Sharkovskii's Theorem Research Articles

ANALYSIS OF SKELLAM MODELS WITH A RIGID HARVESTING STRATEGY

Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time. In the simplest case such equations have form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function. Among such equations the discrete Skellam model are most often used in practice. In the given paper the Skellam model of the form $N_{t+1}=a N_t/ (1+b N_t)$, $N_{t+1}=a N_t^2/ (b^2+ N_t^2)$, $N_{t+1}=a N_t/ (1+b N_t^2)$ is considered, where the parameters $a,b>0$ with taking an effect of harvesting. Positive equilibrium points and conditions for their stability for these equations were found. It is shown in analytical form that these equations do not have periodic solutions with period $T=2$, which means, according by the Sharkovskii theorem, periodic solutions of any periods. In the model with harvesting, only regimes with monotonic stabilization of the population size are observed. Therefore, in all models of Skellam with harvesting, the existence of a critical conception is show, beyond which the population will be completely destroyed. For practice it is important to know the permissible limits of harvesting intensity, which are found in this paper.

From the Sharkovskii theorem to periodic orbits for the Rössler system

We extend Sharkovskii's theorem to the cases of N-dimensional maps which are close to 1D maps, with an attracting n-periodic orbit. We prove that, with relatively weak topological assumptions, there exist also m-periodic orbits for all m▹n in Sharkovskii's order, in the nearby.We also show, as an example of application, how to obtain such a result for the Rössler system with an attracting periodic orbit, for four sets of parameter values. The proofs are computer-assisted.

MODELING HARVESTING PROCESSES FOR POPULATIONS WITH NON-OVERLAPPING GENERATIONS

Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time. In the simplest case such equations have the form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function. Among such equations the discrete logistic equation and Ricker's equation are most often used in practice. In the given paper, these equations are considered width taking into account an effect of harvesting, that is, the equations of the form below are studied $N_{t+1}=r N_t (1- N_t) - c$ and $N_{t+1}= N_t \exp (r(1 - N_t / K )) - c$, where the parameters $r$, $K>0$, $c>0$ are harvesting intensity. Positive equilibrium points and conditions for their stability for these equations were found. These kinds of states are often realized in nature. For practice, periodic solutions are also important, especially with periods $T=2 (N_{t+2} = N_t)$ and $T=3 (N_{t+3} = N_t)$, since, with their existence, by Sharkovskii's theorem, one can do conclusions about the existence of periodic solutions of other periods. For the discrete logistic equation in analytical form, the values that make up the periodic solution with period $T=2$ were found. We used numerical methods in order to find solutions with period $T=3$. For Ricker's model, the question of the existence of periodic solutions can be investigated by computer analysis only. In the paper, a number of computer experiments were conducted in which periodic solutions were found and their stability was studied. For Ricker's model with harvesting, chaotic solutions were also found. As we can see, the study of difference equations gives many unexpected results.

Periodic orbits in the Rössler system

We prove the existence of n-periodic orbits for almost all n∈N in the Rössler system with an attracting periodic orbit, for two sets of parameters. The methods, using covering relations between two-dimensional sets, are similar to the ones used often in the proofs of the Sharkovskii Theorem. The proofs are computer-assisted with the use of C++ library CAPD containing modules for interval arithmetic, differentiation and integration of ODEs.

Simple Permutations with Order $$4n+2$$ 4 n + 2 by Means of Pasting and Reversing

nez-Castiblanco Dedicated to Jes us Hernando P erez (Pelusa), teacher and friend. Abstract. The problem of genealogy of permutations has been solved partially by Stefan (odd order) and Acosta-Hum anez & Bernhardt (power of two). It is well known that Sharkovskii's theorem shows the relationship between the cardinal of the set of periodic points of a continuous map, but simple per- mutations will show the behaviour of those periodic points. Recently Abdulla et al studied the structure of minimal 4n + 2-orbits of the continuous endo- morphisms on the real line. This paper studies some combinatorial dynamics structures of permutations of mixed order 4n + 2, describing its genealogy, using Pasting and Reversing.

