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It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

This paper illustrates an approach to generate multi-wing attractors in coupled Lorenz systems. In particular, novel four-wing (eight-wing) hyperchaotic attractors are generated by coupling two (three) identical Lorenz systems. The paper shows that the equilibria of the proposed systems have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. In analogy with the original Lorenz system, where the two-wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four-wings (eight-wings) of these attractors are located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.

Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubic–quintic Ginzburg–Landau Equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this paper, we address the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. First, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Next, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the starting formulation for the Lagrangian is recent and not well explored. Also, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler–Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well-known stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors yield solitons whose amplitudes and widths are non-stationary or time dependent. In particular, pulsating and snake dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves – their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Snakes and chaotic solitons will be addressed in subsequent papers. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.

This paper deals with the observer-based synchronization in a cascade connection of hyperchaotic discrete-time systems. The paper demonstrates that exact synchronization in finite time is achievable between pairs of drive–response systems using only a scalar synchronizing signal. This “propagated synchronization” starts from the innermost drive–response system pair and propagates toward the outermost drive-system pair. Choosing the drive-system input to be an information signal (encrypted via an arbitrary encryption function) yields a potential application of this architecture in chaos-based communications.

This paper is concerned with the design of a proportional-integral-derivative (PID) controller for synchronization of delayed discrete chaotic models. The evolutionary programming algorithm (EPA) has been considered as a useful technique for finding global optimization solutions for certain complicated functions in recent years. Therefore, in this paper, we attempt to use the EP algorithm in PID control design for deriving optimal or near optimal PID control gains such that a performance index between the master and slave chaotic systems is minimized. A numerical result exemplifies the synchronization procedure.

This paper simulates the non-linear vibration of a gear pair including shaft flexibility. Backlash is included because of the clearance between the teeth and the periodic variation in the stiffness of the mesh produces a parametric excitation. This paper shows the different behaviour that may be observed when the additional degree of freedom is added to the single degree of freedom model previously analysed.

In this talk we present main points of some of Fock papers which were not properly followed when published and review Fock's interpretation of the energy–time uncertainty relation, as well as his ideas on generalization of the concept of physical space published in his last paper.

This paper presents an analysis of the dynamical behaviour of a non-symmetric oscillator with piecewise-linearity. The Chen–Langford (C–L) method is used to obtain the averaged system of the oscillator. Using this method, the local bifurcation and the stability of the steady-state solutions are studied. A Runge–Kutta method, Poincaré map and the largest Lyapunov’s exponent are used to detect the complex dynamical phenomena of the system. It is found that the system with piecewise-linearity exhibits periodic oscillations, period-doubling, period-3 solution and then chaos. When chaos is found, it is detected by examining the phase plane, bifurcation diagram and the largest Lyapunov’s exponent. The results obtained in this paper show that the vibration system with piecewise-linearity do exhibit quite similar dynamical behaviour to the discrete system given by the logistic map.

In this paper we analyse random walk on a fractal structure, specifi- cally fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the corresponding probability distribution which is gaussian like in nature, but shows deviation from the standard behaviour. Moments are calculated in terms of Euclidean distance for a von Koch curve. We also analyse Levy distribution on the same fractal structure, where the dimen- sion of the fractal curve shows significant contribution to the distrubution law by modyfying the nature of moments. The appendix gives a short note on Fourier transform on fractal curves., Comment: 15 pages, 4 figures

In this paper we study the generalized Gause model, with a logistic growth rate for the prey in absence of the predator, a constant death rate for the predator and for several different classes of functional response, all non-analytical. First we consider the piecewise-linear functional response of Holling type I, which essentially has a linear functional response on a bounded interval and a constant functional response for large enough prey density. Next we consider differentiable modifications of this type of functional response, one being a concave down function, the other one being a sigmoidal function. Our main interest is the number of closed orbits of the systems under consideration and the global stability of the system. We compare the generalized Gause model with a functional response that is non-analytical with the generalized Gause model with a functional response that is analytical (e.g., Holling type II or III) and show that the behaviour in the first case is more complicated. As examples of this more complicated behaviour we mention: the co-existence of a stable equilibrium with a stable limit cycle and the existence of a family of closed orbits.

