Uravn.,23, No. 11, 1881–1889, No. 12, 2036–2047 (1987).
A. Yu. Kolesov, Yu. S. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “A relaxation system in the neighborhood of a break-off point: reduction to a regular case,”Usp. Mat. Nauk,43, No. 2, 141–142 (1988).
A. Yu. Kolesov, and E. F. Mishchenko, “Asymptotics of relaxation oscillations,”Mat. Sb.,137, No. 1, 3–18 (1988).
E. F. Mishchenko and A. Yu. Kolesov, “Asymptotic theory of relaxation oscillations,”Trudy Mat. Inst. Akad. Nauk SSSR,117, 3–84 (1991).
A. Yu. Kolesov, “Specific relaxation cycles of a system of the Lotka-Volterra type,”Izv. Akad. Nauk SSSR, Ser. Mat.,55, No. 3, 515–536 (1991).
G. E. Hutchinson, “Circular causal systems in ecology,”Ann. N.Y. Acad. Sci.,50, 221–246 (1948).
A. S. Pikovsky and M. U. Rabinovich, “A simple self-excited oscillator with stochastic behavior,”Dokl. Akad. Nauk SSSR,239, No. 2, 301–304 (1978).
N. N. Chentsova, “Investigation of a model system of quasi-stochastic relaxation oscillations,”Usp. Mat. Nauk,37, No. 5, 205–206 (1982).
N. N. Chentsova, “Investigation of the quasi-stochastic conditions of a self-excited oscillator on a tunnel diode,” in:Methods of the Qualitative Theory of Differential Equations [in Russian], Gorky (1983), pp. 95–118.
A. Yu. Kolesov, Yu. S. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “A mixing attractor in relaxation systems,”Dokl. Akad. Nauk SSSR,309, No. 1, 38–40 (1989).
A. Yu. Kolesov, Yu. S. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “On the chaos phenomena in three-dimensional relaxation systems,” Mat. Zametki,46, No. 2, 153–155 (1989).
A. B. Vasil'eva and V. F. Butusov,Asymptotic Expansion of Solutions of Singularly Perturbed Equations [in Russian], Nauka, Moscow (1973).
Yu. A. Mitropol'sky and O. B. Lykova,Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1973).
L. P. Shil'nikov, “On a certain Poincare-Birkhoff problem,”Mat. Sb.,74, No. 3, 378–397 (1987).
P. Hartman,Ordinary Differential Equations, New York (1964).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 17, Dinamicheskie Sistemy-2, 1994.
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Kolesov, A.Y., Kolesov, Y.S. & Rozov, N.K. Chaos of the broken torus type in three-dimensional relaxation systems. J Math Sci 80, 1533–1545 (1996). https://doi.org/10.1007/BF02363925
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DOI: https://doi.org/10.1007/BF02363925