Solitons Fractals Research Articles

Minimality and distributional chaos in triangular maps

The result of this paper contributes to the classification of triangular maps of the square with zero topological entropy stated by A. N. Sharkovsky in the 1980s. The problem was if a triangular map of the square such that its any ω-limit set contains unique minimal set can be distributionally chaotic. So far such result was disproved only for the class of triangular maps non-decreasing on fibres [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26(2) (2005), pp. 581–589]. In this paper, we solve the problem in negative for all triangular maps of the square, correcting the original result from Balibrea and Smítal [Strong distributional chaos and minimal sets, Topology appl. 156 (2009), pp. 1673–1678].

Dynamical Behavior of Two Coupled Two-Stroke Relaxation Oscillators

Starting from [J.-M. Ginoux et al., Torus breakdown in a uni junction memristor, Int. J. Bifurcation Chaos 28(10) (2018) 1850128; J.-M. Ginoux et al., Torus breakdown in a two-stroke relaxation memristor, Chaos Solitons Fractals 153 (2021) 111594], we define a model which describes the dynamical behavior of a class of two-stroke oscillators (i.e., nonlinear oscillators with two distinct phases per cycle). We highlight some properties of the model still not taken into account, like presence of a global bifurcation and bistability for some regions of the parameters space. We then extend the model to study two coupled two-stroke oscillators, where synchrony and anti-synchrony regimes appear as well as transient chaos and intermittency phenomena. We finally compare the theoretical results on synchrony regimes with the corresponding experimental data obtained from a circuit where two relaxation oscillators, based on the Unijunction Transistor (UJT) electronic component, are coupled together. The comparison shows a good qualitative agreement between the experiments and the theoretical results.

A note on “Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping” [Chaos Solitons Fractals 163 (2022) 112553

In this note, we correct the proof of the global existence of solutions for a class of quasilinear wave equations with structural damping or strong damping obtained in Ding and Zhou (2022).

Topological-numerical analysis of a two-dimensional discrete neuron model.

We conduct computer-assisted analysis of a two-dimensional model of a neuron introduced by Chialvo in 1995 [Chaos, Solitons Fractals 5, 461-479]. We apply the method of rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 [SIAM J. Appl. Dyn. Syst. 8, 757-789] and improved and expanded afterward. Additionally, we introduce a new algorithm to analyze the return times inside a chain recurrent set. Based on this analysis, together with the information on the size of the chain recurrent set, we develop a new method that allows one to determine subsets of parameters for which chaotic dynamics may appear. This approach can be applied to a variety of dynamical systems, and we discuss some of its practical aspects.

Local Distributional Chaos

Distributional chaos was introduced in Schweizer and Smítal (Trans Am Math Soc 344:737–754, 1994) for continuous maps of the interval, as chaotic behavior based on development of distances between the orbits of points in the system. In Balibrea et al. (Chaos Solitons Fractals 23(5):1581–1583, 2005), this phenomenon was generalized to continuous maps of compact metric space and was distinguished into three different forms, chaos DC1, DC2 and DC3. In Loranty and Pawlak (Chaos 29:013104, 2019), the local idea of such behavior is studied, which leads to the definition of distributionally chaotic points (DC-points). It is also proved in Loranty and Pawlak (2019), that for interval maps, positive topological entropy implies existence of DC1-point. In this paper this result for interval maps is strengthened; it is proved that positive topological entropy implies existence of an uncountable set of DC1-points, and moreover this set can be chosen perfect. In greater dimensions than one, we deal with triangular systems on \(I^{2}\). In this case the relationship between topological entropy and different types of distributional chaos is not clearly understood and several different results are possible. In the paper we use an example of map F given by Kolyada (Ergod Theory Dyn Syst 12:749–768, 1992) to prove that the corresponding two dimensional system \((I^ {2}, F)\) has positive topological entropy but without containing DC2-points, proving that there is no concentration of DC2-chaos.

Fractional Modelling and Optimal Control of COVID-19 Transmission in Portugal

A fractional-order compartmental model was recently used to describe real data of the first wave of the COVID-19 pandemic in Portugal [Chaos Solitons Fractals 144 (2021), Art. 110652]. Here, we modify that model in order to correct time dimensions and use it to investigate the third wave of COVID-19 that occurred in Portugal from December 2020 to February 2021, and that has surpassed all previous waves, both in number and consequences. A new fractional optimal control problem is then formulated and solved, with vaccination and preventive measures as controls. A cost-effectiveness analysis is carried out, and the obtained results are discussed.

