A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map
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 i...
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2025-08-01
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| author | Rajanikant A. Metri Bhooshan Rajpathak Kethavath Raghavendra Naik Mohan Lal Kolhe |
| author_facet | Rajanikant A. Metri Bhooshan Rajpathak Kethavath Raghavendra Naik Mohan Lal Kolhe |
| author_sort | Rajanikant A. Metri |
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| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>. 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. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2025-08-01 |
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| spelling | doaj-art-e9b2f582de75491ab8a09ba7c0dd98532025-08-20T03:36:31ZengMDPI AGMathematics2227-73902025-08-011315251810.3390/math13152518A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous MapRajanikant A. Metri0Bhooshan Rajpathak1Kethavath Raghavendra Naik2Mohan Lal Kolhe3Department of Electrical Engineering, Visvesvaraya National Institute of Technology (VNIT), Nagpur 440010, Maharashtra, IndiaDepartment of Electrical Engineering, Visvesvaraya National Institute of Technology (VNIT), Nagpur 440010, Maharashtra, IndiaDepartment of Electrical Engineering, National Institute of Technology (NIT), Jamashedpur 831014, Jharkhand, IndiaFaculty of Engineering and Science, University of Agder, 4630 Kristiansand, NorwayIn 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>. 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.https://www.mdpi.com/2227-7390/13/15/2518border collision bifurcationpiecewise smooth mapdiscontinuous mapchaosbifurcations analysis |
| spellingShingle | Rajanikant A. Metri Bhooshan Rajpathak Kethavath Raghavendra Naik Mohan Lal Kolhe A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map Mathematics border collision bifurcation piecewise smooth map discontinuous map chaos bifurcations analysis |
| title | A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map |
| title_full | A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map |
| title_fullStr | A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map |
| title_full_unstemmed | A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map |
| title_short | A Study of Periodicities in a One-Dimensional Piecewise Smooth Discontinuous Map |
| title_sort | study of periodicities in a one dimensional piecewise smooth discontinuous map |
| topic | border collision bifurcation piecewise smooth map discontinuous map chaos bifurcations analysis |
| url | https://www.mdpi.com/2227-7390/13/15/2518 |
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