For citation:
Jalnine A. Y. On invariant manifolds of stable trajectories in quasiperiodically forced nonlinear dynamical systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2000, vol. 8, iss. 3, pp. 17-26. DOI: 10.18500/0869-6632-2000-8-3-17-26
On invariant manifolds of stable trajectories in quasiperiodically forced nonlinear dynamical systems
In terms of the quasiperiodically driven Henon map we show that two—dimensional stable invariant manifolds of the nodal invariant curve of а smooth invertible three—dimensional map can possess both differentiable and fractal structure. The fractalization of manifolds precedes the destruction of a smooth invariant curve and the birth of а strange nonchaotic attractor in modeling system. An observation of the angle between manifolds along trajectories that belong to the invariant curve after its manifolds fractalization and the strange nonchaotic attractor shows ап existence оf tangencies of the stable manifolds corresponding to different characteristic exponents. This fact means the violation of parabolic structure of manifolds in a small vicinity of the nodal invariant curve.
- Sosnovtseva O, Feudel U, Kurths J, Pikovsky А. Multiband strange nonchaotic attractors in quasiperiodically forced system. Phys. Lett. А. 1996;218(3-6):255-267.
- Grebogi C, Ott E, Pelikan S, Yorke J. Strange attractors that are not chaotic. Phys. D. 1984;13(1-2):261-268.
- Ding M, Grebogi C, Ott E. Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A. 1989;39(5):2593-2598. DOI: doi.org/10.1103/PhysRevA.39.2593.
- Heagy JF, Hammel SM. The birth of strange nonchaotic attractors. Phys. D. 1994;70(1-2):140-153. DOI: 10.1016/0167-2789(94)90061-2.
- Feudel U, Kurths J, Pikovsky А. Strange nonchaotic attractors in а quasiperiodically forced circle mар. Phys. D. 1995;88:176-186.
- Bezruchko BP, Kuznetsov SN, Pikovsky AS, Seleznev EP, Feudel U. On the dynamics of nonlinear systems under external quasi-periodic influence near the end point of the torus doubling line. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(6):3. (in Russian).
- Pikovsky А, Feudel U. Characterizing strange nonchaotic attractors. Chaos. 1995;5(1):253-260. DOI: 10.1063/1.166074.
- Kaneko K. Doubling of torus. Prog. Theor. Phys. 1983;69(6):1806-1810. DOI: 10.1143/PTP.69.1806.
- Lai Y-C, Grebogi C, Yorke J, Kan I. How often are chaotic saddles nonhyperbolic? Nonlinearity. 1993;6(5):779-797.
- 251 reads