Dense Periodic Orbits Research Articles

Hénon–Devaney like maps

We prove a general theorem characterizing the transitivity of homeomorphisms with singularities in the plane. We provide examples where this theorem applies including the classical Hénon–Devaney map (1981 Commun. Math. Phys. 80 465–476). We also prove some results about the plane homeomorphisms satisfying the hypothesis of our theorem namely the Hénon–Devaney like maps. Indeed, we show that the maximal invariant set of a Hénon–Devaney like map is topologically conjugated to a shift map. Furthermore, every Hénon–Devaney like map is fixed point free with dense periodic orbits. This generalizes some constructions by Devaney (1981 Commun. Math. Phys. 80 465–476) and Lenarduzzi (2015 Discrete Continuous Dyn. Syst. 35 1163–1177).

Chaos as a symmetry-breaking phenomenon

Chaos is an ubiquitous and fundamental phenomenon with a wide range of features pointing to a similar phenomenology. Although apparently distinct, it is natural to ask if all these features emerge from a unifying principle. Recently, it was realized that all continuous-time stochastic dynamical systems (DSs) — the most relevant in physics because natural DSs are always subject to noise and time is continuous — possess a topological supersymmetry (TS). It was then suggested that its spontaneous breakdown could be interpreted as the stochastic generalization of deterministic chaos. This conclusion stems from the fact that such phenomenon encompasses features that are traditionally associated with chaotic dynamics such as non-integrability, positive topological entropy, sensitivity to initial conditions and the Poincaré–Bendixson theorem. Here, we strengthen and complete this picture by showing that the remaining hallmarks of deterministic chaos — topological transitivity/mixing and dense periodic orbits — while being consistent with the TS breaking, do not actually admit a stochastic generalization. This is a major limitation, since all physical systems are always noisy to some extent. We, therefore, conclude that spontaneous TS breaking can be considered as the most general definition of continuous-time dynamical chaos. Contrary to the common perception and semantics of the word “chaos,” this phenomenon should then be truly interpreted as the low-symmetry, or ordered phase of the DSs that manifest it. We also argue that the concept of chaos in low-dimensional, discrete-time DSs that do not obey the Poincaré–Bendixson theorem, is related to the explicit TS breaking.

Intransitive Sectional-Anosov Flows on 3-manifolds

For each $$n\in \mathbb {Z}^+$$ , we show the existence of Venice masks (i.e. intransitive sectional-Anosov flows with dense periodic orbits, Bautista and Morales in Lectures on sectional-Anosov flows. http://preprint.impa.br/Shadows/SERIE_D/2011/86.html , 2011; Bautista and Morales in Discrete Contin Dyn Syst 19(4): 761–775, 2007; Lopez Barragan and Sanchez in Bull Braz Math Soc N Ser 48(1): 1–18, 2017, Morales and Pacifico in Pac J Math 216(2): 327–342, 2004) containing n equilibria on certain compact 3-manifolds. These examples are characterized because of the maximal invariant set is a finite union of homoclinic classes. Here, the intersection between two different homoclinic classes is contained in the closure of the union of unstable manifolds of the singularities.

Sectional Anosov Flows: Existence of Venice Masks with Two Singularities

We show the existence of Venice masks (i.e. nontransitive sectional Anosov flows with dense periodic orbits, Bautista and Morales http://preprint.impa.br/Shadows/SERIE_D/2011/86.html ; Bautista et al. Discr Contin Dyn Syst 19(4):761, 2007; Morales and Pacifico Pac J Math 216(2):327–342, 2004, Morales et al. Pac J Math 229(1):223–232, 2007) containing two equilibria on certain compact 3-manifolds. Indeed, we present two type of examples in which the homoclinic classes composing their maximal invariant set intersect in a very different way.