Defining Chaos in the Logistic Map by Sharkovskii's Theorem

Defining Chaos in the Logistic Map by Sharkovskii's Theorem

Sharkovskii's theorem, differential inclusions, and beyond

We explain why the Poincaré translation operators along the trajectories of upper-Carathéodory differential inclusions do not satisfy the exceptional cases, described in our earlier counter-examples, for upper semicontinuous maps. Such a discussion was stimulated by a recent paper of F. Obersnel and P. Omari, where they show that, for Carathéodory scalar differential equations, the existence of just one subharmonic solution (e.g of order $2$) implies the existence of subharmonics of all orders. We reprove this result alternatively just via a multivalued Poincaré translation operator approach. We also establish its randomized version on the basis of a universal randomization scheme developed recently by the first author.

Randomization of Sharkovskii-type theorems

We formulate an abstract scheme for the randomization of Sharkovskii-type theorems via transformation to the deterministic case. In particular, Sharkovskii-type theorems for scalar differential equations can be randomized in this way. A random version of the standard Sharkovskii theorem is presented explicitly. Many remarks, comments and illustrating examples are supplied.

On a Multivalued Version of the Sharkovskii Theorem and Its Application to Differential Inclusions, II

Motivated by the applications to differential equations without uniqueness conditions, we separately prove multivalued versions of the celebrated Sharkovskii and Li–Yorke theorems. These are then applied, via multivalued Poincare operators, to Caratheodory differential inclusions. Thus, besides another, infinitely many subharmonics of all integer orders can be obtained. Unlike in the single-valued case, for example, period three brings serious obstructions. Three counter-examples, related to these complications, are therefore presented as well. In a multivalued setting, new phenomena are so exhibited.

On a multivalued version of the Sharkovskii theorem and its application to differential inclusions, III

An extension of the celebrated Sharkovskiĭ cycle coexisting theorem (see [ Coexistence of cycles of a continuous map of a line into itself , Ukrain. Math. J. 16 (1964), 61–71]) is given for (strongly) admissible multivalued self-maps in the sense of [L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings , Kluwer, Dordrecht, 1999], on a Cartesian product of linear continua. Vectors of admissible self-maps have a triangular structure as in [[P. E. Kloeden, On Sharkovsky’s cycle coexisting ordering , Bull. Austral. Math. Soc. 20 (1979), 171–177]. Thus, we make a joint generalization of the results in [J. Andres, J. Fiser and L. Juttner, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions , Set-Valued Anal. 10 (2002), 1–14], [J. Andres and L. Juttner, Period three plays a negative role in a multivalued version of Sharkovskiĭ’s theorem , Nonlinear Anal. 51 (2002), 1101–1104], [J. Andres, L. Juttner and K. Pastor, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions II ] (a multivalued case), in [P. E. Kloeden, On Sharkovsky’s cycle coexisting ordering , Bull. Austral. Math. Soc. 20 (1979), 171–177] (a multidimensional case), and in [H. Schirmer, A topologist’s view of Sharkovskiĭ’s theorem , Houston, J. Math. 11 (1985), 385–395] (a linear continuum case). The obtained results can be applied, unlike in the single-valued case, to differential equations and inclusions.

A refinement of Sharkovskii's theorem on orbit types characterized by two parameters

The so-called type problem or forcing problem is considered as a way to generalize Sharkovskii's theorem. In this paper, by focusing on certain types of orbits, we obtain a solution of the type problem, which gives a refinement of Sharkovskii's theorem on orbit types characterized by two parameters.

