Fractal basin boundaries and escape dynamics in a multiwell potential
The escape dynamics in a two-dimensional multiwell potential is explored. A thorough numerical investigation is conducted in several types of two-dimensional planes and also in a three-dimensional subspace of the entire four-dimensional phase space in order to distinguish between non-escaping (ordered and chaotic) and escaping orbits. The determination of the location of the basins of escape toward the different escape channels and their correlations with the corresponding escape time of the orbits is undoubtedly an issue of paramount importance. It was found that in all examined cases regions of non-escaping motion coexist with several basins of escape. Furthermore, we monitor how the percentages of all types of orbits evolve when the total orbital energy varies. The larger escape periods have been measured for orbits with initial conditions in the fractal basin boundaries, while the lowest escape rates belong to orbits with initial conditions inside the basins of escape. The Newton–Raphson basins of attraction of the equilibrium points of the system have also been determined. We hope that our numerical analysis will be useful for a further understanding of the escape mechanism of orbits in open Hamiltonian systems with two degrees of freedom.
KeywordsHamiltonian systems Numerical simulations Escapes Fractals
I would like to express my warmest thanks to the four anonymous referees for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.
Compliance with Ethical Standards
The author states that he has not received any research grants.
Conflict of interest
The author declares that he has no conflict of interest.
- 7.Berezovoj, V.P., Bolotin, Yu.L., Ivashkevych, G.I.: Geometrical approach for description of the mixed state in multi-well potentials. XIII International Seminar “Nonlinear Phenomena in Complex Systems”, Minsk, Belarus, May 16–19 (2006)Google Scholar
- 11.Bolotin, YuL, Cherkaskiy, V.A., Ivashkevych, G.I.: Over-barrier decay of a mixed state in 2D multiwell potentials. Ukr. J. Phys. 55, 838–847 (2010)Google Scholar
- 12.Churchill, R.C., et al.: In Como Conference Proceedings on Stochastic Behavior in Classical and Quantum Hamiltonian Systems. Casati, G., Fords, J (ed.) Lecture Notes in Physics, Vol. 93, Springer, Berlin, 76 (1979)Google Scholar
- 32.Lyapunov, A.M.: Problème gènèral de las stabilitè de movement. Ann. Fac. Sci. Toulouse 9, 203–274 (1907)Google Scholar
- 33.Navarro, J.F., Henrard, J.: Spiral windows for escaping stars. Astron. Astrophys. 369, 1112–1121 (2001)Google Scholar