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

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Listed:
  • 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)

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.

Suggested Citation

  • Rajanikant A. Metri & Bhooshan Rajpathak & Kethavath Raghavendra Naik & Mohan Lal Kolhe, 2025. "A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map," Mathematics, MDPI, vol. 13(15), pages 1-34, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:15:p:2518-:d:1718148
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    References listed on IDEAS

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    1. Enas Taha Sayed & Abdul Ghani Olabi & Abdul Hai Alami & Ali Radwan & Ayman Mdallal & Ahmed Rezk & Mohammad Ali Abdelkareem, 2023. "Renewable Energy and Energy Storage Systems," Energies, MDPI, vol. 16(3), pages 1-26, February.
    2. Luis Javier Ontañón-García & Juan Gonzalo Barajas-Ramírez & Eric Campos-Cantón & Daniel Alejandro Magallón-García & César Arturo Guerra-García & José Ricardo Cuesta-García & Jonatan Pena-Ramirez & Jos, 2025. "Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane," Mathematics, MDPI, vol. 13(11), pages 1-20, May.
    3. Gardini, Laura & Tramontana, Fabio, 2011. "Border collision bifurcation curves and their classification in a family of 1D discontinuous maps," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 248-259.
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