We study a dynamical analog of bifurcations of invariant tori for a system of interconnected fast phase variables and slowly varying parameters. It is shown that, in this system, due to the slow evolution of the parameters, we observe the appearance of transient processes (from the damping process to multifrequency oscillations) asymptotically close to motions on the invariant torus.
Similar content being viewed by others
References
N. M. Krylov and N. N. Bogolyubov, Application of the Methods of Nonlinear Mechanics to the Theory of Stationary Oscillations [in Russian], Akad. Nauk Ukr. SSR, Kiev (1934).
Yu. I. Neimark, “On some cases of dependence of periodic motions on parameters,” Dokl. Akad. Nauk SSSR, 129, No. 4, 736–739 (1959).
R. Sacker, “A new approach to the perturbation theory of invariant surfaces,” Comm. Pure Appl. Math., 18, 717–732 (1965).
D. Ruelle and F. Takens, “On the nature of turbulence,” Comm. Math. Phys., 20, 167–192 (1971).
J. Marsden and M. McCracken, Hopf Bifurcation and Its Applications, Springer, New York (1976).
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York (1998).
J. K. Hale, “Integral manifolds of perturbed differential systems,” Ann. Math. (2), 73, No. 3, 496–531 (1961).
N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).
N. Fenichel, “Persistence and smoothness of invariant manifolds and flows,” Indiana Univ. Math., 21, No. 3, 193–226 (1971).
Yu. A. Mitropol’skii and A. M. Samoilenko, “On asymptotic integration of weakly nonlinear systems,” Ukr. Mat. Zh., 28, No. 4, 483–500 (1976); English translation: Ukr. Math. J., 28, No. 4, 372–386 (1976).
A. M. Samoilenko, Elements of the Mathematical Theory of Multi-Frequency Oscillations, Kluwer, Dordrecht (1991).
A. M. Samoilenko, “Perturbation theory of smooth invariant tori of dynamical systems,” Nonlin. Anal., 30, No. 5, 3121–3133 (1997).
A. M. Samoilenko and R. I. Petryshyn, Mathematical Aspects of the Theory of Nonlinear Oscillations [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2004).
W. Langford, “Periodic and steady mode interactions lead to tori,” SIAM J. Appl. Math., 37, 22–48 (1979).
N. K. Gavrilov, “On bifurcations of the equilibrium state with two pairs of pure imaginary roots,” in: Methods of the Qualitative Theory of Differential Equations [in Russian], Izd. Gorkii Gos. Univ., Gorkii (1980), pp. 17–30.
Yu. N. Bibikov, “Bifurcation of a stable invariant torus from the equilibrium state,” Mat. Zametki, 48, Issue 1, 15–19 (1990).
Yu. N. Bibikov, Multifrequency Nonlinear Oscillations and Their Bifurcations [in Russian], Leningradskii Universitet, Leningrad (1991).
Ya. M. Goltser, “On the bifurcation of invariant tori in the mappings, with a spectrum on a unit circle,” Funct. Different. Equat., 5, No. 1–2, 121–138 (1998).
Yu. N. Bibikov and V. R. Bukaty, “Multifrequency oscillations of singularly perturbed systems,” Differents. Uravn., 48, No. 1, 21–26 (2012).
M. A. Shishkova, “Investigation of one system of differential equations with small parameter at higher derivatives,” Dokl. Akad. Nauk SSSR, 209, No. 3, 576–579 (1973).
A. I. Neishtadt, “Asymptotic investigation of the loss of stability of equilibrium as a result of the slow transition of a pair of eigenvalues through the imaginary axis,” Usp. Mat. Nauk, 40, No. 5, 300–301 (1985).
A. I. Neishtadt, “On the delay of the loss of stability for dynamical bifurcations,” Differents. Uravn., 23, No. 12, 2060–2067 (1987); 24, No. 2, 226–233 (1988).
A. Neishtadt, “On stability loss delay for dynamical bifurcations,” Discrete Contin. Dynam. Syst., Ser. 2, 2, No. 4, 897–909 (2009).
É Benoît (editor), Dynamic Bifurcations, Springer, Berlin (1991).
V. F. Butuzov, N. N. Nefedov, and K. R. Schneider, “Singularly perturbed problems in the case of exchange of stabilities,” J. Math. Sci., 121, No. 1, 1973–2079 (2004).
D. Rachinskii and K. Schneider, “Dynamic Hopf bifurcations generated by nonlinear terms,” J. Different. Equat., 210, No. 1, 65–86 (2005).
O. D. Anosova, “Invariant manifolds and dynamical bifurcations,” Usp. Mat. Nauk., 60, No. 1, 157–158 (2005).
P. Cartier, “Singular perturbations of ordinary differential equations and nonstandard analysis,” Usp. Mat. Nauk, 39, No. 2, 57–76 (1984).
S. Liebscher, “Dynamics near manifolds of equilibria of codimension one and bifurcation without parameters,” Electron. J. Different. Equat., 2011, No. 63, 1–12 (2001); URL: http://ejde.math.txstate.edu .
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 7, pp. 890–915, July, 2015.
Rights and permissions
About this article
Cite this article
Samoilenko, A.M., Parasyuk, I.O. & Repeta, B.V. Dynamical Bifurcation of Multifrequency Oscillations in a Fast-Slow System. Ukr Math J 67, 1008–1037 (2015). https://doi.org/10.1007/s11253-015-1133-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1133-x