Finite Dynamical Systems Research Articles

Polynomial-delay generation of functional digraphs up to isomorphism

We describe a procedure for the generation of functional digraphs up to isomorphism; these are digraphs with uniform outdegree 1, also called mapping patterns, finite endofunctions, or finite discrete-time dynamical systems. This procedure is based on a reverse search algorithm for the generation of connected functional digraphs, which is then applied as a subroutine for the generation of arbitrary ones. Both algorithms output solutions with O(n2) delay and require linear space with respect to the number n of vertices.

Factorisation in the semiring of finite dynamical systems

Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct product of their respective graphs; the collection of FDSs endowed with these operations then forms a semiring. The algebraic structure of the product of FDSs is particularly interesting. For instance, an FDS can be factorised if and only if it is composed of two sub-systems running in parallel. In this work, we further the understanding of the factorisation, division, and root finding problems for FDSs. Firstly, an FDS A is cancellative if one can divide by it unambiguously, i.e. AX=AY implies X=Y. We prove that an FDS A is cancellative if and only if it has a fixed point. Secondly, we prove that if an FDS A has a k-th root (i.e. B such that Bk=A), then it is unique. Thirdly, unlike integers, the monoid of FDS product does not have unique factorisation into irreducibles. We instead exhibit a large class of monoids of FDSs with unique factorisation. To obtain our main results, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. This allows us to work with (possibly infinite) trees, where the product is easier to handle than its counterpart for FDSs.

Количество аттракторов и циклических состояний в конечных динамических системах ориентаций полных графов

Graph models occupy an important place in tasks related to information protection and information security, including in the construction of models and methods for managing the continuous operation and recovery of systems, and countering denial of service. We consider a finite dynamical system (Γkn, α), n ⩾ 1, the states of which are all possible orientations of the complete graph Kn, and the evolution function is specified as follows: the dynamic image of the digraph is a digraph obtained from the original one by reorienting all the arcs included in the sinks; there are no other differences between the original digraph and its image. Formulas are obtained for calculating the number of cyclic (belonging to attractors) states of the system; states that are not cyclical; attractors of the system, including various types. The corresponding tables are given for n from 1 to 20 inclusive.

Stochastic Process Leading to Catalan Number Recurrence

Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the system, represented by partitions of a subset of {1,2,…,N}, are derived using basic combinatorics. The existence and uniqueness of a stationary state are shown using Markov chains terminology. A well-defined mean-field type approximation is used to obtain block size distribution and the consistency of the approach is verified. It is shown that this recurrence asymptotically takes the form of Catalan number recurrence for particular dynamics parameters of the system.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields, the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates. The initial growth directions depend on the Jacobians of Poincaré map on that cycle, for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincaré sections. The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems. Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration. A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

Asynchronous, finite dynamical systems

Asynchronous, finite dynamical systems

Covariant influences for finite discrete dynamical systems.

We develop a rigorous theory of external influences on finite discrete dynamical systems, going beyond the perturbation paradigm, in that the external influence need not be a small contribution. Indeed, the covariance condition can be stated as follows: If we evolve the dynamical system for n time steps and then disturb it, then it is the same as first disturbing the system with the same influence and then letting the system evolve for n time steps. Applying the powerful machinery of resource theories, we develop a theory of covariant influences both when there is a purely deterministic evolution and when randomness is involved. Subsequently, we provide necessary and sufficient conditions for the transition between states under deterministic covariant influences and necessary conditions in the presence of stochastic covariant influences, predicting which transitions between states are forbidden. Our approach, for the first time, employs the framework of resource theories, borrowed from quantum information theory, to the study of finite discrete dynamical systems. The laws we articulate unify the behavior of different types of finite discrete dynamical systems, and their mathematical flavor makes them rigorous and checkable.

An algorithmic pipeline for solving equations over discrete dynamical systems modelling hypothesis on real phenomena

This paper provides an algorithmic pipeline for studying the intrinsic structure of a finite discrete dynamical system (DDS) modelling an evolving phenomenon. Here, by intrinsic structure we mean, regarding the dynamics of the DDS under observation, the feature of resulting from the ‘cooperation’ of the dynamics of two or more smaller DDS. The intrinsic structure is described by an equation over DDS which represents a hypothesis over the phenomenon under observation. The pipeline allows solving such an equation, i.e., validating the hypothesis over the phenomenon, as far the asymptotic behaviour and the number of states of the DDS under observation are concerned. The results are about the soundness and completeness of the pipeline and they are obtained by exploiting the algebraic setting for DDS introduced in Dennunzio et al. (2018).

Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions

In this paper, we investigate the complexity of a number of computational problems defined on a synchronous boolean finite dynamical system, where update functions are chosen from a template set of exclusive-or and its negation. We first show that the reachability and path-intersection problems are solvable in logarithmic space-uniform AC1 if the objects execute permutations, while the reachability problem is known to be in P and the path-intersection problem to be in UP in general. We also explore the case where the reachability or intersection are tested on a subset of objects, and show that this hardens complexity of the problems: both problems become NP-complete, and even \({\Pi }^{p}_{2}\)-complete if we further require universality of the intersection. We next consider the exact cycle length problem, that is, determining whether there exists an initial configuration that yields a cycle in the configuration space having exactly a given length, and show that this problem is NP-complete. Lastly, we consider the t-predecessor and t-Garden of Eden problem, and prove that these are solvable in polynomial time even if the value of t is also given in binary as part of instance, and the two problems are in logarithmic space-uniform NC2 if the value of t is given in unary as part of instance.

Counting distinct functional graphs from linear finite dynamical systems

Let Fq be the finite field with q elements and, for each positive integer n, let Aq(n) be the number of non isomorphic functional graphs arising from Fq-linear maps T:Fqn→Fqn. In 2013, Bach and Bridy proved that, for every prime power q, we have that 12≤liminfn→+∞log⁡log⁡Aq(n)log⁡n≤limsupn→+∞log⁡log⁡Aq(n)log⁡n≤1. By combining some ideas from linear algebra, combinatorics and number theory, in this paper we provide sharper estimates on the function Aq(n) and, in particular, we prove that limn→+∞⁡log⁡log⁡Aq(n)log⁡n=1 for every prime power q.

Emergence and algorithmic information dynamics of systems and observers.

One of the challenges of defining emergence is that one observer’s prior knowledge may cause a phenomenon to present itself as emergent that to another observer appears reducible. By formalizing the act of observing as mutual perturbations between dynamical systems, we demonstrate that the emergence of algorithmic information does depend on the observer’s formal knowledge, while being robust vis-a-vis other subjective factors, particularly: the choice of programming language and method of measurement; errors or distortions during the observation; and the informational cost of processing. This is called observer-dependent emergence (ODE). In addition, we demonstrate that the unbounded and rapid increase of emergent algorithmic information implies asymptotically observer-independent emergence (AOIE). Unlike ODE, AOIE is a type of emergence for which emergent phenomena will be considered emergent no matter what formal theory an observer might bring to bear. We demonstrate the existence of an evolutionary model that displays the diachronic variant of AOIE and a network model that displays the holistic variant of AOIE. Our results show that, restricted to the context of finite discrete deterministic dynamical systems, computable systems and irreducible information content measures, AOIE is the strongest form of emergence that formal theories can attain.This article is part of the theme issue ‘Emergent phenomena in complex physical and socio-technical systems: from cells to societies’.

Dynamical systems with benign ghosts

We consider finite and infinite-dimensional ghost-ridden dynamical systems whose Hamiltonians involve nonpositive definite kinetic terms. We point out the existence of three classes of such systems where the ghosts are benign, i.e., systems whose evolution is unlimited in time: (i) systems obtained from the variation of bounded-motion systems; (ii) systems describing motions over certain Lorentzian manifolds and (iii) higher-derivative models related to certain modified Korteweg-de Vries equations.

