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Periodic motion is of paramount importance in science and engineering. The subject is usually covered in courses related to “waves and vibrations”, and Fourier series are frequently used as a tool of analysis of diverse phenomena. In addition, the discrete Fourier transform (DFT) is a means of signal processing in many fields. In this contribution, the periodic patterns of three different biological oscillators are examined using numerical tools suitable for students in the early stages of chemistry, physics, and engineering-oriented careers. The DFT is used to reveal oscillation periods related to the circadian and the menstrual cycles. The time evolution of the segmentation clock, a molecular oscillator operating at the genetic level in the early stages of embryonic growth of vertebrates, is also analyzed. Aimed at both students and instructors, brief descriptions of the methodologies involved are provided and a critical assessment of the results obtained with either of these techniques is carried out.

A word v=wu is a (nontrivial) Duval extension of the unbordered word w, if (u is not a prefix of v and) w is an unbordered factor of v of maximum length. After a short survey of the research topic related to Duval extensions, we show that, if wu is a minimal Duval extension, then u is a factor of w. We also show that finite, unbordered factors of Sturmian words are Lyndon words.

Crochemore's repetitions algorithm introduced in 1981 was the first O(n log n) algorithm for computing repetitions. Since then, several linear-time worst-case algorithms for computing runs have been introduced. They all follow a similar strategy: first compute the suffix tree or array, then use the suffix tree or array to compute the Lempel-Ziv factorization, then using the Lempel-Ziv factorization compute all the runs. It is conceivable that in practice an extension of Crochemore's repetitions algorithm may outperform the linear-time algorithms, or at least for certain classes of strings. The nature of Crochemore's algorithm lends itself naturally to parallelization, while the linear-time algorithms are not easily conducive to parallelization. For all these reasons it is interesting to explore ways to extend the original Crochemore's repetitions algorithm to compute runs. We present three variants of extending the repetitions algorithm to compute runs: two with a worsen complexity of O(n (log n)²), and one with the same complexity as the original algorithm. The three variants are tested for speed of performance and their memory requirements are analyzed. The third variant is tested and analyzed for various memory-saving alterations. The purpose of this research is to identify the best extension of Crochemore's algorithm for further study, comparison with other algorithms, and parallel implementation.

We study equality sets of mappings. In particular, we study D0L equality sets. If s = (s(n))_n≥0 and t = (t(n))_n≥0 are D0L sequences, their equality set is defined by E(s,t) = {n ≥ 0 ∣ s(n) = t(n)}. We study various periodicity and decidability questions concerning these sets. We also study HD0L and DT0L equality sets.

Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality. The goal of this paper is to analyze Fine and Wilf's periodicity theorem with respect to these equivalence relations. Fine and Wilf's theorem tells exactly how long a word with two periods p and q can be without having the greatest common divisor of p and q as a period. Recently, the same question has been studied for abelian periods. In this paper we show that for k-abelian periods the situation is similar to the abelian case: In general, there is no bound for the lengths of such words, but the values of the parameters p, q and k for which the length is bounded can be characterized. In the latter case we provide nontrivial upper and lower bounds for the maximal lengths of such words. In some cases (e.g., for k = 2) we found the maximal length precisely.

In [A. Frid, S. Puzynina and L. Q. Zamboni, On palindromic factorization of words, Adv. in Appl. Math.50 (2013) 737–748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately periodic. We introduce variants of this conjecture and prove this conjecture when the bound is 2. Especially we introduce left and right greedy palindromic lengths. These lengths are always greater than or equals to the initial palindromic length. When the greedy left (or right) palindromic lengths of prefixes of a word are bounded then this word is ultimately periodic.

We prove an inequality for the number of periods in a word $x$ in terms of the length of $x$ and its initial critical exponent. Next, we characterize all periods of the length-$n$ prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of $n$, and use this result to show that our inequality is tight for infinitely many words $x$. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where $x$ is overlap-free or squarefree.