In this paper, we present a novel explicit analytical solution for the normalized state equations of mutually-coupled simple chaotic systems. A generalized analytical solution is obtained for a class of simple nonlinear electronic circuits with two different nonlinear elements. The synchronization dynamics of the circuit systems were studied using the analytical solutions. the analytical results thus obtained have been validated through numerical simulation results. Further, we provide a sufficient condition for synchronization in mutually-coupled, second-order simple chaotic systems through an analysis on the eigenvalues of the difference system. The bifurcation of the eigenvalues of the difference system as functions of the coupling parameter in each of the piecewise-linear regions, revealing the existence of stable synchronized states is presented. The stability of synchronized states are studied using the {\emph{Master Stability Function}}. Finally, the electronic circuit experimental results confirming the phenomenon of complete synchronization in each of the circuit system is presented., 12 pages, 6 figures. arXiv admin note: text overlap with arXiv:1611.04289

In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems. We prove that Li-Yorke pairs of A-coupled-expanding system under some conditions have full Hausdorff dimension in the invariant set. Moreover we give a generalization of the result of [5] which is on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and have a self-similar invariant set, to the case of the dynamical systems topologically semi-conjugate to some kinds of subshifts. Furthermore Hausdorff dimension of “chaotic invariant set” for some kinds of A-coupled-expanding maps is shown.

In this paper we present applications of special numbers obtained from a difference equation of degree three. As a particular case of this difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternion algebras. Using properties of these quaternion elements, we can define a set with an interesting algebraic structure, namely, an order on a generalized rational quaternion algebra. Another presented application is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes).

The risk of propagation of infectious diseases such as avian influenza and COVID-19 is generally controlled or reduced by quarantine measures. Considering this situation, a network-based SIQS (susceptible-infected-quarantined-susceptible) infectious disease model with nonmonotone incidence rate is established and analyzed in this paper. The psychological impact of the transmission of certain diseases in heterogeneous networks at high levels of infection may be characterized by the related nonmonotone incidence rate. The expressions of the basic reproduction number and equilibria of the model are determined analytically. We demonstrate in detail the uniform persistence of system and the global asymptotic stability of the disease-free equilibrium. The global attractivity of the unique endemic equilibrium is discussed by using monotone iteration technique. We obtain that the endemic equilibrium is globally asymptotically stable under certain conditions by constructing appropriate Lyapunov function. In addition, numerical simulations are performed to indicate the theoretical results.

This paper evaluated the correlation among the activations of eyes and brain in visual stimulation. Because of the complex structure of eye fluctuations and brain signals, we employed fractal theory and sample entropy to decode their correlation. Ten subjects looked at four different images with different complexities while we collected their eye fluctuations and EEG signals simultaneously. The results indicated that the variations of the complexity of eye fluctuations and EEG signals are strongly correlated (r = -0.8043 in case of fractal dimension and r = -0.9740 in case of sample entropy), which reflects the correlation among the activations of the eyes and brain. This analysis can be conducted to examine how the activations of other organs are linked to brain activation.

This paper provides a derivation of Zipf-Pareto laws directly from the principle of least effort. A probabilistic functional of efficiency is introduced as the consequence of an extension of the nonadditivity of the efficiency of thermodynamic engine to a large number of living agents assimilated to engines, all randomly distributed over their output. Application of the maximum calculus to this efficiency yields the Zipf-Pareto laws., Comment: 26 pages, 1 figure

Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m − n . Our RME depends on three parameters: the network size n , the size of the smaller set m , and the connectivity between the two sets α , where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r , the Shannon entropy of the eigenvectors S , and the Randic index R . First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ ≡ ξ ( n , m , α ) that scales the average normalized measures X ¯ > (with X representing r , S and R ). Specifically, we show that (i) ξ ∝ α n with a weak dependence on m , and (ii) for ξ 1 / 10 most vertices in the mutualistic network are isolated, while for ξ > 10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1 / 10 ξ 10 . Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves X ¯ > vs. ξ show good correspondence with the properties of real-world networks.

