A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map

Rajanikant A. Metri, Bhooshan Rajpathak, Kethavath Raghavendra Naik and Mohan Lal Kolhe ()
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Rajanikant A. Metri: Department of Electrical Engineering, Visvesvaraya National Institute of Technology (VNIT), Nagpur 440010, Maharashtra, India
Bhooshan Rajpathak: Department of Electrical Engineering, Visvesvaraya National Institute of Technology (VNIT), Nagpur 440010, Maharashtra, India
Kethavath Raghavendra Naik: Department of Electrical Engineering, National Institute of Technology (NIT), Jamashedpur 831014, Jharkhand, India
Mohan Lal Kolhe: Faculty of Engineering and Science, University of Agder, 4630 Kristiansand, Norway

Mathematics, 2025, vol. 13, issue 15, 1-34

Abstract: In this study, we investigate the nonlinear dynamical behavior of a one-dimensional linear piecewise-smooth discontinuous (LPSD) map with a negative slope, motivated by its occurrence in systems exhibiting discontinuities, such as power electronic converters. The objective of the proposed research is to develop an analytical approach. Analytical conditions are derived for the existence of stable period-1 and period-2 orbits within the third quadrant of the parameter space defined by slope coefficients a < 0 and b < 0 . The coexistence of multiple attractors is demonstrated. We also show that a novel class of orbits exists in which both points lie entirely in either the left or right domain. These orbits are shown to eventually exhibit periodic behavior, and a closed-form expression is derived to compute the number of iterations required for a trajectory to converge to such orbits. This method also enhances the ease of analyzing system stability by mapping the state–variable dynamics using a non-smooth discontinuous map. The analytical findings are validated using bifurcation diagrams, cobweb plots, and basin of attraction visualizations.

Keywords: border collision bifurcation; piecewise smooth map; discontinuous map; chaos; bifurcations analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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