حداكثر تعداد سيكل هاي حدي براي برخي دستگاه هاي ديناميكي قطعه اي خطي

چكيده انگليسي :

Maximum Number of Limit Cycles for Certain Picewise Linear Dynamical Systems Athar Hosseinpour atharhoseinpour@gmail com 2019 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Rasoul Asheghi r asheghi@cc iut ac irMSC 34C05 34C07 37G15 Keywords discontinuous differntial system limit cycle piecewise linear differential system Abstract Non smooth dynamical systems emerge in a natural way modeling many real processes and phenomena for instance recently piecewise linear differential equations appeared as idealized models of cell activity Due to that in these lastyears the mathematical community became very interested in understanding the dynamics of these kind of systems Ingeneral some of the main source of motivation to study non smooth systems can be found in control theory impact andfriction mechanics nonlinear oscillations economics and biology In this thesis we are interested in discontinuouspiecewise linear differential systems The study of this particular class of non smooth dynamical systems has startedwith Andronov and coworkers We start with a historical fact Lum and Chua conjectured that a continuous piecewiselinear vector field in the plane with two zones separated by a straight line which is the easiest example of this kindof system has at most one limit cycle This conjecture was proved by Freire et al Even this relatively easy casedemanded a hard work to show the existence of at most one limit cycle In this these we address the problem of Lumand Chua fornon sliding limit cycles extended to the class of discontinuous piecewise linear differential systemsin the plane with two zones separated by a straight line Here a non sliding limit cycle is a limit cycle that doesnot contain any sliding segment in This problem is very related to the Hilbert s 16th problem Limit cyclesof discontinuous piecewise linear differential systems with two zones separated by astraight line have been studiedrecently by several authors Nevertheless the problem of Lum and Chua remains open for this class of differentialequations In this work we give a partial solution for this problem We note the authors proved that if one of the twolinear systems has its singular point on the discontinuity straight line then the number of limit cycles of such a systemis at most 4 these results reduce this upper bound to 2 and additionally we prove that it is reached Our point ofinterest in the Lum and Chua problem is aligned with two directions which face serious technical difficulties First while solutions in each linear region are easy to find the times of passage along the regions are not simple to achieve It means that matching solutions across regions is a very difficult task Second to control all possible configurationsone must generally consider a large number of parameters It was conjectured that a planar piecewise linear differentialsystems with two zones separated by a straight line have at most 2 non sliding limit cycles A negative answer forthis conjecture was provided in via a numerical example having 3 non sliding limit cycles Analytical proofs for theexistence of these 3 limit cycles were given in Finally in it was studied general conditions to obtain 3 non slidinglimit cycles in planar piecewise linear differential systems with two zones separated by a straight line Recently perturbative techniques were used together with newly developed tools on Chebyshev systems to obtain 3 limit cyclesin such systems when they are near to non smooth centers When a general curve of discontinuity is considered insteadof a straight line there is no upper bound for the maximum number of non sliding limit cycles that a system of thisfamily can have It is a consequence of a conjecture stated by Braga and Mello and then proved by Novaes and Ponce In this these we deal with planar vector fields Z expressed as z F z sign x G z where z x y R2 and F and G are linear vector fields in R 2 or equivalently x 0 if X

حسين پور، اطهر

حداكثر تعداد سيكل هاي حدي براي برخي دستگاه هاي ديناميكي قطعه اي خطي

دستگاه ديفرانسيل ناپيوسته , دستگاه ديفرانسيل قطعه اي خطي , سيكل حدي , كران بالا