Recent advances in the field of uncertainty quantification are based on achieving suitable functional representations of the solutions to random systems. This aims at improving the performance of Monte Carlo simulation, at least for low-dimensional problems and moderately large independent variable. One of these functional representations are the so-called generalized polynomial chaos (gPC) expansions, based upon orthogonal polynomial decompositions. When the input random parameters are independent (a germ), a Galerkin projection technique applied to the truncated gPC expansion is usually employed. This approach exhibits fast mean-square convergence for smooth dynamics, whenever applicable. However, the main difficulty arises when solving the Galerkin system for the gPC coefficients, which may rely on different solvers (algorithms and codes) than those for the original system. A recent contribution noticed that, for random Hamiltonian systems, the Galerkin system is Hamiltonian too. Thus, the well-known symplectic integrators can be applied. The present paper investigates random Hamiltonian systems in general, when the input random parameters may be non-independent. In such a case, polynomial expansions based on the canonical basis and an imitation of the Galerkin projection technique are proposed. The Hamiltonian structure of the original system is unfortunately not conserved, but volume preservation is. Hence volume-preserving integrators are of use. Numerical experiments suggest that the proposed polynomial expansions may be useful for fast and accurate uncertainty quantification.

In this paper, the commensurate fractional-order variant of an Hopfield neuronal network is analyzed. The system is integrated with the ABM method for fractional-order equations. Beside the standard stability analysis of equilibria, the divergence of fractional order is proposed to determine the instability of the equilibria. The bifurcation diagrams versus the fractional order, and versus one parameter, reveal a strange phenomenon suggesting that the bifurcation branches generated by initial conditions outside neighborhoods of unstable equilibria are spurious sets although they look similar with those generated by initial conditions close to the equilibria. These spurious sets look ``delayed'' in the considered bifurcation scenario. Once the integration step-size is reduced, the spurious branches maintain their shapes but tend to the branches obtained from initial condition within neighborhoods of equilibria. While the spurious branches move once the integration step size reduces, the branches generated by the initial conditions near the equilibria maintain their positions in the considered bifurcation space. This phenomenon does not depend on the integration-time interval, and repeats in the parameter bifurcation space., accepted in CS&F 2021

The spatial distributions of cities fall into two groups: one is the simple distribution with characteristic scale (e.g. exponential distribution), and the other is the complex distribution without characteristic scale (e.g. power-law distribution). The latter belongs to scale-free distributions, which can be modeled with fractal geometry. However, fractal dimension is not suitable for the former distribution. In contrast, spatial entropy can be used to measure any types of urban distributions. This paper is devoted to generalizing multifractal parameters by means of dual relation between Euclidean and fractal geometries. The main method is mathematical derivation and empirical analysis, and the theoretical foundation is the discovery that the normalized fractal dimension is equal to the normalized entropy. Based on this finding, a set of useful spatial indexes termed dummy multifractal parameters are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The dummy multifractal indexes are applied to the population density distribution of Hangzhou city, China. The calculation results reveal the feature of spatio-temporal evolution of Hangzhou's urban morphology. This study indicates that fractal dimension and spatial entropy can be combined to produce a new methodology for spatial analysis of city development., Comment: 23 pages, 3 figures, 5 tables

In this study, a nonlinear investigation of a periodically driven gas bubble in glycerine is presented. The bifurcation structure of the bubble oscillator (Keller–Miksis equation) is explored in the pressure amplitude-frequency parameter plane of the excitation by means of initial (high resolution bi-parametric plots) and boundary value problem solvers at various ambient temperatures. The range of the applied temperature covers two orders of magnitude difference in the liquid viscosity which is the main damping factor of the system. Therefore, the evolution of the harmonic and ultraharmonic resonances are presented starting with an overdamped behaviour (there are no resonances in the parameter space) and ending up with a fully developed bifurcation superstructure. The results reveal a complex period bubbling mechanism organized in a Farey-tree; inside each bubble a fine substructure of alternating chaotic and periodic bands exist. The description of the bifurcation structure presented throughout the paper can help to understand the mechanism of dissipation on the behaviour of nonlinear systems in more detail.

