Attractors in piecewise increassing linear maps in the real line
DOI:
https://doi.org/10.37135/ns.01.08.03Keywords:
Topological attractor, Global Attractor, Piecewise linear maps, Transitivity, Invariant ergodic measure.Abstract
Piecewise linear maps appear as mathematical models to describe systems from electrical engineering, physical sciences and economics recently also appears in models of neural activity. . It is considered a family with 4 parameters of piecewise increasing linear maps on the real line R using the theory of piecewise continuous increasing transformations on the compact intervals to study the existence of an attractor set and describe the dynamics of the attractor, verifying the existence of periodic orbits, transitivity and the existence of ergodic invariant measures. The different values of the parameter where the family exhibits an attractor interval are specifically demonstrated. The necessary and sufficient conditions are tested for the attractor set to be globally attractor. in fact, in this case it is proved that the dynamics of said attractor behaves like the dynamics of the Poincaré rotation on the unit circle. Also, it is described under what conditions in the parameters the family exhibits a topological attractor. Finally, the existence of absolutely continuous ergodic invariant measures to the Lebesgue measure associated with the family restricted to the attractor is proven, the case in which the measure is equivalent to the Lebesgue measure is even tested.
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Avrutin, V., Gardini, L., Schanz, M., & Sushko, I. (2014). Bifurcations of Chaotic Attractors in One-Dimensional Piecewise Smooth Maps. Http://Dx.Doi.Org/10.1142/S0218127414400124, 24(8).
Avrutin, V., Schanz, M., & Banerjee, S. (2006). Multi-parametric bifurcations in a piecewise–linear discontinuous map. Nonlinearity, 19(8), 1875. https://doi.org/10.1088/0951-7715/19/8/007
Banerjee, S., & Verghese, G. C. (2001). Nonlinear phenomena in power electronics : attractors, bifurcations, chaos, and nonlinear control. 441.
Barrientos, P. G. (2015). A Family of Eventually Expanding Piecewise Linear Maps of the Interval. The American Mathematical Monthly, 122(7), 674–680. https://doi.org/10.4169/amer.math.monthly.122.7.674
Belyaev, A., & Ryazanova, T. (2019). Stochastic sensitivity of attractors for a piecewise smooth neuron model. Https://Doi.Org/10.1080/10236198.2019.1678596, 25(9–10), 1468–1487. https://doi.org/10.1080/10236198.2019.1678596
Boyarsky, A., & Góra, P. (1997). Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension. 420. http://www.amazon.co.uk/Laws-Chaos-Invariant-Probability-Applications/dp/0817640037
Choi, Y. (2004). Attractors from one dimensional lorenz-like maps. Discrete and Continuous Dynamical Systems, 11(2–3), 715–730. https://doi.org/10.3934/DCDS.2004.11.715
Coelho, Z., Lopes, A., & da Rocha, L. F. (1995). Absolutely continuous invariant measures for a class of affine interval exchange maps. Proceedings of the American Mathematical Society, 123(11), 3533–3533. https://doi.org/10.1090/s0002-9939-1995-1322918-6
Du, R. H., Wang, S. J., Jin, T., & Qu, S. X. (2018). Phase order in one-dimensional piecewise linear discontinuous map. Chinese Physics B, 27(10), 100502. https://doi.org/10.1088/1674-1056/27/10/100502
Eslami, P., & Góra, P. (2011). On eventually expanding maps of the interval. American Mathematical Monthly, 118(7), 629–635. https://doi.org/10.4169/amer.math.monthly.118.07.629
Glendinning, P., & Jeffrey, M. R. (2019). An Introduction to Piecewise Smooth Dynamics. http://link.springer.com/10.1007/978-3-030-23689-2
GÓRA, P. (2009). Invariant densities for piecewise linear maps of the unit interval. Ergodic Theory and Dynamical Systems, 29(5), 1549–1583. https://doi.org/10.1017/S0143385708000801
Jain, P., & Banerjee, S. (2003). Border-collision bifurcations in one-dimensional discontinuous maps. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 13(11), 3341–3351. https://doi.org/10.1142/S0218127403008533
Milnor, J. (1985). On the concept of attractor. Communications in Mathematical Physics, 99(2), 177–195. https://doi.org/10.1007/BF01212280
Morales, C. A., & Pujals, E. R. (1997). Singular strange attractors on the boundary of Morse-Smale systems. Annales Scientifiques de l’École Normale Supérieure, 30(6), 693–717. https://doi.org/10.1016/S0012-9593(97)89936-3
NUSSE, H. E., & YORKE, J. A. (1995). BORDER-COLLISION BIFURCATIONS FOR PIECEWISE SMOOTH ONE-DIMENSIONAL MAPS. Http://Dx.Doi.Org/10.1142/S0218127495000156, 05(01), 189–207. https://doi.org/10.1142/S0218127495000156
Parry, W. (1964). Representations for real numbers. Acta Mathematica Academiae Scientiarum Hungaricae, 15(1–2), 95–105. https://doi.org/10.1007/BF01897025
Parry, William. (1979). The lorenz attractor and a related population model. Ergodic Theory. Lecture Notes in Mathematics, 729, 169–187. https://doi.org/10.1007/BFB0063293
Rajpathak, B., Pillai, H. K., & Bandyopadhyay, S. (2012). Analysis of stable periodic orbits in the one dimensional linear piecewise-smooth discontinuous map. Chaos (Woodbury, N.Y.), 22(3), 033126. https://doi.org/10.1063/1.4740061
Rajpathak, B., Pillai, H. K., & Bandyopadhyay, S. (2015). Analysis of unstable periodic orbits and chaotic orbits in the one-dimensional linear piecewise-smooth discontinuous map. Chaos, 25(10). https://doi.org/10.1063/1.4929382
Rényi, A. (1957). Representations for real numbers and their ergodic properties. Acta Mathematica Academiae Scientiarum Hungaricae, 8(3–4), 477–493. https://doi.org/10.1007/BF02020331
Tramontana, F., & Gardini, L. (2011). Border collision bifurcations in discontinuous one-dimensional linear-hyperbolic maps.
Communications in Nonlinear Science and Numerical Simulation, 16(3), 1414–1423. https://doi.org/10.1016/J.CNSNS.2010.06.012
Viana, M., & Oliveira, K. (2016). Foundations of Ergodic Theory. Foundations of Ergodic Theory. https://doi.org/10.1017/CBO9781316422601
Wilkinson, K. M. (1974). Ergodic properties of certain linear mod one transformations. Advances in Math., 14, 64–72.
