Addendum to: ``A fixed point theorem for bounded dynamical systems'' [Illinois J. Math. 46 (2002), no. 2, 491--495]

Abstract This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

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ON SOME NONAUTONOMOUS CHAOTIC SYSTEM ON THE PLANE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127499001322 ◽

1999 ◽

Vol 09 (09) ◽

pp. 1853-1858 ◽

Cited By ~ 6

Author(s):

KLAUDIUSZ WÓJCIK

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Fixed Point Theorem ◽

Chaotic System ◽

Point Theorem ◽

Chaotic Behavior ◽

Topological Methods ◽

Lefschetz Fixed Point Theorem ◽

Nonautonomous Equations ◽

Time Periodic

We prove the existence of the chaotic behavior in dynamical systems generated by some class of time periodic nonautonomous equations on the plane. We use topological methods based on the Lefschetz Fixed Point Theorem and the Ważewski Retract Theorem.

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Controllability of Nonlinear Fractional Dynamical Systems with a Mittag–Leffler Kernel

Mathematics ◽

10.3390/math8122139 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2139

Author(s):

Jiale Sheng ◽

Wei Jiang ◽

Denghao Pang ◽

Sen Wang

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Fixed Point Theorem ◽

Laplace Transform ◽

Point Theorem ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

This paper is concerned with controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel. First, the solution of fractional dynamical systems with a Mittag–Leffler kernel is given by Laplace transform. In addition, one necessary and sufficient condition for controllability of linear fractional dynamical systems with Mittag–Leffler kernel is established. On this basis, we obtain one sufficient condition to guarantee controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel by fixed point theorem. Finally, an example is given to illustrate the applicability of our results.

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The controllability of nonlinear implicit fractional delay dynamical systems

International Journal of Applied Mathematics and Computer Science ◽

10.1515/amcs-2017-0035 ◽

2017 ◽

Vol 27 (3) ◽

pp. 501-513 ◽

Cited By ~ 2

Author(s):

Rajagopal Joice Nirmala ◽

Krishnan Balachandran

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Fixed Point Theorem ◽

Point Theorem ◽

Fractional Derivatives ◽

Sufficient Conditions ◽

Distributed Delays ◽

Multiple Delays ◽

Darbo Fixed Point Theorem ◽

Fractional Delay

AbstractThis paper is concerned with the controllability of nonlinear fractional delay dynamical systems with implicit fractional derivatives for multiple delays and distributed delays in control variables. Sufficient conditions are obtained by using the Darbo fixed point theorem. Further, examples are given to illustrate the theory.

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Counting Periodic Points in Parallel Graph Dynamical Systems

Complexity ◽

10.1155/2020/9708347 ◽

2020 ◽

Vol 2020 ◽

pp. 1-9

Author(s):

Juan A. Aledo ◽

Ali Barzanouni ◽

Ghazaleh Malekbala ◽

Leila Sharifan ◽

Jose C. Valverde

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Lower Bound ◽

Fixed Point Theorem ◽

Fixed Points ◽

Point Theorem ◽

Periodic Points ◽

Point Of View ◽

Dominating Sets ◽

Minimum Number

Let F:0,1n⟶0,1n be a parallel dynamical system over an undirected graph with a Boolean maxterm or minterm function as a global evolution operator. It is well known that every periodic point has at most two periods. Actually, periodic points of different periods cannot coexist, and a fixed point theorem is also known. In addition, an upper bound for the number of periodic points of F has been given. In this paper, we complete the study, solving the minimum number of periodic points’ problem for this kind of dynamical systems which has been usually considered from the point of view of complexity. In order to do this, we use methods based on the notions of minimal dominating sets and maximal independent sets in graphs, respectively. More specifically, we find a lower bound for the number of fixed points and a lower bound for the number of 2-periodic points of F. In addition, we provide a formula that allows us to calculate the exact number of fixed points. Furthermore, we provide some conditions under which these lower bounds are attained, thus generalizing the fixed-point theorem and the 2-period theorem for these systems.

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PERIODICITY AND SHARKOVSKY'S THEOREM FOR RANDOM DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493701000199 ◽

2001 ◽

Vol 01 (03) ◽

pp. 299-338 ◽

Cited By ~ 11

Author(s):

MARC KLÜNGER

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Fixed Point Theorem ◽

Periodic Orbits ◽

Point Theorem ◽

Random Dynamical Systems ◽

Random Fixed Point ◽

One Dimensional ◽

Sharkovsky's Theorem ◽

Random Fixed Point Theorem

We generalize the deterministic notion of periodicity to random dynamical systems, which leads to three different objects, called random periodic orbits, point and cycles. We analyze the relation of these three notions and prove a "random fixed point theorem" for one-dimensional random dynamical systems. Finally we use these notions to prove partial generalizations of Sharkovsky's theorem to random dynamical systems.

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