Next Article in Journal
A Hybrid Personnel Scheduling Model for Staff Rostering Problems
Next Article in Special Issue
A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit
Previous Article in Journal
A Class of Fractional Degenerate Evolution Equations with Delay
Open AccessArticle

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

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
Mathematics 2020, 8(10), 1701; https://doi.org/10.3390/math8101701
Received: 10 September 2020 / Revised: 28 September 2020 / Accepted: 30 September 2020 / Published: 3 October 2020
(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)
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). View Full-Text
Keywords: impulsive differential equations; n-valued maps; Hutchinson-Barnsley operators; multiple periodic solutions; topological fractals; Devaney’s chaos on attractors; Poincaré operators; Nielsen number impulsive differential equations; n-valued maps; Hutchinson-Barnsley operators; multiple periodic solutions; topological fractals; Devaney’s chaos on attractors; Poincaré operators; Nielsen number
Show Figures

Figure 1

MDPI and ACS Style

Andres, J. Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses. Mathematics 2020, 8, 1701.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Search more from Scilit
 
Search
Back to TopTop