This paper focuses on providing a new scheme to find the fuzzy approximate solution of fractional differential equations (FDEs) under uncertainty. The Caputo-type derivative base on the generalized Hukuhara differentiability is approximated by a linearization formula to reduce the corresponding uncertain FDE to an ODE under fuzzy concept. This new approach may positively affect on the computational cost and easily apply for the other types of uncertain fractional-order differential equation. The performed numerical simulations verify the proficiency of the presented scheme.

The spatially discrete-continuous dynamical systems, that are composed of a spatially extended medium coupled with a set of lumped elements, are frequently met in different fields, ranging from electronics to multicellular structures in living systems. Due to the natural heterogeneity of such systems, the calculation of Lyapunov exponents for them appears to be a challenging task, since the conventional techniques in this case often become unreliable and inaccurate. The paper suggests an effective approach to calculate Lyapunov exponents for discrete-continuous dynamical systems, which we test in stability analysis of two representative models from different fields. Namely, we consider a mathematical model of a 1D transferred electron device coupled with a lumped resonant circuit, and a phenomenological neuronal model of spreading depolarization, which involves 2D diffusive medium. We demonstrate that the method proposed is able reliably recognize regular, chaotic and hyperchaotic dynamics in the systems under study.

In this paper, the severe acute respiratory syndrome coronavirus (SARS-CoV-2) or COVID-19 is researched by employing mathematical analysis under modern calculus. In this context, the dynamical behavior of an arbitrary order p and fractal dimensional q problem of COVID-19 under Atangana Bleanu Capute (ABC) operator for the three cities, namely, Santos, Campinas, and Sao Paulo of Brazil are investigated as a case-study. The considered problem is analyzed for at least one solution and unique solution by the applications of the theorems of fixed point and non-linear functional analysis. The Ulam-Hyres stability condition via nonlinear functional analysis for the given system is derived. In order to perform the numerical simulation, a two-step fractional type, Lagrange plynomial (Adams Bashforth technique) is utilized for the present system. MATLAB simulation tools have been used for testing different fractal fractional orders considering the data of aforementioned three regions. The analysis of the results finally infer that, for all these three regions, the smaller order values provide better constraints than the larger order values.

In this paper, we present a new type of Gronwall inequality and discuss some particular cases. We apply these results to investigate the uniqueness and δ -Ulam–Hyers–Rassias stability of mild solutions of a fractional differential equation with non-instantaneous impulses in a P δ -normed Banach space. In this sense, we present an example, in order to elucidate one of the results discussed.

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \xi\sigma (u_t(x)) \dot F(t,x), \end{equation*} where $\xi$ is a positive parameter and $\sigma$ is a globally Lipschitz continuous function. The stochastic forcing term $\dot F(t,x)$ is white in time but possibly colored in space. The operator $\mathcal{L}$ is a non-local operator. We study the behaviour of the solution with respect to the parameter $\xi$, extending the results in \cite{FoonNual} and \cite{Bin}, Comment: 14 pages; updated versionof the original paper titled "Large time behaviour for the non-linear fractional stochastic heat equation"

We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving fractional boundary value problems. As an application, we solve a special class of fractional Euler-Lagrange equations. The method is based on Hale and Townsend algorithm for finding the roots and weights of the fractional Gauss-Jacobi quadrature rule and the predictor-corrector method introduced by Diethelm for solving fractional differential equations. Illustrative examples show that the given method is more accurate than the one introduced in [Comput. Math. Appl. 66 (2013), no. 5, 597--607], which uses the Golub-Welsch algorithm for evaluating fractional directional integrals., Comment: This is a preprint of a paper whose final and definite form is with 'Chaos, Solitons & Fractals', ISSN: 0960-0779. Submitted 1 Dec 2016; Article revised 17 Apr 2017; Article accepted for publication 19 Apr 2017; see [http://dx.doi.org/10.1016/j.chaos.2017.04.034]

This paper derives the stochastic solution of a Cauchy problem for the distribution of a fractional diffusion process. The governing equation involves the Bessel-Riesz derivative (in space) to model heavy tails of the distribution, and the Caputo-Djrbashian derivative (in time) to depicts the memory of the diffusion process. The solution is obtained as Brownian motion with time change in terms of the Bessel-Riesz subordinator on the inverse stable subordinator. This stochastic solution, named fractional Bessel-Riesz motion, provides a method to simulate a large class of stochastic motions with memory and heavy tails.

