In the work nonlinear Duffing oscillator is considered under impulse excitation with two ways of introduction of the random additive term simulating noise, - with help of amplitude modulation and modulation of period of impulses sequence. The scaling properties both in the Feigenbaum scenario and in the tricritical case are shown.

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1. Кузнецов С.П. Динамический хаос (курс лекций). 2-е изд., испр. и доп. М.: Физматлит. 2006. 356 с.

2. Crutchfield J.P., Nauenberg M., Rudnik J. Scaling for external noise at the onset of chaos // Phys. Rev. Lett. 1981. Vol. 46, No 14. P. 933.

3. Hirsch J.E., Nauenberg M., Scalapino D.J. Intermittency in the presence of noise: A renormalization group formulation // Phys. Lett. A. 1982. Vol. 87. P. 391.

4. Gyorgyi G., Tishby N.  ̈ Scaling in stochastic Hamiltonian systems: A renormalization approach // Phys. Rev. Lett. 1987. Vol. 58, No 6. P. 527.

5. Hamm A., Graham R. Scaling for small random perturbations of golden critical circle maps // Phys. Rev. A. 1992. Vol. 46, No 10. P. 6323.

6. Kapustina J.V., Kuznetsov A.P., Kuznetsov S.P., Mosekilde E. Scaling properties of bicritical dynamics in unidirectionally coupled period-doubling systems in presence of noise // Phys. Rev. E. 2001. Vol. 64, 066207.

7. Isaeva O.B., Kuznetsov S.P., Osbaldestin A.H. Effect of noise on the dynamics of a complex map at the period-tripling accumulation point // Phys. Rev. E. 2004. Vol. 69, 036216.

8. Shraiman B., Wayne C.E., Martin P.C. Scaling theory for noisy period-doubling transitions to chaos // Phys. Rev. Lett. 1981. Vol. 46, No14. P. 935.

9. Kuznetsov A.P., Kuznetsov S.P., Turukina L.V., Mosekilde E. Two-parameter analysis of the scaling behavior at the onset of chaos: Tricritical and pseudo-tricritical points // Physica A. 2001. Vol. 300, No 3-4. P. 367.

10. Kuznetsov A.P., Turukina L.V., Mosekilde E. Dynamical systems of different classes as models of the kicked nonlinear oscillators // Int. J. of Bifurcation and Chaos. 2001. Vol. 11, No4. P. 1065.

11. Кузнецов А.П., Тюрюкина Л.В. Динамические системы разных классов как модели нелинейного осциллятора с импульсным воздействием // Известия вузов. Прикладная нелинейная динамика. 2000. Том 8, No 2. C. 31-42.

12. Carr Y., Eilbech Y.C. One-dimensional approximations for a quadratic Ikeda map // Phys. Lett. A. 1984. Vol. 104. P. 59.

13. Кузнецов А.П., Капустина Ю.В. Свойства скейлинга при переходе к хаосу в модельных отображениях с шумом // Известия вузов. Прикладная нелинейная динамика. 2000. Том 8, No 6. C. 78.

14. Marcus M., Hess B. Lyapunov exponents of the logistic map with periodic forcing // Computers & Graphics. 1989. Vol. 13, No 4. P. 553.

15. Rossler J., Kiwi M., Hess B., Marcus M.  ̈ Modulated nonlinear processes and a novel mechanism to induce chaos // Phys. Rev. A. 1989. Vol. 39, No 11. P. 5954.

16. Marcus M. Chaos in maps with continuous and discontinuous maxima // Computers in physics. 1990. September/October. P. 481.

17. Bastos de Figueireido J.C., Malta C.P. Lyapunov graph for two-parameter map: Application to the circle map // Int. J. of Bifurcation and Chaos. 1998. Vol. 8, No 2. P. 281.

18. Kuznetsov A.P., Kuznetsov S.P., Sataev I.R. A variety of period-doubling universality classes in multi-parameter analysis of transition to chaos // Physica D. 1997. Vol. 109. P. 91.

19. Kuznetsov A.P., Kuznetsov S.P., Sataev I.R. Three-parameter scaling for one-dimensional maps // Phys. Lett. A. 1994. Vol. 189. P. 367.

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