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Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 397--406 | DOI:10.5890/DNC.2016.12.005

A.D. Morozov; T.N. Dragunov

Institute of IT, Mathematics and Mechanics, Lobachevsky University of Nizhny Novgorod, 23 Gagarin Ave, Nizhny Novgorod, 603950, Russia

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Abstract

Quasi-periodic two-frequency perturbations are studied in a system which is close to a nonlinear two-dimensional Hamiltonian one. The example of Duffing equation with a saddle and two separatix loops is considered. Several problems are studied: dynamical behavior in a neighborhood of a resonance level of the unperturbed system, conditions for the existence of resonance quasi-periodic solutions (two-dimensional resonance tori), global behavior of solutions inside domains separated from the unperturbed separatrix. In a neighborhood of the unperturbed separatrix the problem of relative position of stable an unstable separatrix manifolds is studied, conditions for the existence of doubly asymptotic solutions are found.

Acknowledgments

Our work was partially supported by the RFFR grant No 14-01-00344, RSCF, grant No 14-41-00044 and the Ministry of Education and Science of Russian Federation, Project 1410.

References

1. [1]	Afraimovich, V.S. and Shil'nikov, L.P. (1974), On small periodic perturbations of autonomous systems, Dokl. Akad. Nauk SSSR (Russia), 214(4), 739-742.

2. [2]	Morozov, A.D. (1976), On total qualitative investigation of the Duffing equation, J. Differentsialnye uravnenia (Russian), 12(2), 241-255.

3. [3]	Morozov, A.D. and Shil'nikov, L.P. (1983), On nonconservative periodic systems similar to two-dimensional Hamiltonian ones, Prikl. Mat. i Mekh. (Russian), 47(3), 385-394.

4. [4]	Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Berlin.

5. [5]	Morozov, A.D. (1998), Quasi-conservative systems: cycles, resonances and chaos, World Sci.: Singopure, in ser. Nonlinear Science, ser. A, V. 30, 325 p.

6. [6]	Mel'nikov, V.K. (1963), On stability of a center under periodic in time perturbations,Works of Moscow Math. Soc., 12, 3-52.

7. [7]	Berger, M.S. and Chen, Y.Y. (1992), Forced Quasiperiodic and Almost Periodic Oscillations of Nonlinear Duffing Equations, Nonlinear Analysis, Theory, Methods and Applications, 19(3), 249-257.

8. [8]	Liu, B. and You, J. (1998), Quasiperiodic solutions of Duffing's Equations, Nonlinear Analysis, 33, 645-655.

9. [9]	Ravichandran, V., Chinnathambi, V. and Rajasekar, S. (2007), Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force, Physica A, 376, 223-236.

10. [10]	Grischenko, A.D. and Vavriv, D.M. (1997), Dynamics of Pendulum with Quasi-periodic excitation, J. Theor. Phys. (in Russian), 67(10),

11. [11]	Jing, Z.J., Huang, J.C., and Deng J. (2007), Complex dynamics in three-well duffing system with two external forcings, Chaos, Solitons and Fractals, 33, 795-812.

12. [12]	Spears, B.K., Hutchings, M., and Szeri, A.J. (2005), Topological Bifurcations of Attracting 2-Tori of Quasiperiodically Driven Oscillators. J. Nonlinear Sci, 15, 423-452.

13. [13]	Morozov, A.D. and Kostromina, O.S. (2014), On Periodic Perturbations of Asymmetric Duffing-Van-der-Pol Equation, International Journal of Bifurcation and Chaos, 24(5).

14. [14]	Bogolyubov, N.N. and Mitropolsky, Yu.A. (1958), Asymptotical methods it the theory of nonlinear oscillations (in Russian), Fizmatgiz, Moscow.

15. [15]	Sanders, J.M. (1980), Melnikov's method and averaging, SIAM J. Math. Anal. 11, 750-770.

16. [16]	Hale, J.K. (1963), Oscillations in nonlinear systems, McGRAW-Hill Book company Inc., New York, Toronto, London.

17. [17]	Shilnikov, L.P. (1967), On a Poincarè-Birkhoff problem, Math. USSR Sb., 74(3), 378-397.