In this paper, we introduce h(x) – Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field K ( K = R , C ) . These polynomials generalize h(x) – Fibonacci quaternion polynomials andh(x) – Fibonacci octonion polynomials. For h(x) – Fibonacci polynomials in an arbitrary algebra, we provide generating function, Binet-style formula, Catalan-style identity, and d’Ocagne-type identity.

This paper focuses on the stability of a hydropower station. First, we established a novel nonlinear mathematical model of a Francis hydro-turbine governing system considering both fractional-order derivative and time delay. The fractional-order α , which is introduced into the penstock system, in the range from 0.82 to 1.00 is on the left side of the model in a incommensurate manner in increment of 0.03 to provide an adjustable degree of system memory. The time delay τ , which exists between the signal and response in the hydraulic servo system, in the range from 0 s to 0.26 s is inserted on the right side of the model in increment of 0.04 s . Utilizing the principle of statistical physics, we respectively explored the effects of the fractional-order α and the time delay τ on the stable region of the system. Furthermore, we exhaustively investigated the nonlinear dynamic behaviors of the system with different governor parameters by using bifurcation diagrams, time waveforms and power spectrums, finding that only under the condition of reasonable collocation of governor parameters the system can maintain stable operation. Finally, all of the above numerical experiments supply new methods for studying the stability of a hydropower station.

We propose a chaotic hash algorithm based on circular shifts with variable parameters in this paper. We exploit piecewise linear chaotic map and one-way coupled map lattice to produce initial values and variable parameters. Circular shifts are introduced to improve the randomness of hash values. We evaluate the proposed hash algorithm in terms of distribution of the hash value, sensitivity of the hash value to slight modifications of the original message and secret keys, confusion and diffusion properties, robustness against birthday and meet-in-the-middle attacks, collision tests, analysis of speed, randomness tests, flexibility, computational complexity, and the results demonstrate that the proposed algorithm has strong security strength. Compared with the existing chaotic hash algorithms, our algorithm shows moderate statistical performance, better speed, randomness tests, and flexibility.

In this paper we study two entropic dynamical models from the viewpoint of information geometry. We study the geometry structures of the associated statistical manifolds. In order to analyse the character of the instability of the systems, we obtain their geodesics and compute their Jacobi vector fields. The results of this work improve and extend a recent advance in this topics studied in [13], Accepted for publication in Chaos, Solitons & Fractals

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.

The uniformly hyperbolic Anosov C-systems defined on a torus have very strong instability of their trajectories, as strong as it can be in principle. These systems have exponential instability of all their trajectories and as such have mixing of all orders, nonzero Kolmogorov entropy and a countable set of everywhere dense periodic trajectories. In this paper we are studying the properties of their spectrum and of the entropy. For a two-parameter family of C-system operators A(N,s), parametrised by the integers N and s, we found the universal limiting form of the spectrum, the dependence of entropy on N and the period of its trajectories on a rational sublattice. One can deduce from this result that the entropy and the periods are sharply increasing with N. We present a new three-parameter family of C-operators A(N,s,m) and analyse the dependence of its spectrum and of the entropy on the parameter m. We developed our earlier suggestion to use these tuneable Anosov C-systems for multipurpos Monte-Carlo simulations. The MIXMAX family of random number generators based on Anosov C-systems provide high quality statistical properties, thanks to their large entropy, have the best combination of speed, reasonable size of the state, tuneable parameters and availability for implementing the parallelisation., 12 pages, 5 figures, references, comments and a note added

In this paper, a reaction-diffusion SIR epidemic model is proposed. It takes into account the individuals mobility, the time periodicity of the infection rate and recovery rate, and the general nonlinear incidence function, which contains a number of classical incidence functions. In our model, due to the introduction of the general nonlinear incidence function, the boundedness of the infected individuals can not be obtained, so we study the existence and nonexistence of periodic traveling wave solutions of original system with the aid of auxiliary system. The basic reproduction number R 0 and the critical wave speed c * are given. We obtained the existence and uniqueness of periodic traveling waves for each wave speed c > c * using the Schauder’s fixed points theorem when R 0 > 1 . The nonexistence of periodic traveling waves for two cases (i) R 0 > 1 and 0 c c * , (ii) R 0 ≤ 1 and c ≥ 0 are also obtained. These results generalize and improve the existing conclusions. Finally, the numerical experiments support the theoretical results. The differences of traveling wave solution between periodic system and general aperiodic coefficient system are analyzed by numerical simulations.

