Measuring complexity in a business cycle model of the Kaldor type

The purpose of this paper is to study the dynamical behavior of a family of two-dimensional nonlinear maps associated to an economic model. Our objective is to measure the complexity of the system using techniques of symbolic dynamics in order to compute the topological entropy. The analysis of the variation of this important topological invariant with the parameters of the system, allows us to distinguish different chaotic scenarios. Finally, we use a another topological invariant to distinguish isentropic dynamics and we exhibit numerical results about maps with the same topological entropy. This work provides an illustration of how our understanding of higher dimensional economic models can be enhanced by the theory of dynamical systems.     

The complexity of a minimal sub-shift on symbolic spaces

Let (Σ,ρ) be a one-sided symbolic space (with two symbols), and let σ be the shift on Σ. In this paper, we prove that there exists a minimal set T⊂Σ such that σT_| is Wiggins chaotic, Martelli chaotic, distributionally chaotic, strictly ergodic, topologically weakly mixing and has zero topological entropy.     

Chaos among self-maps of the Cantor space

Glasner and Weiss have shown that a generic homeomorphism of the Cantor space has zero topological entropy. Hochman has shown that a generic transitive homeomorphism of the Cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense. We show that a generic self-map of the Cantor space has zero topological entropy. Moreover, we show that a generic self-map of the Cantor space has no periodic points and hence is not Devaney chaotic nor Devaney chaotic on any subsystem. However, we exhibit a dense subset of the space of all self-maps of the Cantor space each element of which has infinite topological entropy and is Devaney chaotic on some subsystem.     

一类一维混沌映射的拓扑条件

20世纪中期以来,人们在物理、天文、气象等领域中发现了大量的混沌现象.这些新发现引发了近几十年来对混沌现象的研究.由于它的困难程度和在解决实际问题中的巨大价值,对混沌现象的研究成为动力系统乃至数学中的一个长期的前沿和热点研究方向.混沌现象最本质的特征是初值敏感性,保证有初值敏感性的一个充分条件是系统具有正Lyapunov指数.因此研究系统是否具有正Lyapunov指数成为研究系统是否出现混沌的重要方法.从拓扑角度给出了一类一维映射出现混沌现象的充分条件.从拓扑的角度来研究,将加深对此类映射出现混沌的机理的认识.研究此类映射,最重要的是研究临界点、临界点轨道及它们的相互关系.我们采用临界点的逆像建立拓扑工具,使用这一拓扑工具分析临界点轨道与临界点的复杂关系,研究临界点逆轨道的运动形态、相应开集的拓扑特征,进而导出系统出现混沌的拓扑特征及它与Lyapunov指数之间的关系.     

Reciprocal inhibitory coupling: Measure and control of chaos on a biophysically motivated model of bursting

Bursting activity is an interesting feature of the temporal organization in many cell firing patterns. This complex behavior is characterized by clusters of spikes (action potentials) interspersed with phases of quiescence. As shown in experimental recordings, concerning the electrical activity of real neurons, the analysis of bursting models reveals not only patterned periodic activity but also irregular behavior [1], [2]. The interpretation of experimental results, particularly the study of the influence of coupling on chaotic bursting oscillations, is of great interest from physiological and physical perspectives. The inability to predict the behavior of dynamical systems in presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we focus our attention on a specific class of biophysically motivated maps, proposed in the literature to describe the chaotic activity of spiking–bursting cells [Cazelles B, Courbage M, Rabinovich M. Anti-phase regularization of coupled chaotic maps modelling bursting neurons. Europhys Lett 2001;56:504–9]. More precisely, we study a map that reproduces the behavior of a single cell and a map used to examine the role of reciprocal inhibitory coupling, specially on two symmetrically coupled bursting neurons. Firstly, using results of symbolic dynamics, we characterize the topological entropy associated to the maps, which allows us to quantify and to distinguish different chaotic regimes. In particular, we exhibit numerical results about the effect of the coupling strength on the variation of the topological entropy. Finally, we show that complicated behavior arising from the chaotic coupled maps can be controlled, without changing of its original properties, and turned into a desired attracting time periodic motion (a regular cycle). The control is illustrated by an application of a feedback control technique developed by Romeiras et al. [Romeiras FJ, Grebogi C, Ott E, Dayawansa WP. Controlling chaotic dynamical systems. Physica D 1992;58:165–92]. This work provides an illustration of how our understanding of chaotic bursting models can be enhanced by the theory of dynamical systems.     

Topological sequence entropy for maps of the interval

A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

     

A directed weighted complex network for characterizing chaotic dynamics from time series

We propose a reliable method for constructing a directed weighted complex network (DWCN) from a time series. Through investigating the DWCN for various time series, we find that time series with different dynamics exhibit distinct topological properties. We indicate this topological distinction results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor. Furthermore, we associate different aspects of dynamics with the topological indices of the DWCN, and illustrate how the DWCN can be exploited to detect unstable periodic orbits of different periods. Examples using time series from classical chaotic systems are provided to demonstrate the effectiveness of our approach.     