The convergence characteristics of a single dissipative Hopfield-type neuron with self-interaction under periodic external stimuli are considered. Sufficient conditions are established for associative encoding and recall of the periodic patterns associated with the external stimuli. Both continuous-time-continuous-state and discrete-time-continuous-state models are discussed. It is shown that when the neuronal gain is dominated by the neuronal dissipation on average, associative recall of the encoded temporal pattern is guaranteed and this is achieved by the global asymptotic stability of the encoded pattern.

We estimate the probability that random noise, of several plausible standard distributions, creates a false alarm that a periodicity (or log-periodicity) is found in a time series. The solution of this problem is already known for independent Gaussian distributed noise. We investigate more general situations with non-Gaussian correlated noises and present synthetic tests on the detectability and statistical significance of periodic components. A periodic component of a time series is usually detected by some sort of Fourier analysis. Here, we use the Lomb periodogram analysis, which is suitable and outperforms Fourier transforms for unevenly sampled time series. We examine the false-alarm probability of the largest spectral peak of the Lomb periodogram in the presence of power-law distributed noises, of short-range and of long-range fractional-Gaussian noises. Increasing heavy-tailness (respectively correlations describing persistence) tends to decrease (respectively increase) the false-alarm probability of finding a large spurious Lomb peak. Increasing anti-persistence tends to decrease the false-alarm probability. We also study the interplay between heavy-tailness and long-range correlations. In order to fully determine if a Lomb peak signals a genuine rather than a spurious periodicity, one should in principle characterize the Lomb peak height, its width and its relations to other peaks in the complete spectrum. As a step towards this full characterization, we construct the joint-distribution of the frequency position (relative to other peaks) and of the height of the highest peak of the power spectrum. We also provide the distributions of the ratio of the highest Lomb peak to the second highest one. Using the insight obtained by the present statistical study, we re-examine previously reported claims of "log-periodicity" and find that the credibility for log-periodicity in 2D-freely decaying turbulence is weakened while it is strengthened for fracture, for the ion-signature prior to the Kobe earthquake and for financial markets.

We have studied the influence of the distribution of bimodal bonds on the phase transition in two-dimensional 8-state Potts model by the recently proposed Wang–Landau (WL) and the Swendsen–Wang (SW) algorithm. All simulations and measurements are done for r = 0.5. Physical quantities such as energy density and specific heat are evaluated at all temperatures. We have also obtained the probability distributions of the energy in order to monitor the transitions. We have observed that some cases of the periodically arranged bond distributions show a single peak, and some cases show double or triple peaks in the specific heat. Besides, it seems that the appearing of these peaks in the specific heat relates to a blocking procedure for periodicity. When the number of interaction pairs between the bimodal bonds is increased on the lattice with the blocking procedure, one can observe a single peak, otherwise, one can observe a double or triple peaks in the specific heat. From the point of view of simulation methods, the WL algorithm also works efficiently in the simulation of the system for a periodically arranged bond distribution as well as the SW algorithm.

Transmission of respiratory infectious diseases depends greatly on human close-proximity contacts, making thorough understanding of current and upcoming contacts essential for epidemic containment. Although different devices and software have been developed for contact data collection, there are few effective methods for contact prediction available in the near future as far as the authors know. In this study, we propose an approach to predict human contacts. We first extract human features together with their significances from the human contacts through alternating direction method of multipliers (ADMM), then predict future significances based on periodicity of contacts, and finally construct future contacts from human features and future significances. With the help of contact data collected in a Chinese University, we compare this approach with a trivial method of directly averaging known contacts. The comparison shows that our approach generates contacts deviating less from the true ones.

The Lorenz–Stenflo system is a four-parameter four-dimensional autonomous nonlinear continuous-time dynamical system, derived to model the time evolution of finite amplitude acoustic gravity waves in a rotating atmosphere. In this paper, we propose a modified Lorenz–Stenflo system, where the variable $x$ in the fourth equation of the original Lorenz–Stenflo system was replaced by $ln|x|$. We investigate cross-sections of the parameter-space of this new system, characterizing regions of different dynamical behaviors. We show that the aforementioned replacement may promote the emergence of organized periodic structures in places of these cross-sections, where they did not exist before modification.

