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Mathematical models based on fractional-order differential equations have recently gained interesting insights into epidemiological phenomena, by virtue of their memory effect and nonlocal nature. This paper investigates the nonlinear dynamic behavior of a novel COVID-19 pandemic model described by commensurate and incommensurate fractional-order derivatives. The model is based on the Caputo operator and takes into account the daily new cases, the daily additional severe cases, and the daily deaths. By analyzing the stability of the equilibrium points and by continuously varying the values of the fractional order, the paper shows that the conceived COVID-19 pandemic model exhibits chaotic behaviors. The system dynamics are investigated via bifurcation diagrams, Lyapunov exponents, time series, and phase portraits. A comparison between integer-order and fractional-order COVID-19 pandemic models highlights that the latter is more accurate in predicting the daily new cases. Simulation results, besides to confirming that the novel fractional model well fit the real pandemic data, also indicate that the numbers of new cases, severe cases, and deaths undertake chaotic behaviors without any useful attempt to control the disease. Supplementary Information The online version supplementary material available at 10.1007/s11071-021-06867-5.

The gravitational potentials of realistic galaxy models are in general non-integrable, in the sense that they admit orbits that do not have three independent isolating integrals of motion and are therefore chaotic. However, if chaotic orbits are a small minority in a stellar system, it is expected that they have negligible impact on the main dynamical properties of the system. In this paper we address the question of quantifying the importance of chaotic orbits in a stellar system, focusing, for simplicity, on axisymmetric systems. Chaotic orbits have been found in essentially all (non-St\"ackel) axisymmetric gravitational potentials in which they have been looked for. Based on the analysis of the surfaces of section, we add new examples to those in the literature, finding chaotic orbits, as well as resonantly trapped orbits among regular orbits, in Miyamoto-Nagai, flattened logarithmic and shifted Plummer axisymmetric potentials. We define the fractional contributions in mass of chaotic ($\xi_{\rm c}$) and resonantly trapped ($\xi_{\rm t}$) orbits to a stellar system of given distribution function, which are very useful quantities, for instance in the study of the dispersal of stellar streams of galaxy satellites. As a case study, we measure $\xi_{\rm c}$ and $\xi_{\rm t}$ in two axisymmetric stellar systems obtained by populating flattened logarithmic potentials with the Evans ergodic distribution function, finding $\xi_{\rm c}\sim 10^{-4}-10^{-3}$ and $\xi_{\rm t}\sim 10^{-2}-10^{-1}$., Comment: Accepted for publication in MNRAS. 15 pages, 11 figures

Physical implementation of asynchronous cellular automata networks has shown stably random oscillations under certain conditions. We present two simple mathematical models to describe transient and stationary regimes. The models are based on simple assumptions taking into account several aspects such as number of inputs of the cellular automata, rule balance, and technological frequency limitation. Numerical simulations reveal the possibility of chaotic dynamics of the average transition rate of the cellular automata in a stationary regime. With physical implementations on FPGA (field programmable gate array), preliminary experimental results show very good qualitative agreement with model’s prediction and numerical simulations. Several networks of interconnected 5-input asynchronous cellular automata have been successfully implemented in different FPGA devices, and we present some preliminary experimental results. This work aims at finding fundamental mechanisms of randomness such that the collective behavior of the cellular automata system does not depend on physical implementation details.

Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schr&ouml, dinger equation is solved within the two-state approximation, the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincar&eacute, map and exponential separation of the trajectories of the Bloch vector &sigma, in the Bloch sphere.

This work attempts to characterise the dynamic properties of a nonlinear model in which a monopolist produces a fixed amount of an intermediate good. The firm employs this good in producing two vertically differentiated final commodities, sold in two distinct markets. Consumers’ preferences are described by quasi-linear and quadratic utility functions respectively yielding isoelastic and linear demand functions. In addition, we assume that the monopolist adopts a gradient adjustment rule based on profit variation to adjust its choice about the amount of intermediate good to employ in the production of each final good. Two alternative scenarios emerge: in the first, there exists coexistence of two markets; in the second, the monopolist specialises on the production of a single final good. The dynamics show that the elasticities of market demands play an ambiguous role in affecting the stability of the equilibrium, whereas the speed of adjustment unambiguously destabilises the system. Moreover, the article investigates the role of the productive capacity and the gap in marginal costs in defining the different scenarios. In particular, economic dynamics may be chaotic and/or multiple attractors may appear.

We analyze the time series of the temperature of the sedimentary core MD01-2443 originating from the Iberian Margin with a duration of 420 kyr. The series has been tested for unit-root and a long term trend is estimated. We identify four significant periodicities together with a low climatic activity every 100 kyr, and these were associated with internal and external forcings. Also, we identify a high-frequency fast component that acts on top of a nonlinear, irreversible slow-changing dynamics. We find the presence of chaos in the climate of the Iberian Margin by means of a neural network asymptotic test on the largest Lyapunov exponent. The analysis suggests that the chaotic dynamics is associated with the fast high-frequency component. We also carry out a statistical analysis of the dimensionality of the attractor. Our results confirm the possibility that periodic behavior and chaos may coexist on different time scales. This could lead to different degrees of predictability in the climate system according to the characteristic time scales and/or phase-space locations., Berenice Rojo-Garibaldi is grateful for the CONACYT scholar-ship that supported her study at the Posgrado en Ciencias del Mar yLimnología, Universidad Nacional Autónoma de México.

