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ISSN 2542-1905 (Online)

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Silchenko A. N., Beri S., Luchinsky D. G., McClintock P. Fluctuational transitions across locally-disconnected and locally-connected fractal basin boundaries. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 3, pp. 38-45. DOI: 10.18500/0869-6632-2003-11-3-38-45

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519.6: 537.86:519.2

Fluctuational transitions across locally-disconnected and locally-connected fractal basin boundaries

Silchenko Alexander Nikolaevich, Lancaster University
Beri Stefano, Lancaster University
Luchinsky Dmitrii G., Lancaster University
McClintock Peter Vaughan Elsmere, Lancaster University

We study fluctuational transitions in а discrete dynamical system that has two coexisting attractors in phase space, separated by а fractal basin boundary which may be cither locally-disconnected оr locaily-connected. It is shown that, in each case, transitions оссur via аn accessible point оn the boundary. The complicated structure of paths inside the locally-disconnecied fractal boundary is determined by а hierarchy of homoclinic original saddles. The most probable escape path from а regular attractor to the fractal boundary is found for the each type of boundary using both statistical analyses of fluctuational trajectories and the Hamiltonian theory оf fluctuations.

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The research has been supported by the Engineering аnd Physical Sciences Research Council (UK) and INTAS.
  1. Stratonovich RL. Topics in the Theory of Random Noise (Mathematics and its Applications). Taylor and Francis; 1967. 292 p.
  2. Anishchenko VS, Neiman AB, Vadivasova TE, Schimansky-Geier L, Astakhov VV. Nonlinear Dynamics оf Chaotic and Stochastic Systems. Berlin: Springer-Verlag; 2001. 446 p.
  3. Ott Е. Chaos in Dynamical Systems. Cambridge University Press; 2002. 478 p. DOI: 10.1017/CBO9780511803260.
  4. McDonald SW, Grebogi C, Ott E, Yorke JA. Fractal basin boundaries. Physica D. 1985;17(2):125–153. DOI: 10.1016/0167-2789(85)90001-6.
  5. Cartwright ML, Littlewood JE. Some fixed point theorems. Ann. Math. 1951;54(1):1–37. DOI: 10.2307/1969308; Moon FC, Li G-X. Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential. Phys. Rev. Lett. 1985;55(14):1439–1442. DOI: 10.1103/PhysRevLett.55.1439.
  6. Sommerer JC, Ott Е. A physical system with qualitatively uncertain dynamics. Nature. 1993;365(6442):138–140. DOI: 10.1038/365138a0.
  7. Nusse HE, Yorke JA. Basins оf attraction. Science. 1996;271(5254):1376–1380. DOI: 10.1126/science.271.5254.1376.
  8. Hunt ВR, Оtt E, Rоsа Е. Sporadically fractal basin boundaries of chaotic systems. Phys. Rev. Lett. 1999;82(18):3597–3600. DOI: 10.1103/PhysRevLett.82.3597.
  9. Fradkov AL, Pogromsky AY. Introduction to control of oscillations and chaos. In: Series оn Nonlinear Science А. Vol. 35. Singapore: World Scientific; 1998. P. 408. DOI: 10.1142/3412.
  10. Boccaletti S, Grebogi C, Lai Y-C, Mancini H, Маzа В. The control оf chaos: theory and applications. Phys. Rep. 2000;392(3):103–197. DOI: 10.1016/S0370-1573(99)00096-4.
  11. Shinbrot T, Grebogi C, Ott Е, Yorke JA. Using small perturbations to control chaos. Nature. 1993;363(6428):411–417. DOI: 10.1038/363411a0; Auerbach D, Grebogi C, Оtt E, Yorke JA. Controlling chaos in high dimensional systems. Phys. Rev. Lett. 1992;69(24):3479–3482. DOI: 10.1103/PhysRevLett.69.3479.
