Research of Periodic Orbits and Chaos Produced in 1-D Liner Piecewise-Smooth Maps with Single Discontinuity, Positive and Negative Slopes
Jayant Mane1, K Vadirajacharya2
1Jayant Mane, Department of Electrical Engineering, DBATU, Lonere, Raigad (Maharashtra), India.
2K Vadirajacharya, Department of Electrical Engineering, DBATU, Lonere, Raigad (Maharashtra), India.
Manuscript received on 12 October 2019 | Revised Manuscript received on 21 October 2019 | Manuscript Published on 02 November 2019 | PP: 853-858 | Volume-8 Issue-2S11 September 2019 | Retrieval Number: B1140982S1119/2019©BEIESP | DOI: 10.35940/ijrte.B1140.0982S1119
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© The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: Last four decades have seen a major development in the theory of piece-wise smooth discontinuous maps for the analysis of typical bifurcation phenomena for systems that can be modelled as such. The major focus of this paper is the analysis of 1-D linear piecewise smooth maps with a discontinuity, one positive and another negative slope. Interestingly, this type of map analysis has been carried out by various authors and the results have been reported in literature. For example, the existence of period adding cascade in particular parameter regions specified by the parameters ‘a’ and ‘b’ was proven. The new range of parameters this work presents are a ϵ (0,1), b ϵ (-1, 0) and a ϵ (0,1), b ϵ (- , -1). Elementary algebraic and geometric tools have been used to analyze the periodicities in the 1-D linear piecewise smooth discontinuous map with respect to parameters a, b, µ and l. Various examples have been illustrated along with the plotted bifurcation curves. The analysis of the behaviour of the system with varied parameter ranges indicates that non-trivial cases are present for negative values of one or more parameters. A sample basin of attraction plot is illustrated as well. Further, an analytic proof for the existence of LnR orbits for the region a ϵ (0,1), b ϵ (-1, 0) was successfully produced, which is unpublished till date. The research concentrates on the theoretical results.
Keywords: Single Produced Theoretical Development.
Scope of the Article: Recent Trends & Developments in Computer Networks