Border-collision bifurcations in a generalizedpiecewise linear-power map

Zhiying Qin; Jichen Yang; Soumitro Banerjee; Guirong Jiang

doi:10.3934/dcdsb.2011.16.547

Article Contents

2011, Volume 16, Issue 2: 547-567. Doi: 10.3934/dcdsb.2011.16.547

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Border-collision bifurcations in a generalized piecewise linear-power map

1.
School of Mechanical and Electrical Engineering, Hebei University of Science and Technology, Shijiazhuang 050018
2.
Department of Dynamics and Control, Beihang University, Beijing 100191
3.
Indian Institute of Science Education & Research, Kolkata Mohanpur-741252
4.
Department of Computational Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004

Received: January 31, 2010

Revised: September 30, 2010

Published: August 2011

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Abstract

In this paper a class of generalized piecewise smooth maps is studied, which is linear at one side and nonlinear with power dependence at the other side. According to the value of the power in the term $x^z$, the bifurcations occurring in this map are classified into five types: $z>1$, $z=1$, $0<z<1$, $z=0$, and $z<0$. We derive the occurrence conditions of border collision bifurcation and smooth fold and flip bifurcation, especially the codimension-2 bifurcation points describing the interaction between border collision bifurcation and smooth bifurcation. The general results are then applied to the specific cases of the power $z$, and different bifurcation scenarios are shown for individual cases, from which the period-adding scenario is found to be general for any power.

Keywords:

Mathematics Subject Classification: Primary: 37E05, 37E15, 37G35.

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