Lozi map embedded into the 2D border collision normal form

References

• D. Aharonov, R. Devaney, and U. Elias, The dynamics of a piecewise linear map and its smooth approximation, Int. J. Bif. Chaos 7 (1997), pp. 351–372.
   Web of Science ®Google Scholar
• V. Avrutin, L. Gardini, and M. Schanz, On a special type of border-collision bifurcations occurring at infinity, Phys. D 239 (2010), pp. 1083–1094.
   Web of Science ®Google Scholar
• V. Avrutin, L. Gardini, I. Sushko, and F. Tramontana, Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures, Nonlinear Science Series A, Vol. 95, World Scientific, 2019. https://doi.org/10.1142/8285.
   Google Scholar
• V. Avrutin, M. Schanz, and S. Banerjee, Occurrence of multiple attractor bifurcations in the two-dimensional piecewise linear normal form map, Nonlinear Dyn. 67 (2011), pp. 293–307.
   Web of Science ®Google Scholar
• V. Avrutin, Zh.T. Zhusubaliyev, A. Saha, S. Banerjee, L. Gardini, and I. Sushko, Dangerous bifurcations revisited, Int. J. Bifurcat. Chaos 26 (2017), Article ID 1630040.
   Web of Science ®Google Scholar
• M. Aziz-Alaoui, C. Robert, and C. Grebogi, Dynamics of a Hénon–Lozi-type map, Chaos Solit. Fractals 12 (2001), pp. 2323–2341.
   Web of Science ®Google Scholar
• S. Banerjee and C. Grebogi, Border collision bifurcation in two-dimensional piecewise smooth maps, Phys. Rev. E 59 (1999), pp. 4052–4061.
   Web of Science ®Google Scholar
• S. Banerjee and G.C. Verghese, Nonlinear Phenomena in Power Electronics – Attractors, Bifurcations, Chaos, and Nonlinear Control, IEEE Press, 2001.
   Google Scholar
• S. Banerjee, J.A. Yorke, and C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80 (1998), pp. 3049–3052.
   Web of Science ®Google Scholar
• G. Bischi, C. Chiarella, M. Kopel, and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations, Springer Science & Business Media, 2010.
   Google Scholar
• V. Botella–Soler, J.M. Castelo, J.A. Oteo, and J. Ros, Bifurcations in the Lozi map, J. Phys. A: Math. Theor. 44 (2011), Article ID 305101.
   PubMed Web of Science ®Google Scholar
• B. Brogliato, Nonsmooth Mechanics, Communications and Control Engineering, Springer, 1999.
   Google Scholar
• P. Collet and Y. Levy, Ergodic Properties of the Lozi Mappings, The Theory of Chaotic Attractors, Springer, 1984, pp. 222–242.
   Google Scholar
• M. di Bernardo, C.J. Budd, A.R. Champneys, and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences Vol. 163, Springer, 2008.
   Google Scholar
• M. Dutta, H. Nusse, E. Ott, J.A. Yorke, and G. Yuan, Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems, Phys. Rev. Lett. 83 (1999), pp. 4281–4284.
   Web of Science ®Google Scholar
• Z. Elhadj, Lozi Mappings: Theory and Applications, CRC Press, 2013.
   Google Scholar
• A. Ganguli and S. Banerjee, Dangerous bifurcation at border collision: When does it occur? Phys. Rev. E 71 (2005), Article ID 057202.
   Web of Science ®Google Scholar
• L. Gardini, V. Avrutin, and M. Schanz, Connection between bifurcations on the Poincaré equator and dangerous bifurcations, in Iteration Theory, A. Sharkovsky and I. Sushko, eds., Grazer Math. Ber., 2009, pp. 53–72.
   Google Scholar
• L. Gardini and W. Tikjha, Role of the virtual fixed point in the center bifurcations in a family of piecewise linear maps, Int. J. Bif. Chaos 29 (2019), Article ID 1930041.
