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Articles

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: divergence and bounded dynamics

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Pages 791-824 | Received 21 Mar 2015, Accepted 18 Apr 2015, Published online: 19 Jun 2015

References

  • F. Angulo, M. di Bernardo, E. Fossas, and G. Olivar, Feedback control of limit cycles: A switching control strategy based on nonsmooth bifurcation theory, IEEE Trans. Circuits Syst.-I 52(2) (2005), pp. 366–378. 10.1109/TCSI.2004.841595.
  • V. Avrutin, M. Schanz, and S. Banerjee, Codimension-three bifurcations: explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps, Phys. Rev. E 75(6) (2007), p. 066205. doi:10.1103/PhysRevE.75.066205.
  • V. Avrutin, P.S. Dutta, M. Schanz, and S. Banerjee, Influence of a square-root singularity on the behaviour of piecewise smooth maps, Nonlinearity 23(2) (2010), pp. 445–463. doi:10.1088/0951-7715/23/2/012.
  • M. di Bernardo, C.J. Budd, A.R. Champneys, and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, in Applied Mathematical Sciences, 163, Springer-Verlag, London, 2008.
  • M. di Bernardo, C. Budd, and A. Champneys, Grazing, skipping and sliding: analysis of the non-smooth dynamics of the DC/DC buck converter, Nonlinearity 11(4) (1998), pp. 859–890. doi:10.1088/0951-7715/11/4/007.
  • M. di Bernardo, C.J. Budd, and A.R. Champneys, Corner collision implies border-collision bifurcation, Phys. D 154(3–4) (2001), pp. 171–194. doi:10.1016/S0167-2789(01)00250-0.
  • M. di Bernardo, P. Kowalczyk, and A.B. Nordmark, Bifurcations of dynamical systems with sliding: derivation of normal-form mappings, Phys. D 170(3–4) (2002), pp. 175–205. doi:10.1016/S0167-2789(02)00547-X.
  • L. Gardini, Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding periodic points, Nonlinear Anal. Theory Methods Appl. 23(8) (1994), pp. 1039–1089. doi:10.1016/0362-546X(94)90198-8.
  • L. Gardini, I. Sushko, V. Avrutin, and M. Schanz, Critical homoclinic orbits lead to snap-back repellers, Chaos Solitons Fract. 44 (2011), pp. 33–49.
  • O. Makarenkov, and S.W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Phys. D 241(22) (2012), pp. 1826–1844. doi:10.1016/j.physd.2012.08.002.
  • R. Makrooni, and L. Gardini, Bifurcation structures in a family of one-dimensional linear-power discontinuous maps, Gecomplexity Discussion Paper N.7 ISSN: 2409-7497. Available at http://econpapers.repec.org/paper/cstwpaper/ 2015.
  • R. Makrooni, N. Abbasi, M. Pourbarat, and L. Gardini, Robust unbounded chaotic attractors in 1D discontinuous maps, Chaos Solitons Fract. (in press).
  • R. Makrooni, F. Khellat, and L. Gardini, Border collision and fold bifurcations in a family of piecesiwe smooth maps: Unbounded chaotic sets, J. Diff. Equ. Appl. (in press, 2015)
  • R. Makrooni, L. Gardini, and I. Sushko, Bifurcation structures in a family of 1D discontinuos linear-hyperbolic invertible maps, Submitted for publication 2015.
  • F. Marotto, Snap-back repellers imply chaos in Rn, J. Math. Anal. Appl. 63(1) (1978), pp. 199–223. doi:10.1016/0022-247X(78)90115-4.
  • F. Marotto, On redefining a snap-back repeller, Chaos Solitons Fract. 25(1) (2005), pp. 25–28. doi:10.1016/j.chaos.2004.10.003.
  • A.B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, J. Sound Vibr. 145(2) (1991), pp. 279–297. doi:10.1016/0022-460X(91)90592-8.
  • A.B. Nordmark, Universal limit mapping in grazing bifurcations, Phys. Rev. E 55(1) (1997), pp. 266–270. doi:10.1103/PhysRevE.55.266.
  • Z. Qin, J. Yang, S. Banerjee, and G. Jiang, Border-collision bifurcations in a generalized piecewise linear-power map, Discrete Continuous Dyn. Sys. Ser. B 16(2) (2011), pp. 547–567. doi:10.3934/dcdsb.2011.16.547.
  • Z. Qin, Y. Zhao, and J. Yang, Nonsmooth and smooth bifurcations in a discontinuous piecewise map, Int. J. Bifurcation Chaos 22(5) (2012), 1250112 (7 pages).10.1142/S021812741250112X.

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