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Articles

Bubbling, riddling, blowout and critical curves

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Pages 939-964 | Received 10 Oct 2016, Accepted 28 Oct 2016, Published online: 19 Apr 2017

References

  • R. Abraham, C. Mira, and L. Gardini, Chaos in Discrete Dynamical Systems (A Visual Introduction in Two Dimension), Springer, New York, 1997.
  • J. Alexander, J.A. Yorke, Z. You, and I. Kan, Riddled basins, Int. J. Bifur. Chaos 2 (1992), pp. 795–813.
  • P. Ashwin, J. Buescu, and I. Stewart, Bubbling of attractors and synchronisation of chaotic oscillators, Phys. Lett. A 193 (1994), pp. 126–139.
  • P. Ashwin, J. Buescu, and I. Stewart, From attractor to chaotic saddle: A tale of transverse instability, Nonlinearity 9 (1996), pp. 703–737.
  • G.I. Bischi and L. Cerboni Baiardi, Fallacies of composition in nonlinear marketing models, Commun. Nonlinear Sci. Numer. Simul. 20 (2015), pp. 209–228.
  • G.I. Bischi, L. Cerboni Baiardi, and D. Radi, On a discrete-time model with replicator dynamics in renewable resource exploitation, J. Differ. Equ. Appl. 21 (2015), pp. 954–973.
  • G.I. Bischi, M. Gallegati, and A. Naimzada, Symmetry-breaking bifurcations and representativefirm in dynamic duopoly games, Ann. Oper. Res. 89 (1999), pp. 252–271.
  • G.I. Bischi and L. Gardini, Role of invariant and minimal absorbing areas in chaos synchronization, Phys. Rev. E 58 (1998), pp. 5710–5719.
  • G.I. Bischi and L. Gardini, Global properties of symmetric competition models with riddling and blowout phenomena, Discrete Dyn. Nat. Soc. 5 (2000), pp. 149–160.
  • G.I. Bischi, L. Gardini, and M. Kopel, Analysis of global bifurcations in a market share attraction model, J. Econ. Dyn. Control 24 (2000), pp. 855–879.
  • G.I. Bischi and F. Lamantia, Chaos synchronization and intermittency in a duopoly game with spillover effects, in Oligopoly Dynamics, T. Puu and I. Sushko, eds., Springer, Berlin, 2002, pp. 195–217.
  • G.I. Bischi and F. Lamantia, Nonlinear duopoly games with positive cost externalities due to spillover effects, Chaos Solitons Fractals 13 (2002), pp. 701–721.
  • G.I. Bischi, L. Stefanini, and L. Gardini, Synchronization, intermittency and critical curves in a duopoly game, Math. Comput. Simul. 44 (1998), pp. 559–585.
  • J. Buescu, Exotic Attractors: From Lyapunov Stability to Riddled Basins, Birkhäuser, Basel, 1997.
  • P. Collet and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Springer Science & Business Media, Berlin, 2009.
  • A. Ferretti and N. Rahman, A study of coupled logistic map and its applications in chemical physics, Chem. Phys. 119 (1988), pp. 275–288.
  • H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Prog. Theor. Phys. 69 (1983), pp. 32–47.
  • I. Gumowski and C. Mira, Dynamique Chaotique: Transformations Ponctuelles, Transition Ordre-désordre, Cepadues Editions, Toulouse, 1980.
  • M. Hasler and Y.L. Maistrenko, An introduction to the synchronization of chaotic systems: Coupled skew tent maps, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 44 (1997), pp. 856–866.
  • J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. 40 (2003), pp. 479–519.
  • Y.C. Lai and C. Grebogi, Noise-induced riddling in chaotic systems, Phys. Rev. Lett. 77 (1996), pp. 5047–5051.
  • Y. Maistrenko, T. Kapitaniak, and P. Szuminski, Locally and globally riddled basins in two coupled piecewise-linear maps, Phys. Rev. E 56 (1997), pp. 6393–6399.
  • Y.L. Maistrenko, V. Maistrenko, A. Popovich, and E. Mosekilde, Role of the absorbing area in chaotic synchronization, Phys. Rev. Lett. 80 (1998), pp. 1638–1641.
  • Y.L. Maistrenko, V. Maistrenko, A. Popovich, and E. Mosekilde, Transverse instability and riddled basins in a system of two coupled logistic maps, Phys. Rev. E 57 (1998), pp. 2713–2724.
  • J. Milnor, On the concept of attractor, in The Theory of Chaotic Attractors, Brian R. Hunt, Tien-Yien Li, Judy A. Kennedy and Helena E. Nusse, eds., Springer, 1985, pp. 243–264.
  • C. Mira, Chaotic Dynamics: From the One-dimensional Endomorphism to the Two-dimensional Diffeomorphism, World Scientific, Singapore, 1987.
  • C. Mira, J.P. Carcasses, G. Millérioux, and L. Gardini, Plane foliation of two-dimensional noninvertible maps, Int. J. Bifur. Chaos 6 (1996), pp. 1439–1462.
  • C. Mira, L. Gardini, A. Barugola, and J.C. Cathala, Chaotic Dynamics in Two-dimensional Noninvertible Maps, World Scientific, Singapore, 1996.
  • Y. Nagai and Y.C. Lai, Periodic-orbit theory of the blowout bifurcation, Phys. Rev. E 56 (1997), pp. 4031–4041.
  • E. Ott and J.C. Sommerer, Blowout bifurcations: The occurrence of riddled basins and on-off intermittency, Phys. Lett. A 188 (1994), pp. 39–47.
  • L.M. Pecora and T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990), pp. 821–825.
  • L.M. Pecora and T.L. Carroll, Synchronization of chaotic systems, Chaos 25 (2015), pp. 097611–22.
  • A.S. Pikovsky and P. Grassberger, Symmetry breaking bifurcation for coupled chaotic attractors, J. Phys. A 24 (1991), p. 4587–4597.
  • P.D. Taylor and L.B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci. 40 (1978), pp. 145–156.
  • S.C. Venkataramani, B.R. Hunt, and E. Ott, Bubbling transition, Phys. Rev. E 54 (1996), pp. 1346–1360.
  • S.C. Venkataramani, B.R. Hunt, E. Ott, D.J. Gauthier, and J.C. Bienfang, Transitions to bubbling of chaotic systems, Phys. Rev. Lett. 77 (1996), pp. 5361–5364.
  • A.V. Zimin, B.R. Hunt, and E. Ott, Bifurcation scenarios for bubbling transition, Phys. Rev. E 67 (2003), pp. 016204–16.

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