Impact Factor

    Global Dynamics of Systems Close to Hamiltonian Ones Under Nonconservative Quasi-periodic Perturbation

    2019, Vol. 15, no. 2, pp.  187-198

    Author(s): Morozov A. D., Morozov K. E.

    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, limit cycles, separatrix manifolds
    Citation: Morozov A. D., Morozov K. E., Global Dynamics of Systems Close to Hamiltonian Ones Under Nonconservative Quasi-periodic Perturbation, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp.  187-198

    Download File
    PDF, 2.01 Mb


    [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  crossref  mathscinet  zmath  adsnasa
    [2] Liu, B. and You, J., “Quasiperiodic Solutions of Duffing's Equations”, Nonlinear Anal., 33:6 (1998), 645–655  crossref  mathscinet  zmath
    [3] Jing, Zh., Yang, Zh., and Jiang, T., “Complex Dynamics in Duffing – van der Pol Equation”, Chaos Solitons Fractals, 27:3 (2006), 722–747  crossref  mathscinet  zmath  adsnasa
    [4] Mel'nikov, V. K., “On the Stability of a Center for Time-Periodic Perturbations”, Tr. Mosk. Mat. Obs., 12 (1963), 3–52 (Russian)  mathnet  mathscinet  zmath
    [5] Sanders, J. A., “Melnikov's Method and Averaging”, Celestial Mech., 28:1–2 (1982), 171–181  crossref  mathscinet  zmath  adsnasa
    [6] Wiggins, S., Chaotic Transport in Dynamical Systems, Interdiscip. Appl. Math., 2, Springer, New York, 1992, XIII, 301 pp.  crossref  mathscinet  zmath
    [7] Grischenko, A. D. and Vavriv, D. M., “Dynamics of Pendulum with a Quasiperiodic Perturbation”, Tech. Phys., 42:10 (1997), 1115–1120  crossref; Zh. Tekh. Fiz., 67:10 (1997), 1–7 (Russian)
    [8] Yagasaki, K., “Second-Order Averaging and Chaos in Quasiperiodically Forced Weakly Nonlinear Oscillators”, Phys. D, 44:3 (1990), 445–458  crossref  mathscinet  zmath
    [9] Belogortsev, A. B., “Quasiperiodic Resonance and Bifurcations of Tori in the Weakly Nonlinear Duffing Oscillator”, Phys. D, 59:4 (1992), 417–429  crossref  mathscinet  zmath
    [10] Jing, Zh. and Wang, R., “Complex Dynamics in Duffing System with Two External Forcings”, Chaos Solitons Fractals, 23:2 (2005), 399–411  crossref  mathscinet  zmath  adsnasa
    [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.  mathscinet
    [12] Bogoliubov, N. N. and Mitropolsky, Yu. A., Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon & Breach, New York, 1961  mathscinet
    [13] Mitropolsky, Yu. A. and Lykova, O. B., Integrated Manifolds in the Nonlinear Mechanics, Nauka, Moscow, 1973, 512 pp. (Russian)  mathscinet
    [14] Morozov, A. D. and Morozov, K. E., “Quasiperiodic Perturbations of Two-Dimensional Hamiltonian Systems”, Differ. Equ., 53:12 (2017), 1557–1566  crossref  mathscinet  zmath; Differ. Uravn., 53:12 (2017), 1607–1615 (Russian)  zmath
    [15] Morozov, A. D., Resonances, Cycles and Chaos in Quasi-Conservative Systems, R&C Dynamics, Institute of Computer Science, Moscow – Izhevsk, 2005 (Russian)  mathscinet
    [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  crossref  mathscinet  zmath
    [17] Morozov, A. D., “Resonances and Chaos in Parametric Systems”, J. Appl. Math. Mech., 58:3 (1994), 413–423  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 58:3 (1994), 41–51 (Russian)  mathscinet  zmath
    [18] Morozov, A. D. and Morozov, K. E., “On Synchronization of Quasiperiodic Oscillations”, Russian J. Nonlinear Dyn., 14:3 (2018), 367–376  mathscinet
    [19] Hale, J. K., Oscillations in Nonlinear System, McGraw-Hill, New York, 1963, 192 pp.  mathscinet
    [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  crossref
    [21] Shilnikov, L. P., “On a Poincaré – Birkhoff Problem”, Math. USSR-Sb., 3:3 (1967), 353–371  mathnet  crossref  mathscinet; Mat. Sb. (N. S.), 74(116):3 (1967), 378–397 (Russian)  mathscinet

    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License