In this work, we generalize a (deterministic) mathematical model that anticipates the influence of temperature on dengue transmission incorporating temperature-dependent model parameters. The motivation comes by the epidemiological evidence and several recent studies clearly states fluctuations in temperature, rainfall, and global climate indexes are determinant on the transmission dynamic and epidemic behavior of dengue virus that causes deadly diseases with incidence rates significantly risen worldwide in the past decade. Taking into account the importance of the subject in nowadays and the diversity of fractional calculus operators in mathematical modeling of complex real-world systems, in this paper we investigated the importance of the new model based on Mittag-Leffler kernel as being non-singular kernel. The sensitivity analysis of the generalized model is newly investigated. Numerical simulations are carried out in a comparative sense within the temperature fluctuations for both singular and non-singular fractional operators of different orders.

The effective reproduction number (R) which signifies the number of secondary cases infected by one infectious individual, is an important measure of the spread of an infectious disease. Due to the dynamics of COVID-19 where many infected people are not showing symptoms or showing mild symptoms, and where different countries are employing different testing strategies, it is quite difficult to calculate the R, while the pandemic is still widespread. This paper presents a probabilistic methodology to evaluate the effective reproduction number by considering only the daily death statistics of a given country. The methodology utilizes a linearly constrained Quadratic Programming scheme to estimate the daily new infection cases from the daily death statistics, based on the probability distribution of delays associated with symptom onset and to reporting a death. The proposed methodology is validated in-silico by simulating an infectious disease through a Susceptible-Infectious-Recovered (SIR) model. The results suggest that with a reasonable estimate of distribution of delay to death from the onset of symptoms, the model can provide accurate estimates of R. The proposed method is then used to estimate the R values for two countries.

In this paper we conduct a simulation study of the spread of an epidemic like COVID-19 with temporary immunity on finite spatial and non-spatial network models. In particular, we assume that an epidemic spreads stochastically on a scale-free network and that each infected individual in the network gains a temporary immunity after its infectious period is over. After the temporary immunity period is over, the individual becomes susceptible to the virus again. When the underlying contact network is embedded in Euclidean geometry, we model three different intervention strategies that aim to control the spread of the epidemic: social distancing, restrictions on travel, and restrictions on maximal number of social contacts per node. Our first finding is that on a finite network, a long enough average immunity period leads to extinction of the pandemic after the first peak, analogous to the concept of "herd immunity". For each model, there is a critical average immunity length $L_c$ above which this happens. Our second finding is that all three interventions manage to flatten the first peak (the travel restrictions most efficiently), as well as decrease the critical immunity length $L_c$, but elongate the epidemic. However, when the average immunity length $L$ is shorter than $L_c$, the price for the flattened first peak is often a high second peak: for limiting the maximal number of contacts, the second peak can be as high as 1/3 of the first peak, and twice as high as it would be without intervention. Thirdly, interventions introduce oscillations into the system and the time to reach equilibrium is, for almost all scenarios, much longer. We conclude that network-based epidemic models can show a variety of behaviors that are not captured by the continuous compartmental models., 17 pages text, 27 figures of which 22 in the appendix

This paper analyses the bit error rate (BER) of the spatial modulation orthogonal frequency division multiplex (SM-OFDM) system and derives the optimal power allocation between the data and the pilot symbols by minimizing the upper bound for the BER operating with imperfect channel estimation. Furthermore, we prove the proposed optimal power allocation scheme applies to all generalized linear interpolation techniques with the minimum mean square error (MMSE) channel estimation . Simulation results show that employing the proposed optimal power allocation provides a substantial gain in terms of the average BER performance for the SM-OFDM system compared to its equal-power-allocation counterpart.

