Points Of Discrete Dynamical Systems Research Articles

A predictive TDM-PON resource allocation using the ANN method based on equilibrium points of discrete dynamical systems

Resource allocation is one of the challenges in the time division multiple access (TDM) passive optical network (PON) area. Much research has been done to find an efficient resource allocation method. Using machine learning (ML) and finding proper ML algorithms in predictive dynamic bandwidth allocation remains an open research question. This paper aims to predict resource allocation accurately and efficiently based on historical requests. Artificial neural networks (ANN) with historically allocated bandwidth from different time series are used to build the prediction model. This model is constructed by the normal equation method. The proposed model needs a small dataset for its training phase, while its accuracy is higher than other models. The discrete dynamical system builds to find an equilibrium point, which is a stable point in the system. The proposed model can be implemented as a pre-training, or it can be dynamically refined when the model is running by using some of its output as feedback. The system evaluation verifies that the proposed DBA (EQ-ANN) elevates the quality of service (QoS) metrics, such as system throughput, packet drop probability, and packet means delay. The results show that the system throughput improves by up to 5%, and packet loss improves by up to 15%. Moreover, the prediction accuracy of the system reached 99.48%.

Finding Nontrivial Minimum Fixed Points in Discrete Dynamical Systems: Complexity, Special Case Algorithms and Heuristics

Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the dissemination of undesirable contagions (such as rumors and misinformation), convergence to fixed points with a small number of affected nodes is a desirable goal. Motivated by such considerations, we formulate a novel optimization problem of finding a nontrivial fixed point of the system with the minimum number of affected nodes. We establish that, unless P = NP, there is no polynomial-time algorithm for approximating a solution to this problem to within the factor n^(1 - epsilon) for any constant epsilon > 0. To cope with this computational intractability, we identify several special cases for which the problem can be solved efficiently. Further, we introduce an integer linear program to address the problem for networks of reasonable sizes. For solving the problem on larger networks, we propose a general heuristic framework along with greedy selection methods. Extensive experimental results on real-world networks demonstrate the effectiveness of the proposed heuristics. A full version of the manuscript, source code and data are available at: https://github.com/bridgelessqiu/NMIN-FPE

Pre-Stability of Fixed Point

In this paper, we introduced certain types of stability of the fixed points in discrete dynamical systems which are pre-stability, pre-c-stability, and pre-ic-stability. We studied the relationships among these types of stability, also the relationships among these types of stability and certain types of stability which are stability, c-stability, and ic-stability

The parameterization method for center manifolds

In this paper, we present a generalization of the parameterization method, introduced by Cabré, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system.

The stability of fixed points of discrete dynamical systems in the space conv ℝ n

Conditions for the asymptotic Lyapunov stability of the fixed points of discrete dynamical systems in the space conv ℝn are established.

A classification of slow convergence near parametric periodic points of discrete dynamical systems

We study the phenomenon of slow convergence in families of discrete dynamical systems where the iteration function has a Puiseux series representation. Such occurrence consists in the slow convergence of orbits near non-hyperbolic parametric periodic points. We provide a precise new definition of the slowness of convergence which is based on literature results for the critical exponents associated with parametric periodic points. Such exponents establish a general classification for slow systems and provide a measure of rates of convergence. For dynamical systems whose iteration functions have Taylor series expansions, the new definition is natural with wider applicability. However, it can be also used for iteration functions where a more sophisticated approach, such as a Lagrange expansion, is needed. In addition, we show that even for such iteration functions, the critical exponent can be easily computed. The presented theoretical results are illustrated by numerical examples having different rates of convergence.

Positive circuits and maximal number of fixed points in discrete dynamical systems

We consider a product X of n finite intervals of integers, a map F from X to itself, the asynchronous state transition graph Γ ( F ) on X that Thomas proposed as a model for the dynamics of a network of n genes, and the interaction graph G ( F ) that describes the topology of the system in terms of positive and negative interactions between its n components. Then, we establish an upper bound on the number of fixed points for F , and more generally on the number of attractors in Γ ( F ) , which only depends on X and on the topology of the positive circuits of G ( F ) . This result generalizes the following discrete version of Thomas’ conjecture recently proved by Richard and Comet: If G ( F ) has no positive circuit, then Γ ( F ) has a unique attractor. This result also generalizes a result on the maximal number of fixed points in Boolean networks obtained by Aracena, Demongeot and Goles. The interest of this work in the context of gene network modeling is briefly discussed.

On Finding a Fixed Point in a Boolean Network with Maximum Indegree 2

Finding fixed points in discrete dynamical systems is important because fixed points correspond to steady-states. The Boolean network is considered as one of the simplest discrete dynamical systems and is often used as a model of genetic networks. It is known that detection of a fixed point in a Boolean network with n nodes and maximum indegree K can be polynomially transformed into (K+1)-SAT with n variables. In this paper, we focus on the case of K = 2 and present an O(1.3171n) expected time algorithm, which is faster than the naive algorithm based on a reduction to 3-SAT, where we assume that nodes with indegree 2 do not contain self-loops. We also show an algorithm for the general case of K = 2 that is slightly faster than the naive algorithm.

Mode decomposition of global bifurcation diagram with Gröbner bases

We propose an algebraic method for the mode decomposition of bifurcation diagram in global parameter space. We investigate the bifurcation behavior of some periodic points of discrete dynamical systems. By using Gröbner bases, we can obtain the algebraic representation of bifurcation diagrams which gives global characteristics of bifurcation diagrams. We also show that the factorization of the algebraic representation gives the mode decomposition of global bifurcation diagrams. Further, we demonstrate that the separation removes singularities on bifurcation points. The essential point of our method is to make use of the ideal generated by the equation of a system.

NUMERICAL COMPUTATIONS OF CONNECTING ORBITS IN DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in nonautonomous and autonomous systems of ordinary differential equations. First, the existence and computation of connecting orbits between fixed points in discrete dynamical systems is discussed; then it is shown that the problem of finding connections between equilibria and periodic solutions in continuous systems may be reduced to finding connections between fixed points in a discrete system. Implementation of the method is considered: the choice of a linear solver is discussed and phase conditions are suggested for the discrete system. The paper concludes with some numerical examples: connections for equilibria and periodic orbits are computed for discrete systems and for nonautonomous and autonomous systems, including systems arising from the discretization of a partial differential equation.

Rigorous chaos verification in discrete dynamical systems

In this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems. The existence of such points guarantees that a Smale horseshoe is embedded in the mapping which therefore is chaotic. We take special interest in the 2-dimensional case and describe a method to obtain a computer-assisted proof of chaos for these systems. Numerical results are presented for the standard map.