Intelligent Soft Computing Based Cryptographic Technique Using Chaos Synchronization for Wireless Communication (CSCT)

In this paper a novel intelligent soft computing based cryptographic technique based on synchronization of two chaotic systems (CSCT) between sender and receiver has been proposed to generate session key using Pecora and Caroll (PC) method. Chaotic system has some unique features like sensitive to initial conditions, topologically mixing; and dense periodic orbits. By nature, the Lorenz system is very sensitive to initial conditions meaning that the error between attacker and receiver is going to grow exponentially if there is a very slight difference between their initial conditions. All these features make chaotic system as good alternatives for session key generation. In the proposed CSCT few parameters ( �, b , r , x1 ,y 2 and z 2 ) are being exchanged between sender and receiver. Some of the parameter which takes major roles to form the session key does not get transmitted via public channel, sender keeps these parameters secret. This way of handling parameter passing mechanism prevents any kind of attacks during exchange of parameters like sniffing, spoofing or phishing.

Multidimensional cellular automata: closing property, quasi-expansivity, and (un)decidability issues

In this paper we study the dynamics of D-dimensional cellular automata (CA) by considering them as one-dimensional (1D) CA along some direction (slicing constructions). These constructions allow to give the D-dimensional version of important notions as 1D closing property and lift well-known one-dimensional results to the D-dimensional settings. Indeed, like in one-dimensional case, closing D-dimensional CA have jointly dense periodic orbits and biclosing D-dimensional CA are open. By the slicing constructions, we further prove that for the class of closing D-dimensional CA, surjectivity implies surjectivity on spatially periodic configurations (old standing open problem). We also deal with the decidability problem of the D-dimensional closing. By extending the Kariʼs construction from [31] based on tilings, we prove that the two-dimensional, and then D-dimensional, closing property is undecidable. In such a way, we add one further item to the class of dimension sensitive properties, i.e., properties that are decidable in dimension 1 and are undecidable in higher dimensions. It is well-known that there are not positively expansive CA in dimension 2 and higher. As a meaningful replacement, we introduce the notion of quasi-expansivity for D-dimensional CA which shares many global properties (in the D-dimensional settings) with the 1D positive expansivity. We also prove that for quasi-expansive D-dimensional CA the topological entropy (which is an uncomputable property for general CA) has infinite value. In a similar way as quasi-expansivity, the notions of quasi-sensitivity and quasi-almost equicontinuity are introduced and studied.

Asynchronous cellular automata and dynamical properties

In this article the dynamical behaviour of asynchronous cellular automata (CA) is formally studied. Classical CA properties as surjectivity, injectivity, sensitivity, expansivity, transitivity, dense periodic orbits and equicontinuity have been adapted to the asynchronous case. We also deal with stability of properties with respect to perturbations on some update sequences which produce a significant dynamical behaviour.

Conservation of some dynamical properties for operations on cellular automata

We consider the family of all the Cellular Automata (CA) sharing the same local rule but having different memories. This family contains also all CA with memory m ≤ 0 (one-sided CA) which can act both on A Z and on A N . We study several set theoretical and topological properties for these classes. In particular, we investigate whether the properties of a given CA are preserved when considering the CA obtained by changing the memory of the original one (shifting operation). Furthermore, we focus our attention on the one-sided CA acting on A Z , starting from the one-sided CA acting on A N and having the same local rule (lifting operation). As a particular consequence of these investigations, we prove that the long-standing conjecture [Surjectivity ⇒ Dense Periodic Orbits (DPO)] can be restated in several different (but equivalent) ways. Furthermore, we give some results on properties conserved under the iteration of the CA global map.

On the intersection of homoclinic classes on singular-hyperbolic sets

We know that two different homoclinic classes contained in the samehyperbolic set are disjoint [12]. Moreover, a connectedsingular-hyperbolic attracting set with dense periodic orbits and aunique equilibrium is either transitive or the union of twodifferent homoclinic classes [6]. These results motivate thequestions of if two different homoclinic classes contained in thesame singular-hyperbolic set are disjoint or if the secondalternative in [6] cannot occur. Here we give a negativeanswer for both questions. Indeed we prove that every compact$3$-manifold supports a vector field exhibiting a connectedsingular-hyperbolic attracting set which has dense periodic orbits,a unique singularity, is the union of two homoclinic classes but isnot transitive.