Dynamics of Triangular Transformations of Sequences over Finite Alphabets

Let Σ be a certain class of transformations of a set X , which can be specified by topological (in particular, metric), combinatorial, or algebraic conditions depending on the structure of the set. The study of the dynamics of mappings from Σ substantially depends on description of the sets of positive integers that can appear as lists of all cycle lengths of a certain transformation from Σ. For instance, the classical Sharkovskii result [1] gives a description of such sets in the case where Σ is the class of all continuous real functions or the class of all continuous transformations of a closed interval. Further research in this area resulted in a new branch of the theory of dynamical systems, called combinatorial dynamics (see the monograph [2]). Investigations in combinatorial dynamics brought about, in particular, various generalizations and modifications of the Sharkovskii theorem, say, its extensions to the classes of continuous transformations of a circle [3, 4], certain graphs [5, 6], etc. (see, e.g., [7, 8]). On the other hand, many papers were devoted to the study of this problem for polynomial mappings over various fields, for rational functions, and the like (see the monograph [9] and references there). In this note, we give a description of possible sets of cycle lengths for triangular maps of the set of all infinite sequences over a given alphabet. As a corollary, we obtain a characterization of cycle lengths and orbits for automaton transformations over a finite alphabet, as well as for nonexpansive transformations of the metric space of p-adic integers. It readily follows that an analog of the Sharkovskii theorem does not hold for continuous transformations of the p-adic line.

Period three plays a negative role in a multivalued version of Sharkovskii's theorem

Period three plays a negative role in a multivalued version of Sharkovskii's theorem

Nielsen theory, braids and fixed points of surface homeomorphisms

We study two problems in Nielsen fixed point theory using Artin's braid groups and the Nielsen–Thurston classification of surface homeomorphismsup to isotopy. The first is that of distinguishing Reidemeister classes of free group automorphisms realized by a braid (and thus induced by homeomorphismsof the 2-disc relative to a finite invariant set), for which we give a necessary and sufficient condition in terms of a conjugacy problem in the braid group. Consequently, one may use any braid conjugacy invariant (those of Garside's algorithm, linking numbers, topological entropy, etc.) and any link invariant (Alexander polynomial, splittability, etc.) to distinguish Reidemeisterclasses, giving much stronger criteria than those already known. The second problem is that of deciding when two fixed points of a surface homeomorphismbelong to the same Nielsen fixed point class. We give two criteria, the first in terms of certain reducing curves which can be checked using the Bestvina–Handel algorithm, the second using the multi-variable Alexander polynomial of a link associated with the suspension of the homeomorphism. Finally we consider generalizations of Sharkovskii's theorem on the coexistence of periodic orbits of interval maps to homeomorphismsof the 2-disc. We show that for each n⩾5 there exists a pseudo-Anosovorientation-preservinghomeomorphismof the 2-disc relative to a periodic orbit of period n that does not have periodic orbits of all periods, with an analogous result for the 2-sphere.

A Simple Special Case of Sharkovskii's Theorem

In this note I c R is a bounded closed interval and f: I -> I is a continuous map; fn denotes the n-fold composition of f with itself. A point x E I is a periodic point for f with period p if fP(x) = x and has least period p if in addition fk(x) # x for 1 < k < p - 1. The note presents a short proof of the following result.

A Simple Special Case of Sharkovskii's Theorem

A Simple Special Case of Sharkovskii's Theorem

Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps

We present Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps. We show that if an unperturbed one-dimensional map has a point of period $n$, then sufficiently close multidimensional perturbations of this map have periodic points of all periods which are allowed by Sharkovskii's theorem.

Sharkovskii theorem for multidimensional perturbations of one-dimensional maps II

We present a multidimensional generalization of the Sharkovskii Theorem concerning the appearance of periodic points for the self-maps on the real line.

Rotation numbers, twists and a Sharkovskii–Misiurewicz-type ordering for patterns on the interval

AbstractWe introducerotation numbersandpairscharacterizing cyclic patterns on an interval and a special order among them; then we prove the theorem which specializes the Sharkovskii theorem in this setting.

Minimal dynamical system with givenD-function and topological entropy

TheD-function is a new topological invariant introduced by the author in [3] to classify the minimal dynamical system and to generalize Sharkovskii's theorem on the coexistence of periodic orbits. We show that theD-function and the topological entropy are independent.