## Quasiperiodic perturbations of two-dimensional Hamiltonian systems.(English. Russian original)Zbl 1394.34088

Differ. Equ. 53, No. 12, 1557-1566 (2017); translation from Differ. Uravn. 53, No. 12, 1607-1615 (2017).
The paper studies quasiperiodic nonconservative perturbations of two-dimensional Hamiltonian systems of the form $\dot x = \frac{\partial H(x,y)}{\partial y}+\varepsilon g(x,y,t), \quad \dot y = \frac{\partial H(x,y)}{\partial x}+\varepsilon f(x,y,t),$ where $$\varepsilon$$ is a small positive parameter and the Hamiltonian $$H$$ and the functions $$g$$ and $$f$$ are sufficiently smooth (analytic) and, together with their partial derivatives of order $$\leq 2$$, uniformly bounded in a domain $$G\subset \mathbb R^2$$ or $$G \times \mathbb S^1$$. Additionally, the functions $$g$$ and $$f$$ are continuous and almost periodic in $$t$$ uniformly with respect to $$(x,y)\in G$$ with rationally independent frequencies. The behavior of the solutions to the considered system in a neighborhood of resonance and nonresonance levels is considered. Conditions for the existence of $$m$$-dimensional resonance tori are derived, and the global behavior of solutions in domains separated from the unpertubed separatrices is discussed. The results of the paper are illustrated by means of the Duffing equation $\dot x = y, \quad \dot y = x-x^3 + \varepsilon(p_1 y + F(t)),$ where either $$F(t)=p_2 \sin(\omega_1 t)+p_3 \sin(\omega_2 t)$$ or $$F(t)=p_2 \sin(\omega_1 t) \sin(\omega_2 t)$$ and $$p_k$$, $$k=1,2,3$$ are parameters.

### MSC:

 34C46 Multifrequency systems of ordinary differential equations 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34C45 Invariant manifolds for ordinary differential equations

### Keywords:

Hamiltonian system; quasiperiodic oscillations
Full Text:

### References:

 [1] Mitropol’skii, Yu.A. and Lykova, O.B., Integral’nye mnogoobraziya v nelineinoi mekhanike (Integral Manifolds in Nonlinear Mechanics), Moscow: Nauka, 1973. [2] Berger, M.S. and Chen, Y.Y., Forced quasiperiodic and almost periodic oscillations of nonlinear Duffing equations, Nonlinear Anal., 1992, vol. 19, no. 3, pp. 249-257. · Zbl 0765.34031 [3] Liu, Bin and You, Jiangong, Quasiperiodic solutions of Duffing’s Equations, Nonlinear Anal., 1998, vol. 33, no. 6, pp. 645-655. · Zbl 0947.34029 [4] Ravichandrana, V., Chinnathambia, V., and Rajasekarb, S., Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force, Phys. A, 2007, vol. 376, pp. 223-236. [5] Grishchenko, A.D. and Vavriv, D.M., Dynamics of a pendulum with a quasiperiodic excitation, Zh. Tekh. Fiz., 1997, vol. 67, no. 10, pp. 1-7. [6] Morozov, A.D. and Shil’nikov, L.P., On nonconservative periodic systems close to two-dimensional Hamiltonian, J. Appl. Math. Mech., 1983, vol. 47, no. 3, pp. 327-334. · Zbl 0564.70022 [7] Morozov, A.D., Rezonansy, tsikly i khaos v kvazikonservativnykh sistemakh (Resonances, Cycles, and Chaos in Quasiconservative Systems), Moscow; Izhevsk: Inst. Komp. Issled., 2005. [8] Bogolyubov, N.N. and Mitropol’skii, Yu.A., Asimptoticheskie metody v teorii nelineinykh kolebanii (Asymptotic Methods in the Theory of Nonlinear Oscillations), Moscow: Gos. Izd. Fiz. Mat. Lit., 1958. · Zbl 0083.08101 [9] Volosov, V.M. and Morgunov, B.I., Metody osredneniya v teorii nelineinykh kolebatel’nykh sistem (Averaging Methods in the Theory of Nonlinear Oscillation Systems), Moscow: Mosk. Gos. Univ., 1971. · Zbl 0232.70021 [10] Hale, J., Oscillations in Nonlinear Systems, New York: Dover, 1992. · Zbl 0115.07401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.