On the dynamics of semilattice networks

Finite dynamical systems are used to model problems from different branches of sciences such as biological networks, computer process, physics, and engineered control systems. An important combinatorial tool for studying limit cycles of a finite dynamical system is dependency graph. In Veliz-Cuba and Laubenbacher (2019), cycle structure of semilattice networks has completely been characterized when the dependency graph is strongly connected. This characterization is in terms of the loop number of the dependency graph.In this paper, we use the methods developed in Jarrah et al. (2010); Veliz-Cuba and Laubenbacher (2019) and we study cycle structure of a semilattice network with an arbitrary dependency graph. We show that the period of any limit cycle divides the loop number. Next, we prove that all periodic points of a semilattice network are fixed points if and only if the loop number of dependency graph is 1. We also find some sufficient conditions under which system has periodic points of any period that divides the loop number. Then, using the same idea as Chen et al. (2020), we give a reduction process in studying cycle structure of a semilattice networks. Finally, we completely characterized the set of fixed points of a semilattice networks in terms of the set of isotone maps between two certain partially ordered sets. Using this characterization, we give a sharp lower bound for the number of fixed points. The results in this paper apply to certain types of Boolean networks and diffusion models, and in particular extends and recovers some of the results of Chen et al. (2020); Jarrah et al. (2010).

Factorisation in the Semiring of Finite Dynamical Systems

Factorisation in the Semiring of Finite Dynamical Systems

Existence, coexistence and uniqueness of fixed points in parallel and sequential dynamical systems over directed graphs

In this work, we solve the fixed-point existence, coexistence and uniqueness problems in the context of homogeneous parallel and sequential dynamical systems on maxterm and minterm Boolean functions over directed dependency graphs. More specifically, we give characterizations for the existence of fixed points. Likewise, we provide a sufficient condition and a necessary one for the non-coexistence of other periods in the dynamics of the system. In addition, we state a necessary and sufficient condition for the uniqueness of a fixed point, what allows us to establish a Fixed-Point Theorem in the sense of Banach for this kind of systems. Finally, we give an advance in the analysis of the number of fixed points of such systems, finding a lower bound for this number in the case of the most simplest maxterm and minterm. Numerical examples are given to show the feasibility of the results. Thus, this work completes the study of the fixed-point problems for homogeneous Boolean finite dynamical systems on maxterm and minterm functions.

Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

AbstractWe generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

Isometries of the hypercube: A tool for Boolean regulatory networks analysis

Boolean finite dynamical systems (FDS) are commonly used in systems biology to model the dynamics of intracellular regulatory networks and interpret the emergence of cellular behaviors. Given a Boolean FDS, we can compute the corresponding regulatory network, that is a directed signed graph representing all the interactions between components (genes), endowed with logical rules specifying the dynamical behavior of the system. We consider the asynchronous trajectories generated by this Boolean FDS, supported by the hypercube. The exploration and analysis of this dynamics is a challenging task because of the combinatorial explosion that we face. A way to approach this problem is to exploit the links between the regulatory graph and the dynamics.We use the isometries of the hypercube to define classes gathering all the isometric Boolean FDS. Thus, we classify the set of Boolean FDS on the basis of those isometries, and emphasize their common features through regulatory graphs and logical rules. We can then restrict the dynamical analysis of all the Boolean FDS to one representative per class, and thereby considerably improve the efficiency of analysis of all the Boolean FDS. Relying on invariants properties, we propose a constructive method to provide, given a FDS, a representative regulatory graph of its class.We illustrate the efficiency of the method in concrete situations. For instance, the motif analysis (Remy et al., 2003; Remy et al., 2016; Didier and Remy, 2012) is strongly improved thanks to this classification. We also revisit the negative Thomas’ rule (Remy and Ruet, 2008; Richard, 2010) by establishing a new demonstration.

Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system

<p style='text-indent:20px;'>At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system.</p><p style='text-indent:20px;'>For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs.</p><p style='text-indent:20px;'>In this paper we give a positive aftermath by proving the existence of pullback <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-attractor. The key point is to find a bounded family of pullback <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback <inline-formula><tex-math id="M4">\begin{document}$ w $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M5">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.</p>

Classification of maps on a finite set under permutation

We characterize when two maps f,g:X→X on a finite set are conjugated, that is, when there is a permutation σ:X→X such that f=σ−1gσ. We build a signature sgn (f) for every map f such that f and g are conjugated if and only if sgn (f)=sgn (g). This signature is an array that includes information about the cycles of the map and the noncyclic elements, called transients, besides data about the insertion of the transients in the cycle. The transient elements form several tree shape graphs. Our characterization is a generalization of the known fact that two permutations are conjugated if and only if they have the same cycle structure. We use elementary facts about finite dynamical systems and about the canonical labeling of unordered trees. The signature can be built by an algorithm of complexity O(n), being n the cardinality of X.