Theory Of Symbolic Dynamics Research Articles

On the Dynamical Complexity of a Seasonally Forced Discrete SIR Epidemic Model with a Constant Vaccination Strategy

In this article, we consider the discretized classical Susceptible‐Infected‐Recovered (SIR) forced epidemic model to investigate the consequences of the introduction of different transmission rates and the effect of a constant vaccination strategy, providing new numerical and topological insights into the complex dynamics of recurrent diseases. Starting with a constant contact (or transmission) rate, the computation of the spectrum of Lyapunov exponents allows us to identify different chaotic regimes. Studying the evolution of the dynamical variables, a family of unimodal‐type iterated maps with a striking biological meaning is detected among those dynamical regimes of the densities of the susceptibles. Using the theory of symbolic dynamics, these iterated maps are characterized based on the computation of an important numerical invariant, the topological entropy. The introduction of a degree (or amplitude) of seasonality, ε, is responsible for inducing complexity into the population dynamics. The resulting dynamical behaviors are studied using some of the previous tools for particular values of the strength of the seasonality forcing, ε. Finally, we carry out a study of the discrete SIR epidemic model under a planned constant vaccination strategy. We examine what effect this vaccination regime will have on the periodic and chaotic dynamics originated by seasonally forced epidemics.

Matrix approach to trajectory control of higher‐order k ‐valued logical control networks

This study investigates the trajectory controllability of higher-order k-valued logical control networks, and presents a number of new results. First, the higher-order k-valued logical control networks are considered as the mappings from the space of input trajectories to the space of output trajectories, based on which the continuity and surjectivity of higher-order k-valued logical control networks are analysed via the theory of symbolic dynamics. Second, as the concept for trajectory controllability of higher-order k-valued logical control networks is defined, an equivalent test criterion is presented for the trajectory controllability via the semi-tensor product method. Moreover, an effective method is proposed to find the control sequence for the trajectory controllability. Finally, an illustrative example is worked out to support the obtained new results.

Invertibility of higher order k-valued logical control networks and its application in trajectory control

This paper investigates the invertibility of higher order k-valued logical control networks, and presents a number of new results. First, the higher order k-valued logical control networks are considered as the mappings from the space of input trajectories to the space of output trajectories, based on which the continuity, injectivity and surjectivity of higher order k-valued logical control networks are analyzed via the theory of symbolic dynamics. Second, as the concept for invertibility of higher order k-valued logical control networks is defined, an equivalent test criterion for invertibility and a method of constructing the inverse system are given via the semi-tensor product method. Third, as the concept for trajectory controllability of higher order k-valued logical control networks is defined, the invertibility is proved to be a sufficient condition for the trajectory controllability. Finally, an illustrative example is worked out to support the obtained new results.

Topological chaos of the spatial prisoner׳s dilemma game on regular networks

The spatial version of evolutionary prisoner׳s dilemma on infinitely large regular lattice with purely deterministic strategies and no memories among players is investigated in this paper. Based on the statistical inferences, it is pertinent to confirm that the frequency of cooperation for characterizing its macroscopic behaviors is very sensitive to the initial conditions, which is the most practically significant property of chaos. Its intrinsic complexity is then justified on firm ground from the theory of symbolic dynamics; that is, this game is topologically mixing and possesses positive topological entropy on its subsystems. It is demonstrated therefore that its frequency of cooperation could not be adopted by simply averaging over several steps after the game reaches the equilibrium state. Furthermore, the chaotically changing spatial patterns via empirical observations can be defined and justified in view of symbolic dynamics. It is worth mentioning that the procedure proposed in this work is also applicable to other deterministic spatial evolutionary games therein.

Dynamic Data-Driven Prediction of Lean Blowout in a Swirl-Stabilized Combustor

This paper addresses dynamic data-driven prediction of lean blowout (LBO) phenomena in confined combustion processes, which are prevalent in many physical applications (e.g., land-based and aircraft gas-turbine engines). The underlying concept is built upon pattern classification and is validated for LBO prediction with time series of chemiluminescence sensor data from a laboratory-scale swirl-stabilized dump combustor. The proposed method of LBO prediction makes use of the theory of symbolic dynamics, where (finite-length) time series data are partitioned to produce symbol strings that, in turn, generate a special class of probabilistic finite state automata (PFSA). These PFSA, called D-Markov machines, have a deterministic algebraic structure and their states are represented by symbol blocks of length D or less, where D is a positive integer. The D-Markov machines are constructed in two steps: (i) state splitting, i.e., the states are split based on their information contents, and (ii) state merging, i.e., two or more states (of possibly different lengths) are merged together to form a new state without any significant loss of the embedded information. The modeling complexity (e.g., number of states) of a D-Markov machine model is observed to be drastically reduced as the combustor approaches LBO. An anomaly measure, based on Kullback-Leibler divergence, is constructed to predict the proximity of LBO. The problem of LBO prediction is posed in a pattern classification setting and the underlying algorithms have been tested on experimental data at different extents of fuel-air premixing and fuel/air ratio. It is shown that, over a wide range of fuel-air premixing, D-Markov machines with D > 1 perform better as predictors of LBO than those with D = 1.

