Structurally unstable regular dynamics in 1D piecewise smooth maps,and circle maps |
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| Authors: | Laura Gardini Fabio Tramontana |
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| Institution: | 1. Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India;2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA;3. Department of Electrical Engineering and Bharti School of Telecommunication, Indian Institute of Technology Delhi, New Delhi, India;1. Department DESP,University of Urbino, Italy;2. Faculty of Science and Technology, Pibulsongkram Rajabhat University and Centre of Excellence in Mathematics, PERDO, CHE, Thailand |
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| Abstract: | In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X′ = aX if ∣X∣ < z, X′ = bX if ∣X∣ > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X′ = F(X) if ∣X∣ < z, X′ = G(X) if ∣X∣ > z is also given, proving the conditions leading to a homeomorphism of the circle. |
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