scholarly journals Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

2008 ◽  
Vol 1 (3) ◽  
pp. 487-505 ◽  
Author(s):  
Sven Kosub
Keyword(s):  
Fixed Point ◽  
Dynamical Systems

scholarly journals On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1088 ◽  
Author(s):  
Juan A. Aledo ◽  
Ali Barzanouni ◽  
Ghazaleh Malekbala ◽  
Leila Sharifan ◽  
Jose C. Valverde
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Periodic Structure ◽  
Boolean Functions ◽  
Directed Graphs ◽  
General Pattern ◽  
General Situation ◽  
Point System ◽  
Local Functions

In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system.

scholarly journals Global Dynamical Systems Involving Generalized -Projection Operators and Set-Valued Perturbation in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yun-zhi Zou ◽  
Xi Li ◽  
Nan-jing Huang ◽  
Chang-yin Sun
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Equilibrium Points ◽  
Projection Operators ◽  
Initial Value ◽  
New Class ◽  
The Fixed Point Theorem ◽  
Generalized Projection Operators ◽  
Generalized Dynamical Systems

A new class of generalized dynamical systems involving generalizedf-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.

scholarly journals On the controllability of fractional dynamical systems

2012 ◽  
Vol 22 (3) ◽  
pp. 523-531 ◽  
Author(s):  
Krishnan Balachandran ◽  
Jayakumar Kokila
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Matrix Function ◽  
Sufficient Conditions ◽  
Schauder’S Fixed Point Theorem ◽  
Finite Dimensional ◽  
Linear And Nonlinear ◽  
Controllability Grammian

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.

scholarly journals Rigidity theory for C∗-dynamical systems and the “Pedersen rigidity problem”

2018 ◽  
Vol 29 (03) ◽  
pp. 1850016 ◽  
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Dynamical Systems ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Alternative Formulation ◽  
Crossed Products ◽  
Locally Compact ◽  
Fixed Point Algebra ◽  
Multiplier Algebras

Let [Formula: see text] be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of [Formula: see text] on [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of [Formula: see text] and [Formula: see text] in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of [Formula: see text] and [Formula: see text]. There is an alternative formulation of the problem: an action of the dual group [Formula: see text] together with a suitably equivariant unitary homomorphism of [Formula: see text] give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of [Formula: see text] and two such equivariant unitary homomorphisms of [Formula: see text] that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of [Formula: see text] and [Formula: see text] is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if [Formula: see text] is discrete, this will be the case for all actions of [Formula: see text].

ON LOCAL CONTROL OF CHAOS: THE NEIGHBOURHOOD SIZE

1996 ◽  
Vol 06 (01) ◽  
pp. 169-178 ◽  
Author(s):  
MIRKO PASKOTA
Keyword(s):  
Functional Analysis ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Local Control ◽  
Linear Control ◽  
Control Law ◽  
Control Of Chaos ◽  
Stability Margins ◽  
Chaotic Dynamical Systems ◽  
The Stability

In this paper, conditions for effective local control of nonlinear, chaotic dynamical systems are analyzed. The aim is to determine the size of a neighbourhood of a fixed point in which the system remains stable when subjected to a linear control law. The stability margins are obtained using ideas from functional analysis, and ways of implementing the obtained results in practice are proposed. An implementation of obtained conditions is illustrated on a real application.

DIFFUSION, INTERMITTENCY, AND NOISE-SUSTAINED METASTABLE CHAOS IN THE LORENZ EQUATIONS: EFFECTS OF NOISE ON MULTISTABILITY

2008 ◽  
Vol 18 (06) ◽  
pp. 1749-1758 ◽  
Author(s):  
WEN-WEN TUNG ◽  
JING HU ◽  
JIANBO GAO ◽  
VINCENT A. BILLOCK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Stable Fixed Point ◽  
Lorenz Equations ◽  
Periodic Forcing ◽  
Parameter Region ◽  
Interesting Phenomenon ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors

Multistability is an interesting phenomenon of nonlinear dynamical systems. To gain insights into the effects of noise on multistability, we consider the parameter region of the Lorenz equations that admits two stable fixed point attractors, two unstable periodic solutions, and a metastable chaotic "attractor". Depending on the values of the parameters, we observe and characterize three interesting dynamical behaviors: (i) noise induces oscillatory motions with a well-defined period, a phenomenon similar to stochastic resonance but without a weak periodic forcing; (ii) noise annihilates the two stable fixed point solutions, leaving the originally transient metastable chaos the only observable; and (iii) noise induces hopping between one of the fixed point solutions and the metastable chaos, a three-state intermittency phenomenon.

scholarly journals SPECIAL CURVES AND POSTCRITICALLY FINITE POLYNOMIALS

2013 ◽  
Vol 1 ◽  
Author(s):  
MATTHEW BAKER ◽  
LAURA DE MARCO
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Moduli Space ◽  
Dense Subset ◽  
Rational Curves ◽  
Rational Maps ◽  
Complex Polynomial ◽  
Polynomial Dynamical Systems ◽  
Postcritically Finite ◽  
Cubic Polynomials

AbstractWe study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.

Partial and complete linearisations at stationary points of infinite-dimensional dynamical systems with foliations and applications

1996 ◽  
Vol 126 (5) ◽  
pp. 1067-1085
Author(s):  
W. M. Rivera
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Coupled Systems ◽  
Stationary Points ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Weakly Coupled ◽  
Beam Equations ◽  
Weakly Coupled Systems

In this paper we discuss C1-linearisations of diffeomorphisms and flows on Banach spaces. Strong foliations of the neighbourhood of the fixed point composed of leaves based on successively larger subspaces (similar to those in [14]) are constructed. Generalised gap conditions which involve the width and separation of vertical bands containing the spectrum of a linear operator are imposed to achieve maximal smoothness. The method of proof generalises that of Hartman and of Mora and Solá-Morales. Our theorems apply to weakly coupled systems of damped wave and beam equations.

Sign in / Sign up

Export Citation Format

Share Document