Invertibility and nonsingularity of Boolean control networks

Invertibility is an interesting and classical control-theoretic problem. However, there has been no result for the invertibility of Boolean control networks (BCNs) so far. We first adopt the theory of symbolic dynamics to characterize it. First, it is shown that a BCN generates a continuous mapping from the space of input trajectories to the space of output trajectories. Based on it, the concepts of nonsingularity and invertibility of BCNs are first defined as the injectivity and bijectivity of the mapping, respectively. Second, combined symbolic dynamics with the semi-tensor product (STP) of matrices, an equivalent test criterion for invertibility is given; easily computable algorithms to construct the inverse BCN for an invertible BCN are presented; and it is proved that invertibility remains invariant under coordinate transformations. Third, an equivalent test criterion for nonsingularity is given via defining a novel directed graph that is called weighted pair graph. Lastly, as an application of invertibility to systems biology, we prove that the BCN model proposed in Fauré et al. (2006) is not invertible, i.e., we prove that arbitrarily controlling mammalian cell cycles is unfeasible at the theoretical level.

An Alternative Proof for the Topological Entropy of the Motzkin Shift

A Motzkin shift is a mathematical model for constraints on genetic sequences. In terms of the theory of symbolic dynamics, the Motzkin shift is nonsofic, and therefore, we cannot use the PerronFrobenius theory to calculate its topological entropy. The Motzkin shift M(M,N) which comes from language theory, is defined to be the shift system over an alphabet A that consists of N negative symbols, N positive symbols and M neutral symbols. For an x in the full shift, x will be in the Motzkin subshift M(M,N) if and only if every finite block appearing in x has a non-zero reduced form. Therefore, the constraint for x cannot be bounded in length. K. Inoue has shown that the entropy of the Motzkin shift M(M,N) is log(M + N + 1). In this paper, a new direct method of calculating the topological entropy of the Motzkin shift is given without any measure theoretical discussion. Keywords—Motzkin shift, topological entropy.

The Spatial Complexity of Inhomogeneous Multi-layer Neural Networks

Inhomogeneous multi-layer neural networks (IHMNNs) have been applied in various fields, for example, biological and ecological contexts. This work studies the learning problem of IHMNNs with an activation function $$f(x) = \dfrac{1}{2} (|x+1| - |x-1|)$$f(x)=12(|x+1|-|x-1|) that derives from cellular neural networks, which can be adapted to the study of the vision systems of mammals. Applying the well-developed theory of symbolic dynamics, the explicit formulae of the topological entropy of the output and hidden spaces are given. We also demonstrate that, for any $$\lambda \in [0, \log 2]$$??[0,log2] and $$\epsilon > 0$$∈>0, parameters such that the topological entropy $$h$$h of the hidden/output space of IHMNN that satisfies $$|h - \lambda | < \epsilon $$|h-?|<∈ exists. This means that the collection of topological entropies is dense in the closed interval $$[0, \log 2]$$[0,log2], which leads to the fact that IHMNNs are universal machines in some sense and hence are more efficient in learning algorithms. This paper aims to provide a mathematical foundation for the illustration of the capability of machine learning, while the method we have adopted can be extended to the investigation of multi-layer neural networks with other activation functions.

Category theory of symbolic dynamics

We study the central objects of symbolic dynamics, that is, subshifts and block maps, from the perspective of basic category theory, and present several natural categories with subshifts as objects and block maps as morphisms. Our main goals are to find universal objects in these symbolic categories, to classify their block maps based on their category theoretic properties, to prove category theoretic characterizations for notions arising from symbolic dynamics, and to establish as many natural properties (finite completeness, regularity etc.) as possible. Existing definitions in category theory suggest interesting new problems in symbolic dynamics. Our main technical contributions are the solution to the dual problem of the Extension Lemma and results on certain types of conserved quantities, suggested by the concept of a coequalizer.