A spectral decomposition for singular-hyperbolic sets

We extend the Spectral Decomposition Theorem for hyperbolic sets to singular-hyperbolic sets on 3-manifolds. We prove that an attracting singular-hyperbolic set with dense periodic orbits and a unique equilibrium of a Cr vector field, where r = 1, is a finite union of transitive sets; the union is disjoint or the set contains finitely many distinct homoclinic classes. If the vector field is Cr-generic, the union is in fact disjoint.

Nonuniform hyperbolicity for singular hyperbolic attractors

In this paper we show nonuniform hyperbolicity for a class of attractors of C 2 flows in dimension three. These attractors are partially hyperbolic with central direction being volume expanding, contain dense periodic orbits and hyperbolic singularities of the associated vector field. Classical expanding Lorenz attractors are the main examples in this class.

Investigating topological chaos by elementary cellular automata dynamics

We apply the two different definitions of chaos given by Devaney and by Knudsen for general discrete time dynamical systems (DTDS) to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A DTDS is chaotic according to the Devaney's definition of chaos iff it is topologically transitive, has dense periodic orbits, and it is sensitive to initial conditions. A DTDS is chaotic according to the Knudsen's definition of chaos iff it has a dense orbit and it is sensitive to initial conditions. We enucleate an easy-to-check property (left or rightmost permutivity) of the local rule associated with a cellular automaton which is a sufficient condition for D-chaotic behavior. It turns out that this property is also necessary for the class of elementary cellular automata. Finally, we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata.

Additive one-dimensional cellular automata are chaotic according to Devaney's definition of chaos

We study the chaotic behavior of a particular class of dynamical systems: cellular automata. We specialize the definition of chaos given by Devaney for general dynamical systems to the case of cellular automata. A dynamical system ( X, F) is chaotic according to Devaney's definition of chaos if its transition map F is sensitive to the initial conditions, topologically transitive, and has dense periodic orbits on X. Our main result is the proof that all the additive one-dimensional cellular automata defined on a finite alphabet of prime cardinality are chaotic in the sense of Devaney.

TOPOLOGICAL CHAOS FOR A CLASS OF LINEAR MODELS

We construct an example of linear rate equation in the Banach space of summable sequences, l1, that exhibits the three properties required as signature of topological chaos, namely: (i) topological transitivity, (ii) dense periodic orbits, and (iii) positive Lyapunov exponents. The example is based on the properties of the backward shift operator on the Banach space l1. Since linear chaos in the sense described above can occur only in an infinite-dimensional setting, possible finite-dimensional approximate manifestations are investigated. The relationship between the linear backward shift and the nonlinear Bernoulli shift is also discussed.

Chaotic Solutions of Systems of First Order Partial Differential Equations

We consider the system of partial differential equations \[ \frac{{\partial {\bf u}}}{{\partial t}} + ({\bf c}({\bf x})\cdot \nabla ){\bf u} = {\bf f}({\bf x},{\bf u}),\]\[ {\bf u}(t = 0,{\bf x}) = {\bf v}({\bf x})\] where ${\bf x} \in \Delta $ , a compact connected subset of $R^m $, ${\bf u} \in R^n $, and $t \geqq 0$. This is a generalization of a problem investigated by Lasota [6] and Brunovsky [2], and the methods used are similar to those used by these authors. Imposing certain conditions on ${\bf c}$ and ${\bf f}$, we show that for ${\bf v}$ in a certain subset of the phase space the solution uniformly approaches a limit set of dimension $n - 1$. We also show that in another subset of the phase space there are both (a) dense orbits and (b) periodic orbits of all periods. Each of (a) and (b) leads to a proof of the existence of a type of chaos in this subset.