Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n -Valued Impulses

Jan Andres
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Jan Andres: Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

Mathematics, 2020, vol. 8, issue 10, 1-21

Abstract: Ordinary differential equations with n -valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).

Keywords: impulsive differential equations; n-valued maps; Hutchinson-Barnsley operators; multiple periodic solutions; topological fractals; Devaney’s chaos on attractors; Poincaré operators; Nielsen number (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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