Modelling of chaos in smooth piecewise dynamical systems with one discontinuous point

Document Type : Original Paper

Authors

1 Department of Mathematics, Shhid Beheshti University, Tehran, Iran

2 Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran

3 Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract

In this paper, we provide conditions on the smooth piecewise dynamical systems that guarantee the existence of Devaney chaos. In fact, we show that if f is a generalized semi-baker map with two branches and its derivative greater than or equal √2, then the dynamical system related to that is chaotic in the sense of Devaney. Such conditions on the dynamical systems with more than one discontinues point essentially does not satisfy this result.

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References

[1]    ‎Banerjee‎, S. and‎ ‎Verghese, G.C‎. (2001)‎. ‎Nonlinear Phenomena in Power‎ Electronics‎: ‎Attractors‎, ‎Bifurcations‎, ‎Chaos‎, ‎and Nonlinear Control‎, ‎IEEE‎ Press‎, ‎New York‎‎.
[2]    ‎Bischi‎, ‎G.L‎. and ‎Chiarella‎, ‎C. and ‎Kopel‎, ‎M. and ‎Szidarovszky‎, F. (2009). Nonlinear oligopolies‎: ‎Stability and bifurcations‎, ‎Heidelberg‎, Springer‎.
[3]    Devaney, R. (1986). An introduction to chaotic dynamical systems. The Benjamin, Cummings Publishing Corporation, Menlo Park, California.
[4]    Bernardo, M.D and Budd, C.J. and Champneys, A.R. and Kowalczyk, P. (2008). Piecewise-smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences.
[5]    Grosse-Erdmann, K. G. and Manguillot, A. P. (2011). Linear Chaos, Springer Verlag, London.
[6]    ‎Makrooni‎, ‎R. and ‎Gardini‎, ‎L. and ‎Sushko‎, I. (2015). ‎Bifurcation structures in‎ a family of 1D discontinuos linear-hyperbolic invertible maps‎, ‎Int‎. ‎J‎. ‎Bifurcation and Chaos, 25, 1530039
[7]    Makrooni‎, R. and Khellat, F. and L. Gardini, L. (2015). Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: Unbounded chaotic sets, Journal of Difference Equations and Applications, 21, 660-695.
[8]    ‎Makrooni‎, R. and ‎Khellat, F. and ‎Gardini‎, L. (2015). ‎Border collision and fold bifurcations in a family of piecesiwe smooth maps‎: ‎Divergence and Bounded Dynamics, 21, 791-824.
[9]    Makrooni, R. and Abbasi, N. and Pourbarat, M. and Gardini, L. (2015). Robust unbounded chaotic attractors in 1D discontinuous maps, Chaos Solitons & Fractals, 77, 310-318.
[10]           Nordmark, A.B. (1991). Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145, 279-297.
[11]           Nordmark‎, A.B. (1997). ‎Universal limit mapping in grazing‎ bifurcations‎, ‎Physical Review E‎, ‎55, 266-270‎.
[12]           Puu‎, ‎T‎. and ‎Sushko‎, I. (2002). ‎Oligopoly Dynamics‎, ‎Models and Tools‎, Springer Verlag‎, ‎New York‎‎.
[13]           Puu‎, ‎T. ‎Sushko‎,I. (2006) ‎Business Cycle Dynamics‎, ‎Models and Tools‎, Springer Verlag‎, ‎New York‎.
[14]           Tramontana‎, ‎F. and ‎Gardini‎, ‎L‎. and ‎Ferri‎, P. ‎(2010). The dynamics of the NAIRU‎ model with two switching regimes, J‎. ‎Econ‎. ‎Dyn‎. ‎Control, 34, 681-695‎.
[15]           Tramontana‎, ‎F‎. ‎and Westerhoff‎, ‎F. and ‎Gardini‎, L. (2010). ‎On the complicated‎ price dynamics of a simple one-dimensional discontinuous financial market‎ model with heterogeneous interacting traders‎, ‎J‎. ‎Econ‎. ‎Behav‎. ‎Organ‎. ‎74, 187-205‎.
Journal of Advanced Mathematical Modeling
Volume 9, Issue 2
September 2019
Pages 93-105

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History

  • Receive Date: 26 May 2018
  • Revise Date: 06 February 2019
  • Accept Date: 11 March 2019

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