Jose S. Cánovas

Li–Yorke chaos in a class of nonautonomous discrete systems

Let be a sequence of continuous interval maps which converges uniformly to a continuous map f. We study the limit behaviour of sequences with the form , , and whether the simplicity (respectively chaoticity) of f implies the simplicity (respectively chaoticity) of .

Periodic structure of alternating continuous interval maps

Given two continuous interval maps f and g, we study the periodic structure for the set of sequences generated by f and g by the rule . It is shown that this structure is intimately related to that of Sharkovskys Theorem, although some differences appear with the periods that are coprime with 2.

Topological entropy of fuzzified dynamical systems

A discrete dynamical system is given by a compact metric space X and any continuous self-map defined on X. This discrete dynamical system can be naturally extended to the space of fuzzy sets on X. In this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart. Among other things, we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system. However, we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system.

Topological dynamic classification of duopoly games

Abstract We classify Cournot maps on the unit square by using topological dynamics. More precisely, we prove that the following properties are equivalent: (1) zero topological entropy, (2) UR ( F )= R ( F ), (3) type less than or equal to 2 ∞ and (4) AP ( F )={( x , y )∈ I 2 :lim n →∞ F 2 n ( x , y )=( x , y )}. These results allow us to decide when the behavior of these two-dimensional dynamical systems is complicated.

On ω-limit sets of antitriangular maps☆

Abstract We give a topological characterization of ω-limit sets of continuous antitriangular maps, that is, maps F :[0,1] 2 →[0,1] 2 with the form F(x,y)=(f2(y),f1(x)), (x,y)∈I2. We also point out some differences between ω-limit set of antitriangular and one-dimensional maps.

On the topological entropy on the space of fuzzy numbers

In the main result of this article, we prove that the topological entropies of a given interval map and its Zadehs extension (fuzzification) to the space of fuzzy numbers (i.e., the space of fuzzy sets with connected α-cuts) are the same. This result is in contrast with our previous result, which claimed that the topological entropy of the Zadehs extension significantly increases for a majority of simple interval maps. In addition, we prove some properties of the limit sets of trajectories that are generated by iterating the fuzzy set valued function on connected fuzzy sets; for instance, we specify the shapes of the possible limit sets. Furthermore, the presented topics are studied for set-valued (induced) discrete dynamical systems. The main results are proved with a variational principle that describes the relations between topological entropy and measure-theoretical entropy.

Distributional chaos of tree maps

Abstract Let T be a finite tree and let f :T→T be a continuous map such that any vertex of T is a fixed point of f . We prove that f is distributionally chaotic if and only if its topological entropy is positive.

Reducing competitors in a Cournot–Theocharis oligopoly model

We study how firms disappear from the market in a Cournot–Theocharis oligopoly model. We find necessary and sufficient conditions for the global convergence of the system to a monopoly or a duopoly. In particular, we prove that when the number of firms increases, it is more difficult to eliminate a firm from the market.

Estimating topological entropy from individual orbits

The relationship between the topological entropy of a continuous piecewise monotone interval map and the permutation entropy of its orbits is given. We use this relationship to estimate the topological entropy of the map.

Topological sequence entropy of interval maps

We establish a full classification of chaotic and non-chaotic interval maps from the point of view of topological sequence entropy. This completes the papers of Franzova and Smital (1991 Positive sequence topological entropy characterizes chaotic maps Proc. Am. Math. Soc. 112 1083–6) and Hric (1999 Topological sequence entropy for maps of the interval Proc. Am. Math. Soc. 127 2045–52). Moreover, with reference to interval maps, this paper establishes an analogous result to Pickels result on metric sequence entropy (1969 Some properties of A-entropy Mat. Zametki 5 327–34 (in Russian)), and partially solves a question of Goodman (1974 Topological sequence entropy Proc. Lond. Math. Soc. 29 331–50).