  12. Onsager L, Machlup S. Fluctuations and irreversible processes. Phys. Rev. 1953;91(6):1505–1512. DOI: 10.1103/PhysRev.91.1505.
  13. Dykman М, McClintock PVE, Smelyanskiy VN, Stein ND, Stocks NG. Optimal paths and the prehistory problem for large fluctuations in noise driven systems. Phys. Rev. Lett. 1992;68(18):2718–2721. DOI: 10.1103/PhysRevLett.68.2718.
  14. Luchinsky DG, Maier RS, Mannella R, McClintock PVE, Stein DL. Experiments on critical phenomena in a noisy exit problem. Phys. Rev. Lett. 1997;79(17):3109–3112. DOI: 10.1103/PhysRevLett.79.3109; Luchinsky DG. On the nature of large fluctuations in equilibrium systems: observation оf аn optimal force. J. Phys. А. 1997;30:L577; Luchinsky DG, McClintock PVE. Irreversibility оf classical fluctuations studied in analogue electrical circuits. Nature. 1997;389(6650):463–466. DOI: 10.1038/38963.
  15. Freidlin MI, Wentzel AD. Random Perturbations in Dynamical Systems. New York: Springer; 1984. 328 p. DOI: 10.1007/978-1-4684-0176-9.
  16. Kaurz RL. Activation energy for thermally induced escape from а basin оf attraction. Phys. Lett. А. 1987;125(6–7):315–319. DOI: 10.1016/0375-9601(87)90151-4.
  17. Beale PD. Noise-induced escape from attractor in one-dimensional maps. Phys. Rev. А. 1989;40(7):3998–4003. DOI: 10.1103/PhysRevA.40.3998.
  18. Grassberger P. Noise-induced escape from attractors. J. Phys. A: Math. Gen. 1989;22(16):3283. DOI: 10.1088/0305-4470/22/16/018.
  19. Graham R, Hamm А, Tel Т. Nonequilibrium potentials for dynamical systems with fractal attractors оr repellers. Phys. Rev. Lett. 1991;66(24):3089–3092. DOI: 10.1103/PhysRevLett.66.3089.
  20. Soskin SM, Arrayds M, Mannella R, Silchenko AN. Strong enhancement of noise-induced escape by nonadiabatic periodic driving due to transient chaos. Phys. Rev. Е. 2001;63(5):051111. DOI: 10.1103/PhysRevE.63.051111.
  21. Holmes P. A nonlinear oscillator with а strange attractor. Phil. Trans. В. Soc. A. 1979;292(1394):419–448. DOI: 10.1098/rsta.1979.0068.
  22. Grebogi C, Ott E, Yorke JА. Basin boundaries metamorphoses – changes in accessible boundary orbits. Physica D. 1987;24(1–3):243–262. DOI: 10.1016/0167-2789(87)90078-9.
  23. Grebogi C, Ott E, Yorke JА. Unstable periodic orbits and the dimensions оf multifractal chaotic attractors. Phys. Rev. А. 1988;37(5):1711–1724. DOI: 10.1103/PhysRevA.37.1711.
  24. Dhamala M, Lai Y-С. The natural measure оf nonattracting chaotic sets and in representation by unstable periodic orbits. Int. J. Bifurc. Chaos. 2002;12(12):2991–3005. DOI: 10.1142/S0218127402006308.
  25. Silchenko AN, Luchinsky DG, McClintock PVE. Noise-induced escape through a fractal basin boundaries. Physica A. 2003;327(3–4):371–377. DOI: 10.1016/S0378-4371(03)00265-6.
  26. Gardini L, Mira C, Barugola J, Cathala JC. Chaotic Dynamics in Two-Dimensional Noninvertible Maps. World Scientific Publishing; 1996. 632 p. DOI: 10.1142/2252.
  27. Devanye RL. Introduction to Chaotic Dynamical Systems. New-York: Addison-Wesley; 1989. 336 p.
  28. Bhattacharjee R, Devaney RL. Tying hairs for the structurally stable exponentials. Ergodic Theory and Dynamical Systems. 2000;20(6):1603–1617. DOI: 10.1017/S0143385700000882.
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