Alternate bearing or masting is a general yield variability phenomenon in perennial tree crops. This paper first presents a theoretical modeling and simulation study of the mechanism for this dynamics in citrus, and then provides a test of the proposed models using data from a previous 16-year experiment in a citrus orchard. Our previous studies suggest that the mutual effects between vegetative and reproductive growths caused by resource allocation and budgeting in plant body might be considered as a major factor responsible for the yield oscillations in citrus. Based on the resource budget model proposed by Isagi et al. (J Theor Biol. 1997;187:231-9), we first introduce the new leaf growth as a major energy consumption component into the model. Further, we introduce a nonlinear Ricker-type equation to replace the linear relationship between costs for flowering and fruiting used in Isagi's model. Model simulations demonstrate that the proposed new models can successfully simulate the reproductive behaviors of citrus trees with different fruiting dynamics. These results may enrich the mechanical dynamics in tree crop reproductive models and help us to better understand the dynamics of vegetative-reproductive growth interactions in a real environment.

In this paper the spectral problem of third order for the inverse scattering transform (IST) method is solved. For the discrete part of the spectral data, the two-multiple poles are taken into account. The line spectrum of continuum states for the IST method is examined as well. The suggested spectrum approximates in first order the step-function. The scope for the suggested spectral data is demonstrated through the analysis of the Vakhnenko–Parkes equation that allows new solutions to be obtained. The account of the time-dependence is different from the standard procedure.

In this paper, we present the solutions of 1-D and 2-D non-linear partial differential equations with initial conditions. We approach the solutions in time domain using two methods. We first solve the equations using Fourier spectral approximation in the spatial domain and secondly we compare the results with the approximation in the spatial domain using orthogonal functions such as Legendre or Chebyshev polynomials as their basis functions. The advantages and the applicability of the two different methods for different types of problems are brought out by considering 1-D and 2-D nonlinear partial differential equations namely the Korteweg-de-Vries and nonlinear Schrodinger equation with different potential function. (C) 2015 Elsevier Ltd. All rights reserved.

a mathematical model depicting the spread of covid-19 epidemic and implementation of population covid-19 intervention in Italy. The model has 8 components leading to system of 8 ordinary differential equations. In this paper, we investigate the model using the concept of fractional differential operator. A numerical method based on the Lagrange polynomial was used to solve the system equations depicting the spread of COVID-19. A detailed investigation of stability including reproductive number using the next generation matrix, and the Lyapunov were presented in detail. Numerical simulations are depicted for various fractional orders.

An exact classification of the different dynamical behaviors that exhibits the phase space of a reversible and conservative cellular automaton, the so-called Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of the Ising model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequence of the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point or a cycle. In this work, we classify them in four types accordingly to well differentiated topological characteristics. Three of them –which we call of type S-I, S-II, and S-III– share a symmetry property, while the fourth, which we call of type AS does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous types of cycles. Moreover, at a combinatorial level, we can determine the number of cycles for some small periods which are almost always present in the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space and, in addition, we realize an exhaustive study of a small Ising system (4 × 4) which is thoroughly analyzed under this new framework, and where simple mathematical tools are introduced in order to have a more direct understanding of the Q2R dynamics and to rediscover known properties like the energy conservation.

The proof that an attractor is chaotic is not trivial. A single-scroll attractor in the Chen system with time delay is investigated through both theory and simulation. The Chen system with time delay is infinite dimensional parameterized by the time delay τ. The detailed procedure operations for finding topological horseshoe in the delay differential equation are different from that one in ordinary differential equation. We show the existence of chaos by using both the topological horseshoe theory and its corollary, and the Smale horseshoe construction if the geometry of the attractor on the 2D plane section satisfies certain conditions. This paper presents both methods for establishing the presence of the horseshoe in a Chen system with time delay. In the first method, we select two quadrilaterals in the 2D transversal section, and calculate the relationship of the quadrilaterals under the map. In the second method, we select quadrilaterals in the neighborhood of a short unstable periodic orbit in the section and obtain the approximate location of the quadrilaterals under the map, yielding the Smale horseshoe. By using the above two methods, the geometrical expansion of the quadrilaterals under the map satisfies the Topological Horseshoe Corollary and also the Smale horseshoe construction, thus showing that the time-delayed single-scroll attractor is indeed chaotic.

