Abstract:
After investigating by examples the unusual and striking elementary properties of the Penrose tilings and the Arnold cat map, we associate a finite symbolic dynamics with finite grammar rules to each of them. Instead of studying these Markovian systems with the help of set-topology, which would give only pathological results, a noncommutative approximately finite C*-algebra is associated to both systems. By calculating the K-groups of these algebras it is demonstrated that this noncommutative point of view gives a much more appropriate description of the phase space structure of these systems than the usual topological approach. With these specific examples it is conjectured that the methods of noncommutative geometry could be successfully applied to a wider class of dynamical systems.

Abstract:
A tiling is a cover of R^d by tiles such as polygons that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in tilings. From a tiling, we can construct a dynamical system which encodes the nature of the tiling. In the literature, properties of this dynamical system were investigated by studying how patches distribute in each tiling. In this article we conversely research distribution of patches from properties of the corresponding dynamical systems. We show periodic structures are hidden in tilings which are not necessarily periodic. Our results throw light on inverse problem of deducing information of tilings from information of diffraction measures, in a quite general setting.

Abstract:
In this case-study, we examine the effects of linear control on continuous dynamical systems that
exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or
controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak
and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can
yield the desired numerical results with two different continuous higher order dynamical systems
that exhibit chaotic behavior, the Lorenz equations and the R？ssler attractor.

Abstract:
The chaotic synchronization regime in coupled dynamical systems is considered. It has been shown, that the onset of synchronous regime is based on the appearance of the phase relation between interacting chaotic oscillators frequency components of Fourier spectra. The criterion of synchronization of spectral components as well as the measure of synchronization have been discussed. The universal power law has been described. The main results are illustrated by coupled R\"ossler systems, Van-der-Pol and Van-der-Pol-Duffing oscillators.

Abstract:
Explosive synchronization (ES), as one kind of abrupt dynamical transitions in nonlinearly coupled systems, has become a hot spot of modern complex networks. At present, many results of ES are based on the networked Kuramoto oscillators and little attention has been paid to the influence of chaotic dynamics on synchronization transitions. Here, the unified chaotic systems (Lorenz, Lü and Chen) and R?ssler systems are studied to report evidence of an explosive synchronization of chaotic systems with different topological network structures. The results show that ES is clearly observed in coupled Lorenz systems. However, the continuous transitions take place in the coupled Chen and Lü systems, even though a big shock exits during the synchronization process. In addition, the coupled R?ssler systems will keep synchronous once the entire network is completely synchronized, although the coupling strength is reduced. Finally, we give some explanations from the dynamical features of the unified chaotic systems and the periodic orbit of the R?ssler systems.

Abstract:
Amplitude death can occur in chaotic dynamical systems with time-delay coupling, similar to the case of coupled limit cycles. The coupling leads to stabilization of fixed points of the subsystems. This phenomenon is quite general, and occurs for identical as well as nonidentical coupled chaotic systems. Using the Lorenz and R\"ossler chaotic oscillators to construct representative systems, various possible transitions from chaotic dynamics to fixed points are discussed.

Abstract:
In the chaotic Lorenz system, Chen system and R\"ossler system, their equilibria are unstable and the number of the equilibria are no more than three. This paper shows how to construct some simple chaotic systems that can have any preassigned number of equilibria. First, a chaotic system with no equilibrium is presented and discussed. Then, a methodology is presented by adding symmetry to a new chaotic system with only one stable equilibrium, to show that chaotic systems with any preassigned number of equilibria can be generated. By adjusting the only parameter in these systems, one can further control the stability of their equilibria. This result reveals an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of a chaotic system.

Abstract:
In this paper we numerically investigate the dynamical behavior of fractional differential systems. By utilizing the fractional Adams method, we numerically find the smallest “efficient dimensions” of the fractional Lorenz system and R¨ossler system such that they are chaotic.

Abstract:
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of aperiodic tiling systems. In [7], we considered tilings of the euclidean spaces of finite dimension, and isomorphism was defined up to translation. Here, we consider, more generally, tilings of a metric space, and isomorphism is defined modulo an arbitrary group of isometries. In Section 1, we define the relational structures associated to tilings. The results of Section 2 concern local isomorphism, the extraction preorder and the characterization of relational structures which can be represented by tilings of some given type. In Section 3, we show that the notions of periodicity and invariance through a translation, defined for tilings of the euclidean spaces of finite dimension, can be generalized, with appropriate hypotheses, to relational structures, and in particular to tilings of noneuclidean spaces. The results of Section 4 are valid for uniformly locally finite relational structures, and in particular tilings, which satisfy the local isomorphism property. We characterize among such structures those which are locally isomorphic to structures without nontrivial automorphism. We show that, in an euclidean space of finite dimension, this property is true for a tiling which satisfies the local isomorphism property if and only if it is not invariant through a nontrivial translation. We illustrate these results with examples, some of them concerning aperiodic tilings systems of euclidean or noneuclidean spaces.

Abstract:
This paper deals with the problem of determining the conditions under which fractional order R？ssler toroidal system can give rise to chaotic behavior. Based on the harmonic balance method, four detailed steps are presented for predicting the existence and the location of chaotic motions. Numerical simulations are performed to verify the theoretical analysis by straightforward computations. 1. Introduction The concept of fractional calculus has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and L’H？spital in 1695. In the past three decades or so, fractional calculus gained considerable popularity and importance, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering [1–3]. In particular, by utilizing fractional calculus technique, many investigations were devoted to the chaotic behaviors and chaotic control of dynamical systems involving the fractional derivative, called fractional-order chaotic system [4–8]. For example, it has been shown that Chua circuit of the order as low as 2.7 can behave in chaotic manner [4]. In [5], the nonautonomous Duffing systems of the order less than 2 can still produce a chaotic attractor. In [6], chaotic behavior of the fractional-order Lorenz system was further studied. In [7], chaos and hyperchaos in the fractional-order R？ssler equations were also studied, in which the authors showed that chaos can exist in the fractional-order R？ssler equation with the order as low as 2.4, and hyperchaos can exist in the fractional-order R？ssler hyperchaos equation with the order as low as 3.8. Later on, the chaotic behavior and its control in the fractional-order Chen system were investigated in [8]. And recently, more dynamic behaviors of fractional order chaotic systems were analyzed by using different approaches; we refer the readers to [9–12]. However, to our knowledge, the conditions for chaos existence in dynamical systems (including integer-order system and fractional-order system) are still incomplete. For a given dynamical system, can we decide (without invoking numerical simulations) whether and in what parameter ranges, chaotic behavior might exist? It is still an open problem [13]. Even though some theorems such as Melnikov’s criteria [14] and Shil’nikov’s theorem [15] may be helpful in some special cases, it seems that a powerful tool is not generally available to determine the accurate parameter ranges for chaos existence in a given dynamical system. Moreover, the fact that