This paper proposes a new dynamical system derived from the Chen system. It is designed by replacing the linear term $(y-x)$ in the $ẋ$ equation of the Chen system, by the nonlinear term $sinh(y-x)$. Three cross-sections of the three-dimensional parameter-space of this new system, also called parameter planes, are used in order to investigate numerically the influence of replacing on solutions. It is shown that most of the chaotic solutions in parameter planes of the Chen system are suppressed by replacing, giving rise to periodic solutions. Also, it is shown that most of the unbounded solutions become periodic solutions.

In this paper, we analyze quantum walks on cycles with an absorbing wall. We set the absorbing wall on cycles with N vertices (where N is an even number), and divide $|\phi (t)〉$ into two parts, $L(t)$ and $R(t)$. Due to the periodicity of the cycles, the condition $L(M+N)=L(M)$ (or $R(M+N)=R(M)$) is applied to $L(t)$ and $R(t)$, then the transmission probability $P_M$ and reflection probability $Q_M$ at the absorbing wall $n=M(M=\frac{N}{2})$ at time t are obtained. Furthermore, we show that over time, the absorbing wall absorbs less and less.

In this work, a new cubic-like smooth nonlinearity is generated by modifying Chua’s piecewise-linear segmental nonlinear function using logarithmic cos-hyperbolic function implementation. A logarithmic cos-hyperbolic function possessing smooth symmetric nonlinear characteristics is implemented through CMOS-based circuit design using the current mode approach. The nonlinear design is then incorporated in a new third-order chaotic oscillator configuration to produce chaotic oscillations. This chaotic circuit is tuned to develop different attractors through the bifurcation parameter. Moreover, the dynamics of chaos such as antimonotonicity and coexistence of attractors are also depicted in circuit simulation by tuning various controlling parameters. Additionally, some numerical analyses are performed on this dynamic system to justify the existence of chaoticity and attractors’ development. This design has been optimized for low-voltage and moderately high dominant frequency of oscillations. Simulations are done using 180-nm CMOS technology in Cadence Virtuoso. Experimental results are presented to verify the workability of the proposed chaotic system.

In this paper, SEIS epidemic models with varying population size are considered. Firstly, we consider the case when births of population are throughout the year. A threshold σ is identified, which determines the outcome of disease, that is, when σ < 1, the disease dies out; whereas when σ > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable; when σ = 1, bifurcation occurs and leads to "the change of stability". Two other thresholds σ′ and are also identified, which determine the dynamics of epidemic model with varying population size, when the disease dies out or it is endemic. Secondly, we consider the other case, birth pulse. The population density is increased by an amount B(N)N at the discrete time nτ, where n is any non-negative integer and τ is a positive constant, B(N) is density-dependent birth rate. By applying the corresponding stroboscopic map, we obtain the existence of infection-free periodic solution with period τ. Lastly, through numerical simulations, we show the dynamic complexities of SEIS epidemic models with varying population size, there is a sequence of bifurcations, leading to chaotic strange attractors. Non-unique attractors also appear, which implies that the dynamics of SEIS epidemic models with varying population size can be very complex.

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

High-order Lorenz systems with five, six, eight, nine, and eleven equations are derived by choosing different numbers of Fourier modes upon truncation. For the original Lorenz system and for the five high-order Lorenz systems, solutions are numerically computed, and periodicity diagrams are plotted on $(\sigma ,r)$ parameter planes with $\sigma ,r\in [0,1000]$, and $b=8/3$. Dramatic shifts of patterns are observed among periodicity diagrams of different systems, including the appearance of expansive areas of period 2 in the fifth-, eighth-, ninth-, and 11th-order systems and the disappearance of the onion-like structure beyond order 5. Bifurcation diagrams along with phase portraits offer a closer look at the two phenomena.

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

In this paper, we investigate a three-parameter four-dimensional dynamical system, which is modeled by a set of four first-order nonlinear ordinary differential equations, each of which contains a crossed cubic term. Dynamical behaviors are characterized in the parameter space of the model. In fact, we use some cross-sections of a three-dimensional parameter-space, namely three related parameter planes, to locate regular and chaotic regions, as well as multistability regions. Lyapunov exponents spectra, bifurcation diagrams, and phase-space portraits are used to complete the analysis.