A family of classical mathematical problems considers the visibility properties of geometric figures in the plane, e.g. curves or polygons. In particular, the {\it domination problem} tries to find the minimum number of points that are able to dominate the whole set, the so called, {\it domination number}. Alternatively, other problems try to determine the subsets of points with a given cardinality, that maximize the basin of domination, the {\it partial dominating set}. Since a discrete time series can be viewed as an ordered set of points in the plane, the dominating number and the partial dominating set can be used to obtain additional information about the visibility properties of the series; in particular, the total visibility number and the partial visibility set. In this paper, we apply these two concepts to study times series that are generated from the logistic map. More specifically, we focus this work on the description of the Feigenbaum cascade to the onset of chaos. We show that the whole cascade has the same total visibility number, $v_T=1/4$. However, a different distribution of the partial visibility sets and the corresponding partial visibility curves can be obtained inside both periodic and chaotic regimes. We prove that the partial visibility curve at the Feigenbaum accumulation point $r_{\infty} \approx 3.5699$ is the limit curve of the partial visibility curves ($n+1$-polygonals) that correspond to the periods $T=2^{n}$ for $n=1,2,\ldots$. We analytically calculate the length of these $n+1$-polygonals and, as a limit, we obtain the length of the partial visibility curve at the onset of chaos, $L_{\infty} = L(r_{\infty}) \approx 1.0414387863$. Finally, we compare these results with those obtained from the period 3-cascade, and with the partial visibility curve of the chaotic series at the crossing point $r_c \approx 3.679$., Comment: 10 pages, 7 figures

In this paper, we compare the ordinary and fractional versions of a prospective model in the prostate cancer literature to be able to verify the hypothesis that fractional differentiation can improve stability results and data fitting ability of the ordinary version. For that, the mean square error values for androgen and prostate-specific antigen for the first 1.5 cycles of intermittent androgen suppression therapy administered to 62 selected patients from the Vancouver Prostate Center are computed. Stability analyses of the fractional models are also studied and showed how the fractional differentiation contributes to improve the stability results of an ordinary prostate model. Additionally, our findings are supported by PSA and androgen simulations of three different class of patients. Finally, with a conclusion part, which hints for future works should be taken into account are pointed out. (C) 2019 Elsevier Ltd. All rights reserved.

Regularity has always been attributed to periodicity. However, there has been a spurt of interest in another unique type of regularity called anitperiodicity. In this paper we have presented results of antiperiodic oscillations obtained from a forced duffing equation with negative linear stiffness wherein the increase in the number of peaks in anti periodic oscillation with the forcing strength has been observed. Similarity function has been used to identify the antiperiodic oscillation and further the bifurcation diagram has been plotted and stability analysis of the fixed points have been carried out to understand its dynamics. An analog electronic circuit governed by the forced Duffing equation has been designed and developed to investigate the dynamics of the antiperiodic oscillations. The circuit is quite robust and stable to enable the comparison of its analog output with the numerically simulated data. Power spectrum analysis obtained by fast Fourier transform has been corroborated using a nonlinear statistical technique called rescale range analysis method. By this technique we have estimated the Hurst exponents and detected the coherent frequencies present in the system. (C) 2015 Elsevier Ltd. All rights reserved.

In this paper we prove the existence of full measure unbounded chaotic attractors which are persistent under parameter perturbation (also called robust). We show that this occurs in a discontinuous piecewise smooth one-dimensional map f, belonging to the family known as Nordmark’s map. To prove the result we extend the properties of a full shift on a finite or infinite number of symbols to a map, here called Baker-like map with infinitely many branches, defined as a map of the interval I = [ 0 , 1 ] into itself with infinitely branches due to expanding functions with range I except at most the rightmost one. The proposed example is studied by using the first return map in I, which we prove to be chaotic in I making use of the border collision bifurcations curves of basic cycles. This leads to a robust unbounded chaotic attractor, the interval ( − ∞ , 1 ] , for the map f.