This article revisits the classical work of Puu (Chaos Soliton Fract 1(6):573–581, 1991) on duopoly dynamics by gathering two distinct aspects of the functioning of markets: production of goods requires time and is subject to some gestation lags, but trading takes place continuously. Dynamics are characterized by a two-dimensional system of delay differential equations. The main aim of this work is to show that regular and non-regular fluctuations may emerge endogenously because of the existence of heterogeneous interacting agents that choose production over time in a myopic way. Chaotic dynamics in the discrete-time model of Puu (Chaos Soliton Fract 1(6):573–581, 1991) appear to be close enough to the origin of axes (implying that quantities produced by both firms are close to zero). In contrast, in our continuous-time version of the model with discrete delays, the dynamic system is more suitable of generating complex dynamics far enough from the origin when marginal costs vary. This is because of the role played by time delays and inertia. From a mathematical point of view, we show the existence of Hopf bifurcations and detect how time delays and inertia affect the stability of the system by using the recent techniques of stability crossing curves introduced by Gu et al. (J Math Anal Appl 311(1):231–253, 2005) and generalized by Lin and Wang (Can Appl Math Quart 20(4):519–533, 2012). The article also provides some findings about global bifurcations and chaotic dynamics by combining analytical studies and simulation exercises.

Good quality entropy sources are indispensable in most modern cryptographic protocols. Unfortunately, many currently deployed networked devices do not include them and may be vulnerable to Random Number Generator (RNG) attacks. Since most of these systems allow firmware upgrades and have serial communication facilities, the potential for retrofitting them with secure hardware-based entropy sources exists. To this aim, very low-cost, robust, easy to deploy solutions are required. Here, a retrofittable, sub 10$ entropy source based on chaotic dynamics is illustrated, capable of a 32 kbit/s rate or more and offering multiple serial communication options including USB, I2C, SPI or USART. Operation is based on a loop built around the Analog to Digital Converter (ADC) hosted on a standard microcontroller., Comment: 4 pages, 6 figures. Pre-print from conference proceedings; IEEE 21th International Conference on Electronics, Circuits, and Systems (ICECS 2014), pp. 175-178, Dec. 2014

This paper concerns recent developments on the discrete logistic equation and its application to social life. The relative simplicity of the logistic map in studying chaotic behavior made this equation the basis in models in several fields as cryptography, traffic control, tourism models and so on. The aim of our discussion is the possibility to choose the simpler of these models and use them in a lab directed to high school students.

In the present paper the conditions that give rise to chaotic motions in a rigid rotor on short journal bearings are investigated and determined. A suitable symmetry was given to the rotor, to the supporting system, to the acting system of forces and to the system of initial conditions, in order to restrict the motions of the rotor to translatory whirl. For an assigned distance between the supports, the ratio between the transverse and the polar mass moments of the rotor was selected conveniently small, with the aim of avoiding conical instability. Since the theoretical analysis of a system's chaotic motions can only be carried out by means of numerical investigation, the procedure here adopted by the authors consists of numerical integration of the rotor's equations of motion, with trial and error regarding the three parameters that characterise the theoretical model of the system: m, the half non-dimensional mass of the rotor, σ, the modified Sommerfeld number relating to the lubricated bearings, and ρ, the dimensionless value of rotor unbalance. In the rotor's equations of motion, the forces due to the lubricating film are written under the assumption of isothermal and laminar flow in short bearings. The number of numerical trials needed to find the system's chaotic responses has been greatly reduced by recognition of the fact that chaotic motions become possible when the value of the dimensionless static eccentricity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabew7aLnaaBaaaleaacaWGZbaabeaaaaa!4046!\[\varepsilon _s \] is greater than 0.4. In these conditions, non-periodic motions can be obtained even when rotor unbalance values are not particularly high (ρ=0.05), whereas higher values (ρ>0.4) make the rotor motion periodic and synchronous with the driving rotation. The present investigation has also identified the route that leads an assigned rotor to chaos when its angular speed is varied with prefixed values of the dimensionless unbalance ρ. The theoretical results obtained have then been compared with experimental data. Both the theoretical and the experimental data have pointed out that in the circumstances investigated chaotic motions deserve more attention, from a technical point of view, than is normally ascribed to behaviours of this sort. This is mainly because such behaviours are usually considered of scarce practical significance owing to the typically bounded nature of chaotic evolution. The present analysis has shown that when the rotor exhibits chaotic motions, the centres of the journals describe orbits that alternate between small and large in an unpredictable and disordered manner. In these conditions the thickness of the lubricating film can assume values that are extremely low and such as to compromise the efficiency of the bearings, whereas the rotor is affected by inertia forces that are so high as to determine severe vibrations of the supports.

This paper examines the interactions between economic activity and consumption externalities in an overlapping generations model. Existence of multiple steady states is studied from a global point of view, and possible mechanisms producing cycles and chaotic behavior are analyzed. Wealth-sensitive positionality is found to be able to generate persistent endogenous fluctuations, whereas wealth-neutral positionality doesn’t. Under wealth-sensitive positionality, the economy may moreover experience complex dynamics, with sensitivity to initial conditions, leading to poverty traps even when starting from relatively large endowments of capital assets.

Chaotic dynamics in a model of a long Josephson junction (LJJ) is studied via standard techniques of non-linear maps. A characterization of chaos in such objects in terms of Lyapunov exponents and Poincare sections is given. Finally the occurrence of chaos in map dynamics is compared with preliminary results of full numerical integration of the perturbed sine-Gordon equation (PSGE).

We propose a method for filling arbitrarily wide gaps in deterministic time series. Crucial to the method is the ability to apply Takens' theorem in order to reconstruct the dynamics underlying the time series. We introduce a functional to evaluate how compatible is a filling sequence of data with the reconstructed dynamics. An algorithm for minimizing the functional with a reasonable computational effort is then discussed., 14 pages (REVTeX preprint), 4 figures