Symbolic dynamics-based error analysis on chaos synchronization via noisy channels

In this study, symbolic dynamics is used to research the error of chaos synchronization via noisy channels. The theory of symbolic dynamics reduces chaos to a shift map that acts on a discrete set of symbols, each of which contains information about the system state. Using this transformation, a coder-decoder scheme is proposed. A model for the relationship among word length, region number of a partition, and synchronization error is provided. According to the model, the fundamental trade-off between word length and region number can be optimized to minimize the synchronization error. Numerical simulations provide support for our results.

A novel construction of chaotic dynamics base gliders in diffusion rule B2/S7

The dynamical behaviors of gliders (mobile localizations) in diffusion rule B2/S7 are quantitatively analyzed from the theory of symbolic dynamics in two-dimensional symbolic sequence space. Their intrinsic complexity is demonstrated by exploiting the relationship between one-dimensional and two-dimensional subshifts. Based on this rigorous approach and technique, the underlying chaos of the extant gliders and their combinations is characterized in a subtle way. It is demonstrated that they can be identified to distinct subsystems with very rich and complicated dynamics; that is, diffusion rule is topologically mixing and possesses positive topological entropy on each subsystem. This analytical assertion provides the fact that diffusion rule is covered with complex subsystems “almost everywhere”. Finally, it is worth mentioning that the procedure proposed in this paper is also applicable to all other gliders arising from the two-dimensional cellular automata therein. It is an extended discovery in both cellular automata and chaos theory.

Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps

In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.

TOPOLOGICAL COMPLEXITY AND PREDICTABILITY IN THE DYNAMICS OF A TUMOR GROWTH MODEL WITH SHILNIKOV'S CHAOS

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

A Theory of Symbolic Dynamics for Hybrid Systems

AbstractIn this paper, we introduce a new semantics for hybrid systems called symbolic dynamics. The semantics is constructed in the tradition of denotational semantics of programming languages, which defines the meaning of a construct by a function. The novelty of this approach is that these semantic functions are defined on complex state spaces carrying algebraic and analytic structures. As in the denotational semantics, we also define operations with symbolic dynamics corresponding to the primitives of the hybrid behaviours like stopping and switching. This semantics can describe both deterministic and stochastic systems. The paper is self-contained and it does not require any apriori background in computer science.

Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincaré return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing δ1. However, for higher values of δ1 the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter ζ) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.

Chaos and crises in a model for cooperative hunting: A symbolic dynamics approach

In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1-sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K,C(0)) and (K,sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity K(c) decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters.

Schmidt games and Markov partitions

Let T be a C2-expanding self-map of a compact, connected, C∞, Riemannian manifold M. We correct a minor gap in the proof of a theorem from the literature: the set of points whose forward orbits are nondense has full Hausdorff dimension. Our correction allows us to strengthen the theorem. Combining the correction with Schmidt games, we generalize the theorem in dimension one: given a point x0 ∊ M, the set of points whose forward orbit closures miss x0 is a winning set. Finally, our key lemma, the no matching lemma, may be of independent interest in the theory of symbolic dynamics or the theory of Markov partitions.

Topological invariants in the study of a chaotic food chain system

The study of ecological systems has generated deep interest in exploring the complexity of chaotic food chains. The role of chaos in ecosystems is not entirely understood. One approach to have a better comprehension of ecological chaos is by analyzing it in mathematical models of basic food chains. In this article it is considered a classical chaotic food chain model from the literature. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of kneading sequences associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The topological entropy allows us to distinguish different chaotic states in some realistic system parameter region. Another numerical invariant is introduced in order to characterize isentropic dynamics. Studying a set of maps with the same topological entropy, we exhibit numerical results about the relation between the second topological invariant and each of the control parameters in consideration. This work provides an illustration of how our understanding of ecological models can be enhanced by the theory of symbolic dynamics.

An estimate of topological entropy of N-buffer fluid networks

In this paper, we give an estimate of the topological entropy of a dynamical system defined on a simplex derived from the model of an N-buffer fluid network using an elementary theory of symbolic dynamics. Our treatment is easy to catch on even for readers with a less than good background of ergodic theory in dynamical systems.

Information flow in chaos synchronization: fundamental tradeoffs in precision, delay, and anticipation.

We use symbolic dynamics to examine the flow of information in unidirectionally coupled chaotic oscillators exhibiting synchronization. The theory of symbolic dynamics reduces chaos to a shift map that acts on a discrete set of symbols, each of which contains information about the system state. Using this transformation we explore so-called achronal synchronization, in which the response lags or leads the drive by a fixed amount of time. We find fundamental tradeoffs between the precision to which the drive state is detected, the quality of synchronization attained, and the delay or anticipation exhibited by the response system. To illustrate these tradeoffs, we provide a physical example using electronic circuits.