Abstract
Whether Wada basin boundaries can occur typically in dynamical systems other than smooth systems has been an open question. We verify the existence of Wada basin boundaries in a switched Hénon map. We combine two basin boundaries (non-Wada property) but when they are alternated in periodic manners the Wada basin boundaries can be created. We give some mathematically rigorous results guaranteeing these emerging Wada basin boundaries by the auxiliary dynamics method. It suggests that switching can also induce the consequence of the existence of a high number of possible final states.
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Li, G.X., Moon, F.C.: Fractal basin boundaries in a two-degree-of-freedom nonlinear system. Nonlinear Dyn. 1, 209–219 (1990)
Mcdonald, S.W., Grebogi, C., Ott, E., Yorke, J.A.: Fractal basin boundaries. Physica D 17, 125–153 (1985)
Yue, X., Xu, W., Zhang, Y.: Global bifurcation analysis of Rayleigh–Duffing oscillator through the composite cell coordinate system method. Nonlinear Dyn. 69, 437–457 (2012)
Hong, L., Xu, J.: Chaotic saddles in Wada basin boundaries and their bifurcations by the generalized cell-mapping digraph (GCMD) method. Nonlinear Dyn. 32, 371–385 (2003)
Moon, F.C., Li, G.X.: Fractal basin boundaries and homoclinic orbits for periodic motions in a two-well potential. Phys. Rev. Lett. 55, 1439–1442 (1985)
Kennedy, J., Yorke, J.A.: Basin of Wada. Physica D 51, 213–225 (1991)
Aguirre, J., Viana, R.L., Sanjuán, M.A.F.: Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81, 333–386 (2009)
Aguirre, J., Sanjuán, M.A.F.: Unpredictable behavior in the Duffing oscillator: Wada basins. Physica D 171, 41–51 (2002)
Zhang, Y., Luo, G.: Unpredictability of the Wada property in the parameter plane. Phys. Lett. A 376, 3060–3066 (2012)
Viana, R.L., Da Silva, E.C., Kroetz, T., Caldas, I.L., Roberto, M., Sanjuán, M.A.F.: Fractal structures in nonlinear plasma physics. Philos. Trans. R. Soc. Lond. A 369, 371–395 (2011)
Bellido, F., Ramirez-Malo, J.B.: Periodic and chaotic dynamics of a sliding driven oscillator with dry friction. Int. J. Non-Linear Mech. 41, 860–871 (2006)
Vandermeer, J.: Wada basins and qualitative unpredictability in ecological models: a graphical interpretation. Ecol. Model. 176, 65–74 (2004)
Seoane, J.M., Sanjuán, M.A.F.: New developments in classical chaotic scattering. Rep. Prog. Phys. 76, 016001 (2013)
Liberzon, D.: Switching in System and Control. Birkhauser, Cambridge (2003)
Afraimovich, V., Ashwin, P., Kirk, V.: Robust heteroclinic and switching dynamics. Dyn. Syst. 25, 285–286 (2010)
Kirk, V., Lane, E., Postlethwaite, C.M., Rucklidge, A.M., Silber, M.: A mechanism for switching near a heteroclinic network. Dyn. Syst. 25, 323–349 (2010)
Harmer, G.P., Abbott, D.: Losing strategies can win by Parrondo’s paradox. Nature 402, 864 (1999)
Allison, A., Abbott, D.: Control systems with stochastic feedback. Chaos 11, 715–724 (2001)
Parrondo, J.M.R., Dinis, L.: Brownian motion and gambling: from ratchets to paradoxical games. Contemp. Phys. 45, 147–157 (2004)
Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order? Physica D 200, 124–132 (2005)
Maier, M.P.S., Peacock-Lopez, E.: Switching induced oscillations in the logistic map. Phys. Lett. A 374, 1028–1032 (2010)
Zhang, Y., Luo, G.: A special type of codimension two bifurcation and unusual dynamics in a phase-modulated system with switched strategy. Nonlinear Dyn. 67, 2727–2734 (2012)
Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69–77 (1976)
Hénon, M.: Numerical study of quadratic area preserving maps. Q. Appl. Math. 27, 291–312 (1969)
Nusse, H.E., Yorke, J.A.: Wada basin boundaries and basin cells. Physica D 90, 242–261 (1996)
Nusse, H.E., Ott, E., Yorke, J.A.: Saddle-node bifurcations on fractal basin boundaries. Phys. Rev. Lett. 75, 2482–2485 (1995)
Breban, R., Nusse, H.E.: On the creation of Wada basins in interval maps through fixed point tangent bifurcation. Physica D 207, 52–63 (2005)
Zhang, Y., Kong, G.: Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system. Phys. Lett. A 374, 208–213 (2009)
Huisman, J., Weissing, F.J.: Fundamental unpredictability in multispecies competition. Am. Nat. 157, 488–494 (2001)
Aguirre, J., Vallejo, J.C., Sanjuan, M.A.F.: Wada basins and unpredictability in Hamiltonian and dissipative systems. Mod. Phys. Lett. B 24, 4171–4175 (2003)
Alligood, K., Yorke, J.A.: Accessible saddles on fractal basin boundaries. Ergod. Theory Dyn. Syst. 12, 377–400 (1992)
Nusse, H.E., Yorke, J.A.: Dynamics: Numerical Explorations, 2nd edn. Springer, New York (1997)
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The author thanks the anonymous reviewers for their helpful comments and suggestions, which led to an improvement of this paper. Some computations have been made using the Software DYNAMICS [32]. This work was supported by the National Natural Science Foundation of China (No. 11002092).
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Zhang, Y. Switching-induced Wada basin boundaries in the Hénon map. Nonlinear Dyn 73, 2221–2229 (2013). https://doi.org/10.1007/s11071-013-0936-2
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DOI: https://doi.org/10.1007/s11071-013-0936-2