For citation:
Golokolenov A. V. Dynamics of weakly dissipative self-oscillatory system at external pulse influence, which amplitude is depending polynomially on the dynamic variable. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 3, pp. 86-98. DOI: 10.18500/0869-6632-2019-27-3-86-98
Dynamics of weakly dissipative self-oscillatory system at external pulse influence, which amplitude is depending polynomially on the dynamic variable
Topic and aim. In this work, we study the dynamics of the kicked van der Pol oscillator with the amplitude of kicks depending nonlinearly on the dynamic variable. We choose the expansions of the function cos x in a Taylor series near zero, as functions describing this dependence. It is known that such a system demonstrates the existence of a Hamiltonian-type critical point in the case when the dependence of the amplitude of an external force on a dynamic variable is described by a quadratic polynomial, and when choosing a dependence in the form of cos x – a stochastic web in the conservative limit. Investigated models. The investigation is conducted for the original flow system and for an approximate discrete mapping. Results. We have investigated the changes in the structure of the parameter space and the phase space when changing the form of the function of external force. It is shown that the complication of the form of the function leads to an increase in the number of saddle-node bifurcations occurring in the system with a decrease in the dissipation parameter.
- Schuster H.G., Just W. Deterministic Chaos, WILEY-VCH Verlag, Weinheim, 2005, 287 p
- Reichl L.E. The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations. New York: Springer-Verlag, 1992, 551 p.
- Zisook A.B. Universal effects of dissipation in two-dimensional mappings // Physical Review A. 1982. T. 24, No 3. P. 1640–1642.
- Morozov A.D. Resonances, Cycles and Chaos in Quasi-Conservative Systems, Moscow; Izhevsk: Regular and Chaotic Dynamics, 2005, 424 p. (in Russian).
- Kuznetsov S.P., Kuznetsov A.P., Sataev I.R. Multiparameter critical situations, universality and scaling in two-dimensional period-doubling maps // Journal of Statistical Physics. 2005. T. 121, No 5–6. P. 697–748.
- Kuznetsov A.P., Kuznetsov S.P., Savin A.V., Savin D.V. Autooscillating system with compensated dissipation: Dynamics of approximate discrete map. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, no. 5, pp. 127–138 (in Russian).
- Kuznetsov A.P., Kuznetsov S.P., Savin A.V., Savin D.V. On the possibility for an autooscillatory system under external periodic drive action to exhibit universal behavior characteristic of the transition to chaos via period-doubling bifurcations in conservative systems. Technical Physics Letters, 2008, vol. 34, pp. 985–988.
- Savin D.V., Savin A.V., Kuznetsov A.P., Kuznetsov S.P., Feudel U. The self-oscillating system with compensated dissipation – the dynamics of the approximate discrete map // Dynamical Systems: An International Journal. 2012. Vol. 27. P. 117–129.
- Feudel U., Grebogi C., Hunt B.R., Yorke J.A. Map with more than 100 coexisting low-period periodic attractors // Physical Review E. 1996. Vol. 54, no. 1. P. 71–81.
- Feudel U., Grebogi C., Poon L., Yorke J.A. Dynamical properties of a simple mechanical system with a large number of coexisting periodic attractors // Chaos, Solitons & Fractals. 1998. Vol.9, no. 1–2. P. 171–180.
- Blazejczyk-Okolewska B., Kapitaniak T. Coexisting attractors of impact oscillator // Chaos, Solitons & Fractals. 1998. Vol. 9, no. 8. P. 1439–1443.
- Feudel U., Grebogi C. Why are chaotic attractors rare in multistable systems? // Physical Review Letters. 2003. Vol. 91, no. 13. 134102.
- de Freitas A.S.T., Viana R.L., Grebogi C. Multistability, basin boundary structure, and chaotic behavior in a suspension bridge model // International Journal of Bifurcation and Chaos. 2004. Vol. 14, no. 3. P. 927–950.
- Rech P., Beims M., Gallas J. Basin size evolution between dissipative and conservative limits //Physical Review E. 2005. Vol. 71, no. 1. 017202.
- Kolesov A.Yu. , Rozov N.Kh., The nature of the bufferness phenomenon in weakly dissipative systems. Theoret. and Math. Phys., 2006, vol. 146, no. 3, pp. 376–392.
- Kuznetsov A.P., Savin A.V., Savin D.V. Ikeda map: From a dissipative to conservative cases of the Ikeda map. Izvestiya VUZ. Applied Nonlinear Dynamics, 2006, vol. 14, no. 2, pp. 94–106 (in Russian).
- Kuznetsov A.P., Savin A.V., Savin D.V. Features in dynamics of an almost conservative Ikeda map. Technical Physics Letters, 2007, vol. 33, iss. 2, pp. 122–124.
- Kuznetsov A.P., Savin A.V., Savin D.V. On some properties of nearly conservative dynamics of Ikeda map // Nonlinear Phenomena in Complex Systems. 2007. Vol. 10, no. 4. P. 393–400.
- Feudel U. Complex dynamics in multistable systems // International Journal of Bifurcation and Chaos. 2008. Vol. 18, no. 6. P. 1607–1626.
- Kuznetsov A.P., Savin A.V., Savin D.V. On some properties of nearly conservative dynamics of Ikeda map and its relation with the conservative case // Physica A. 2008. Vol. 387. no. 7. P. 1464–1474.
- Moser J. KAM-Theory and Stability Problems. Moscow; Izhevsk: Regular and Chaotic Dynamics, 2001, 448 p. (in Russian).
- Zaslavsky G.M. Stochasticity of Dynamical Systems. Moscow: Nauka, 1984, 272 p. (in Russian)
- Savin A.V., Savin D.V. Structure of attraction basins of coexisting attractors of weakly dissipative «web-map». Nelinejnii Mir, 2010, vol. 8, no. 2, pp. 70–71 (in Russian).
- Felk E.V. The effect of weak nonlinear dissipation on structures of the «stochastic web» type. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, no. 3, pp. 72–79 (in Russian).
- Felk E.V., Kuznetsov A.P., Savin A.V. Multistability and transition to chaos in the degenerate Hamiltonian system with weak nonlinear dissipative perturbation // Physica A. 2014. Т. 410. P. 561–572.
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