   Web of Science ®Google Scholar
• M. Hassouneh, E. Abed, and H. & Nusse, Robust dangerous border-collision bifurcations in piecewise smooth systems, Phys. Rev. E 92 (2004), Article ID 070201.
   Web of Science ®Google Scholar
• M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys. 50 (1976), pp. 69–77.
   Web of Science ®Google Scholar
• Y. Ishii, Towards a kneading theory for Lozi mappings. II: Monotonicity of the topological entropy and hausdorff dimension of attractors, Commun. Math. Phys. 190 (1997), pp. 375–394.
   Web of Science ®Google Scholar
• R. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, Vol. 18, Springer Science & Business Media, 2013.
   Google Scholar
• H. Li, K. Li, M. Chen, and B. Bao, Coexisting infinite orbits in an area-preserving Lozi map, Entropy 22 (2020), p. 1119. https://doi.org/10.3390/e22101119.
   PubMed Web of Science ®Google Scholar
• R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon, J. De Phys. Colloque 39 (1978), pp. C5–9.
   Google Scholar
• M. Misiurewicz, Strange attractors for the Lozi mappings, Ann. N. Y. Acad. Sci. 357 (1980), pp. 348–358.
   Google Scholar
• M. Misiurewicz and S. Štimac, Symbolic dynamics for Lozi maps, Nonlinearity 29 (2016), pp. 3031–3046.
   Web of Science ®Google Scholar
• E. Mosekilde and J.L. Laugesen, Nonlinear dynamic phenomena in the beer model, Syst. Dyn. Rev. 23 (2007), pp. 229–252.
   Web of Science ®Google Scholar
• H.E. Nusse and J.A. Yorke, Border-collision bifurcations including 'period two to period three' bifurcation for piecewise smooth systems, Phy. D 57 (1992), pp. 39–57.
   Web of Science ®Google Scholar
• H.E. Nusse and J.A. Yorke, Border-collision bifurcations for piecewise smooth one dimensional maps, Int. J. Bif. Chaos 5 (1995), pp. 189–207.
   Web of Science ®Google Scholar
• D. Simpson, Sequences of periodic solutions and infinitely many coexisting attractors in the border-collision normal form, Int. J. Bif. Chaos 24 (2014), Article ID 1430018.
   Web of Science ®Google Scholar
• D. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, Nonlinear Science A, Vol. 70, World Scientific, 2010.
   Google Scholar
• D. Simpson and J. Meiss, Neimark–Sacker bifurcations in planar, piecewise-smooth, continuous maps, SIAM J. Appl. Dyn. Syst. 7 (2008), pp. 795–824.
   Web of Science ®Google Scholar
• I. Sushko, V. Avrutin, and L. Gardini, Bifurcation structure in the skew tent map and its application as a border collision normal form, J. Differ. Equ. Appl. 22 (2016), pp. 1040–1087.
   Web of Science ®Google Scholar
• I. Sushko, V. Avrutin, and L. Gardini, Center bifurcation in the Lozi map, Int. J. Bif. Chaos 31 (2021), Article ID 2130046.
   Web of Science ®Google Scholar
• I. Sushko and L. Gardini, Center bifurcation for a two-dimensional border-collision normal form, Int. J. Bif. Chaos 18 (2008), pp. 1029–1050.
   Web of Science ®Google Scholar
• I. Sushko and L. Gardini, Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps, Int. J. Bif. Chaos 20 (2010), pp. 2045–2070. https://doi.org/10.1142/S021812741-0026927.
   Web of Science ®Google Scholar
• Zh. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, Nonlinear Science A, Vol. 44, World Scientific, 2003.
   Google Scholar
• Zh. Zhusubaliyev, E. Mosekilde, S.M. Maity, S. Mohanan, and S. Banerjee, Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation, Chaos 16 (2006), Article ID 023122.
   PubMed Web of Science ®Google Scholar