Topological entropy of catalytic sets: Hypercycles revisited

The dynamics of catalytic networks have been widely studied over the last decades because of their implications in several fields like prebiotic evolution, virology, neural networks, immunology or ecology. One of the most studied mathematical bodies for catalytic networks was initially formulated in the context of prebiotic evolution, by means of the hypercycle theory. The hypercycle is a set of self-replicating species able to catalyze other replicator species within a cyclic architecture. Hypercyclic organization might arise from a quasispecies as a way to increase the informational containt surpassing the so-called error threshold. The catalytic coupling between replicators makes all the species to behave like a single and coherent evolutionary multimolecular unit. The inherent nonlinearities of catalytic interactions are responsible for the emergence of several types of dynamics, among them, chaos. In this article we begin with a brief review of the hypercycle theory focusing on its evolutionary implications as well as on different dynamics associated to different types of small catalytic networks. Then we study the properties of chaotic hypercycles with error-prone replication with symbolic dynamics theory, characterizing, by means of the theory of topological Markov chains, the topological entropy and the periods of the orbits of unimodal-like iterated maps obtained from the strange attractor. We will focus our study on some key parameters responsible for the structure of the catalytic network: mutation rates, autocatalytic and cross-catalytic interactions.     

A Note on a Chaotic but not S-S Chaotic Subshift

In this paper we prove that the minimal chaotic but not S-S chaotic subshifts are also uinquely ergodic and have zero topological entropy.     

A topological horseshoe in a fractional-order Qi four-wing chaotic system

A fractional-order Qi four-wing chaotic system is present based on the Grunwald-Letnikov denition. The existence of topological horseshoe in a fractional chaotic system is analyzed by utilizing topological horseshoe theory. A Poincare section is properly chosen to obtain the Poincare map which is proved to be semi-conjugate to a 2-shift map, implying that the fractional-order Qi four-wing chaotic system exhibits chaos.     

Smooth interval maps have symbolic extensions

We prove that if X denotes the interval or the circle then every transformation T:X→X of class C ^r , where r>1 is not necessarily an integer, admits a symbolic extension, i.e., every such transformation is a topological factor of a subshift over a finite alphabet. This is done using the theory of entropy structure. For such transformations we control the entropy structure by providing an upper bound, in terms of Lyapunov exponents, of local entropy in the sense of Newhouse of an ergodic measure ν near an invariant measure μ (the antarctic theorem). This bound allows us to estimate the so-called symbolic extension entropy function on invariant measures (the main theorem), and as a consequence, to estimate the topological symbolic extension entropy; i.e., a number such that there exists a symbolic extension with topological entropy arbitrarily close to that number. This last estimate coincides, in dimension 1, with a conjecture stated by Downarowicz and Newhouse [13, Conjecture 1.2]. The passage from the antarctic theorem to the main theorem is applicable to any topological dynamical system, not only to smooth interval or circle maps.     

A Devaney Chaotic System Which Is Neither Distributively nor Topologically Chaotic

Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.     

混沌群作用

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｉｓｔｈｅｓｅｑｕｅｌｔｏ[1 ,2 ].Ｃａｉｒｎｓｅｔａｌ.[1] ｉｎｔｒｏｄｕｃｅｄｔｈｅｎｏｔｉｏｎｏｆａｃｈａｏｔｉｃｇｒｏｕｐａｃｔｉｏｎａｓａｇｅｎｅｒａｌｉｚａｔｉｏｎｏｆｃｈａｏｔｉｃｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓ(ｓｅｅｄｅｆｉｎｉｔｉｏｎｂｅｌｏｗ) .Ｔｈｅｙｓｈｏｗｅｄｔｈａｔｔｈｅｃｉｒｃｌｅｄｏｅｓｎｏｔａｄｍｉｔａｃｈａｏｔｉｃａｃｔｉｏｎｏｆａｎｙｇｒｏｕｐ ,ａｎｄｃｏｎｓｔｒｕｃｔｅｄａｃｈａｏｔｉｃａｃｔｉｏｎｏｆＧ =Ｚ×…     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

The Hausdorff dimension of chaotic sets for interval self-maps

We obtain two sufficient conditions for an interval self-map to have a chaotic set with positive Hausdorff dimension. Furthermore, we point out that for any interval Lipschitz maps with positive topological entropy there is a chaotic set with positive Hausdorff dimension.     

Classification problems for shifts on modules over a principal ideal domain

In this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain, considering topological shifts which are also submodules. Our main result is the classification, up to algebraic and topological conjugacy, of the torsion-free, transitive, finite memory shifts.

     

Horseshoe chaos in a class of simple Hopfield neural networks

A class of new simple Hopfield neural networks is revisited. To confirm the chaotic behavior in these Hopfield neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of existence of horseshoe chaos by virtue of topological horseshoes theory and estimates of topological entropy in the derived Poincaré maps.     

The Symbolic Extension Theory in Topological Dynamics

In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.     

The set of recurrent points of a continuous self-map on an interval and strong chaos

In this paper, we discuss a continuous self-map of an interval and the existence of an uncountable strongly chaotic set. It is proved that if a continuous self-map of an interval has positive topological entropy, then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.     

Topological invariants in forced piecewise-linear FitzHugh–Nagumo-like systems

Mathematical models for periodically-forced excitable systems arise in many biological and physiological contexts. Chaotic dynamics of a forced piecewise-linear Fitzhugh–Nagumo-like system under large-amplitude forcing was identified by Othmer and Xie in their work [J. Math. Biol. 39 (1999) 139]. Using kneading theory we study the topological entropy of some chaotic return maps associated with a singular system. Finally we introduce a new topological invariant to distinguish isentropic dynamics and we exhibit numerical results about maps with the same topological entropy, that suggest the existence of a relation between the parameters A and θ, when T is fixed.