PERIODIC ORBITS IN TWO CLASSES OF PIECEWISE SMOOTH MAPS WITH POSITIVE NONLINEAR PARTS

Zhengdong Du

doi:10.11948/2017013

2017 Volume 7 Issue 1

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Zhengdong Du. PERIODIC ORBITS IN TWO CLASSES OF PIECEWISE SMOOTH MAPS WITH POSITIVE NONLINEAR PARTS[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 189-206. doi: 10.11948/2017013

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Zhengdong Du. PERIODIC ORBITS IN TWO CLASSES OF PIECEWISE SMOOTH MAPS WITH POSITIVE NONLINEAR PARTS[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 189-206. doi: 10.11948/2017013

PERIODIC ORBITS IN TWO CLASSES OF PIECEWISE SMOOTH MAPS WITH POSITIVE NONLINEAR PARTS

Zhengdong Du

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

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Abstract

Abstract

In this paper we consider two classes of one dimensional piecewise smooth continuous maps that have been derived as normal forms for grazing bifurcations of piecewise smooth dynamical systems. These maps are linear on one side of the phase space and nonlinear on the other side. The case of nonlinear parts with negative coefficients has been studied previously and it is proved that period-adding scenarios are generic in this case. In contrast to this result, in our analytical and numerical results, the period-adding scenarios are not observed when the nonlinear parts have positive coefficients. Furthermore, our results suggest that the typical bifurcation scenario is period doubling cascade leading to chaos in this case, which is similar to that of the smooth logistic map.
- Piecewise smooth map /
- period doubling bifurcation /
- grazing bifurcation /
- chaos
MSC: 37E05;37G10

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