Geodesic
Global Dynamics of Systems Close to Hamiltonian Ones Under Nonconservative Quasi-periodic Perturbation
Russian journal of nonlinear dynamics, Tome 15 (2019) no. 2, pp. 187-198.

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We study quasi-periodic nonconservative perturbations of two-dimensional Hamiltonian systems. We suppose that there exists a region $D$ filled with closed phase curves of the unperturbed system and consider the problem of global dynamics in $D$. The investigation includes examining the behavior of solutions both in $D$ (the existence of invariant tori, the finiteness of the set of splittable energy levels) and in a neighborhood of the unperturbed separatrix (splitting of the separatrix manifolds). The conditions for the existence of homoclinic solutions are stated. We illustrate the research with the Duffing – Van der Pole equation as an example.
Keywords: resonances, quasi-periodic, periodic, averaged system, phase curves, equilibrium states, separatrix manifolds.
Mots-clés : limit cycles 
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     title = {Global {Dynamics} of {Systems} {Close} to {Hamiltonian} {Ones} {Under} {Nonconservative} {Quasi-periodic} {Perturbation}},
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A. D. Morozov; K. E. Morozov. Global Dynamics of Systems Close to Hamiltonian Ones Under Nonconservative Quasi-periodic Perturbation. Russian journal of nonlinear dynamics, Tome 15 (2019) no. 2, pp. 187-198. http://geodesic.mathdoc.fr/item/ND_2019_15_2_a7/

[1] Jing, Zh., Huang, J., and Deng, J., “Complex Dynamics in Three-Well Duffing System with Two External Forcings”, Chaos Solitons Fractals, 33:3 (2007), 795–812 | DOI | MR | Zbl

[2] Liu, B. and You, J., “Quasiperiodic Solutions of Duffing's Equations”, Nonlinear Anal., 33:6 (1998), 645–655 | DOI | MR | Zbl

[3] Jing, Zh., Yang, Zh., and Jiang, T., “Complex Dynamics in Duffing – van der Pol Equation”, Chaos Solitons Fractals, 27:3 (2006), 722–747 | DOI | MR | Zbl

[4] Mel'nikov, V. K., “On the Stability of a Center for Time-Periodic Perturbations”, Tr. Mosk. Mat. Obs., 12 (1963), 3–52 (Russian) | MR | Zbl

[5] Sanders, J. A., “Melnikov's Method and Averaging”, Celestial Mech., 28:1–2 (1982), 171–181 | DOI | MR | Zbl

[6] Wiggins, S., Chaotic Transport in Dynamical Systems, Interdiscip. Appl. Math., 2, Springer, New York, 1992, XIII, 301 pp. | DOI | MR | Zbl

[7] Zh. Tekh. Fiz., 67:10 (1997), 1–7 (Russian) | DOI

[8] Yagasaki, K., “Second-Order Averaging and Chaos in Quasiperiodically Forced Weakly Nonlinear Oscillators”, Phys. D, 44:3 (1990), 445–458 | DOI | MR | Zbl

[9] Belogortsev, A. B., “Quasiperiodic Resonance and Bifurcations of Tori in the Weakly Nonlinear Duffing Oscillator”, Phys. D, 59:4 (1992), 417–429 | DOI | MR | Zbl

[10] Jing, Zh. and Wang, R., “Complex Dynamics in Duffing System with Two External Forcings”, Chaos Solitons Fractals, 23:2 (2005), 399–411 | DOI | MR | Zbl

[11] Morozov, A. D., Quasi-Conservative Systems: Cycles, Resonances and Chaos, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 30, World Sci., River Edge, N.J., 1999, 340 pp. | MR

[12] Bogoliubov, N. N. and Mitropolsky, Yu. A., Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon Breach, New York, 1961 | MR

[13] Mitropolsky, Yu. A. and Lykova, O. B., Integrated Manifolds in the Nonlinear Mechanics, Nauka, Moscow, 1973, 512 pp. (Russian) | MR

[14] Differ. Uravn., 53:12 (2017), 1607–1615 (Russian) | DOI | MR | Zbl

[15] Morozov, A. D., Resonances, Cycles and Chaos in Quasi-Conservative Systems, R Dynamics, Institute of Computer Science, Moscow – Izhevsk, 2005 (Russian) | MR

[16] Morozov, A. D., “On the Structure of Resonance Zones and Chaos in Nonlinear Parametric Systems”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4:2 (1994), 401–410 | DOI | MR | Zbl

[17] Prikl. Mat. Mekh., 58:3 (1994), 41–51 (Russian) | DOI | MR | Zbl

[18] Morozov, A. D. and Morozov, K. E., “On Synchronization of Quasiperiodic Oscillations”, Russian J. Nonlinear Dyn., 14:3 (2018), 367–376 | MR

[19] Hale, J. K., Oscillations in Nonlinear System, McGraw-Hill, New York, 1963, 192 pp. | MR

[20] Morozov, A.D. and Dragunov,T.N., “On Quasi-periodic Perturbations of Duffing Equation”, Interdiscip. J. Discontin. Nonlinearity Complex, 5:4 (2016), 377–386 | DOI

[21] Mat. Sb. (N. S.), 74(116):3 (1967), 378–397 (Russian) | DOI | MR