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Chaos in the fractional order nonlinear Bloch equation with delay

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dc.contributor.author Baleanu, Dumitru
dc.contributor.author Magin, Richard
dc.contributor.author Bhalekar, Sachin
dc.contributor.author Daftardar-Gejji, Varsha
dc.date.accessioned 2017-04-18T11:33:06Z
dc.date.available 2017-04-18T11:33:06Z
dc.date.issued 2015-08
dc.identifier.citation Baleanu, D...et al. (2015). Chaos in the fractional order nonlinear Bloch equation with delay. Communications In Nonlinear Science And Numerical Simulation, 25(1-3), 41-49. http://dx.doi.org/10.1016/j.cnsns.2015.01.004 tr_TR
dc.identifier.issn 1007-5704
dc.identifier.uri http://hdl.handle.net/20.500.12416/1528
dc.description.abstract The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (alpha) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, tau = 0, we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at alpha = 0.8548 with subsequent period doubling that leads to chaos at alpha = 0.9436. A periodic window is observed for the range 0.962 < alpha < 0.9858, with chaos arising again as a nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value alpha = 0.8532, and the transition from two to four cycles at alpha = 0.9259. With further increases in the fractional order, period doubling continues until at alpha = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at alpha = 0.8441, and alpha = 0.8635, respectively. However, the system exhibits chaos at much lower values of a (alpha - 0.8635). A periodic window is observed in the interval 0.897 < alpha < 0.9341, with chaos again appearing for larger values of a. In general, as the value of a decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a delay-dependent model that predicts instability in this non-linear fractional order system consistent with the experimental observations of spin turbulence. tr_TR
dc.language.iso eng tr_TR
dc.publisher Elsevier Science BV tr_TR
dc.relation.isversionof 10.1016/j.cnsns.2015.01.004 tr_TR
dc.rights info:eu-repo/semantics/closedAccess
dc.subject Bloch Equation tr_TR
dc.subject Fractional Calculus tr_TR
dc.subject Chaos tr_TR
dc.subject Delay tr_TR
dc.subject Magnetic Resonance tr_TR
dc.subject Relaxation tr_TR
dc.title Chaos in the fractional order nonlinear Bloch equation with delay tr_TR
dc.type article tr_TR
dc.relation.journal Communications In Nonlinear Science And Numerical Simulation tr_TR
dc.identifier.volume 25 tr_TR
dc.identifier.issue 1-3 tr_TR
dc.identifier.startpage 41 tr_TR
dc.identifier.endpage 49 tr_TR
dc.contributor.department Çankaya Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bilgisayar Bölümü tr_TR


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