On the nonlinear dynamics of a cardiac electrical conduction system model: theoretical and experimental study

The heart has a leading and vital role in the human body, scientific researches dealing with this theme are sources of growing interest. In recent years, several mathematical models of the heart electrical conduction system have appeared in the scientific literature. In the present paper, the nonlinear dynamics of a heart’s electrical conduction system model is investigated. The model consists of three coupled nonlinear oscillators with time delay coupling as earlier proposed by Gois et al [S R F S M Gois and M A Savi, 2009 ‘An analysis of heart rhythm dynamics using a three-coupled oscillator model,’ Chaos, Solitons Fractals, 41, 2553-2565]. Our study here is both theoretical and experimental; theoretical study includes fixed points analysis, bifurcations analysis where the effect of some parameters on the system dynamics is investigated, coexisting bifurcations and hysteresis phenomenon are found for some range of the system parameters, Lyapunov exponents and basin of attraction calculation. The experimental study includes sizing an analog implementation of the mathematical model using OrCAD/Pspice software, as well as the practical realisation of the model. Experimental and practical results are then presented in order to confirm theoretical predictions.

Fractional-Order Modelling and Optimal Control of Cholera Transmission

A Caputo-type fractional-order mathematical model for “metapopulation cholera transmission” was recently proposed in [Chaos Solitons Fractals 117 (2018), 37–49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem is formulated and numerically solved. A cost-effectiveness analysis is performed to assess the relevance of studied control measures. Moreover, such analysis allows us to assess the cost and effectiveness of the control measures during intervention. We conclude that the FOC system is more effective only in part of the time interval. For this reason, we propose a system where the derivative order varies along the time interval, being fractional or classical when more advantageous. Such variable-order fractional model, that we call a FractInt system, shows to be the most effective in the control of the disease.

A note on symmetry breaking in a non linear marketing model

In this paper, we consider the nonlinear discrete-time dynamic model proposed by Bischi and Baiardi (Chaos Solitons Fractals 79:145-156, 2015a). The model considers players with adaptive adjustment mechanisms towards the best reply and a form of inertia in adopting such mechanism. Moreover, we formulate an extension of the original model, where endogenous market size is considered. Through numerical simulations, we show that multiple attractors may exist in the presence of homogeneous agents and the emergence of non-synchronized trajectories both in the short (on-off intermittency) and long (global riddling) run. Therefore, the article highlights that strategic contexts exist in which the players’ knowledge of the market and the adoption of the best reply do not always allow the use of the representative agent’s rhetoric to describe the dynamics of the model.

Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part [Altybay et al. Fractional Klein-Gordon equation with singular mass. Chaos Solitons Fractals. 2021;143:Article ID 110579] which was devoted to the classical Euclidean Klein-Gordon equation.

Stochastic modification of Newtonian dynamics and induced potential—Application to spiral galaxies and the dark potential

Using the formalism of stochastic embedding developed by Cresson and Darses [J. Math. Phys. 48, 072703 (2007)], we study how the dynamics of the classical Newton equation for a force deriving from a potential is deformed under the assumption that this equation can admit stochastic processes as solutions. We focus on two definitions of a stochastic Newton equation called differential and variational. We first prove a stochastic virial theorem that is a natural generalization of the classical case. The stochasticity modifies the virial relation by adding a potential term called the induced potential, which corresponds in quantum mechanics to the Bohm potential. Moreover, the differential stochastic Newton equation naturally provides an action functional that satisfies a stochastic Hamilton–Jacobi equation. The real part of this equation corresponds to the classical Hamilton–Jacobi equation with an extra potential term corresponding to the induced potential already observed in the stochastic virial theorem. The induced potential has an explicit form depending on the density of the stochastic process solutions of the stochastic Newton equation. It is proved that this density satisfies a nonlinear Schrödinger equation. Applying this formalism for the Kepler potential, one proves that the induced potential coincides with the ad hoc “dark potential” used to recover a flat rotation curve of spiral galaxies. We then discuss the application of the previous formalism in the context of spiral galaxies following the proposal and computations given by Da Rocha and Nottale [Chaos, Solitons Fractals, 16(4), 565–595 (2003)] where the emergence of the “dark potential” is seen as a consequence of the fractality of space in the context of the scale relativity theory.

Comment on: “Stokes’ first problem for heated flat plate with Atangana–Baleanu fractional derivative” [Chaos Solitons Fractals 117 (2018) 68

In the sense of distributions, the derivative of the Heaviside unit step function $H(t)$ is a generalized Dirac-$\delta$ distribution. If the velocity $V(t)$ of a flat plate is impulsive, as $V(t)=H(t)$ (i.e., it is suddenly set into motion with unit velocity at $t=0^+$), then its acceleration is $V'(t)=\delta(t)$. The Dirac-$\delta$ distribution has no point values. However, when the Dirac-$\delta$ is the forcing term of an ODE (in $t$), it contributes to the solution. The recently published paper [Chaos Solitons Fractals 117 (2018) 68] incorrectly treats the Dirac-$\delta$ function as being identically 0. This Comment analyzes the source of this error, and provides guidance on how to correct it (based on the established literature). The mathematical error identified is in addition to some issues about rheological models with fractional derivatives, which are also noted. That is to say, whether or not an "Atangana--Baleanu fractional derivative" is used in [Chaos Solitons Fractals 117 (2018) 68], the solution to Stokes' first problem provided therein is not correct.

Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation

Abstract In this article, the Lie group of transformation method via one-dimensional optimal system is proposed to obtain some more exact solutions of the (4+1)-dimensional Fokas equation. Lie infinitesimal generators, possible vector fields, and their commutative and adjoint relations are presented by employing the Lie symmetry method. An optimal system of the one-dimensional subalgebras is also constructed using Lie vectors. Meanwhile, based on the optimal system, Lie symmetry reductions of the Fokas equation is obtained. A repeated process of Lie symmetry reductions, using the single, double, triple, quadruple, and quintuple combinations between the considered vectors, transforms the Fokas equation into nonlinear ordinary differential equations which produce abundant group-invariant solutions. The same problem was studied by Sadat et al. (Chaos Solitons Fractals 140:110134, 2020) using the same Lie symmetry technique via commutative product approach but with the less number of vector fields and therefore could obtain only three exact solutions as compared to the number of analytic solutions in this paper. In order to provide rich localized structures, some solutions are supplemented via numerical simulation, which produces some breather-type solitons, oscillating multi-solitons on the parabolic-shaped surface, fractal dromions, lump-type solitons, and annihilation of different parabolic multi-solitons profiles. The dynamical behaviors of excitation-localized structures are demonstrated graphically via 3D plots for suitable values of the arbitrary free parameters and independent arbitrary functions.

Global Hopf bifurcation for two zooplankton-phytoplankton model with two delays

In this paper, we study the global existence of a bifurcating periodic solution for a two zooplankton-phytoplankton model with two delays. First, we demonstrate that the bifurcating periodic solution exists when one delay increases and the other delay remains unchanged. Second, we give simulation to describe the bifurcating periodic solution when one delay increases. Our work answers the question in Sect. 5 (Shi and Yu in Chaos Solitons Fractals 100:62–73, 2017).

Corrigendum to "Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan" [Chaos Solitons Fractals 135 (2020), 109846

We correct some numerical results of [Chaos Solitons Fractals 135 (2020), 109846], by providing the correct numbers and plots. The conclusions of the paper remain, however, the same. In particular, the numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China. This time all our computer codes are provided, in order to make all computations reproducible. The authors would like to apologize for any inconvenience caused.

Novel analysis of the fractional glucose–insulin regulatory system with non-singular kernel derivative

Diabetes mellitus, at the forefront of the diseases of our age, is a type of disease that plays a leading role in the formation of many deadly diseases and is very common all over the world. In this work, based on the system in Shabestari et al. (Chaos Solitons Fractals 112:44–51, 2018), we give the fractional glucose–insulin regulatory system for the first time by Caputo–Fabrizio derivative. We present special solutions, stability analysis, uniqueness of the solution for this fractional system with the aid of Banach fixed point theory. Taking various values belonging to the fractional order into consideration, we present some numerical results according to the relation between glucose, insulin and $$\beta $$ -cells.

Self-similar analysis of a viscous heated Oberbeck–Boussinesq flow system

One of the simplest model to couple viscous flow to heat conduction is the Oberbeck–Boussinesq (OB) system which were also investigated by E N Lorenz. In our former studies—2015 Chaos Solitons Fractals 78 249, 2017 Chaos Solitons Fractals 103 336—we derived analytic solutions for the velocity, pressure and temperature fields. Additionally, we gave a possible explanation of the Rayleigh–Bénard convection cells with the help of the self-similar Ansatz. Now we generalize the OB hydrodynamical system, including a viscous source term in the heat conduction equation. Our analysis show that the viscous heating term smooths out any kind of Bénard oscillations and stabilizes the flow. All the velocity, pressure and temperature distributions are free of oscillations. These results may attract interest in micro or nanofluidics.

Complex dynamics of a novel 3D autonomous system without linear terms having line of equilibria: coexisting bifurcations and circuit design

In recent years, an increasing interest has been devoted to the discovery of new chaotic systems with special properties. In this paper, a novel and singular 3D autonomous system without linear terms is introduced. The singularity of the model is that it is dissipative, possesses rotation symmetry and line of equilibria thus displays complex dynamics. The nonlinear behaviour of the introduced model is studied in terms of bifurcation diagrams, Lyapunov exponent plots, time series, frequency spectra two-parameter diagrams, Lyapunov stability diagrams as well as basins of attraction. Some interesting phenomena are found including, for instance, periodic oscillations, chaotic oscillations, periodic windows, symmetry restoring crises scenario, the coexistence of multiple bifurcations and offset-boosting property while monitoring the system parameters. Coexistence of attractors discovered in this work includes up to six competing attractors with one point attractor. Compared to some few cases previously reported (system without linear terms) (Xu and Wang in Opt Int J Light Electron Opt 125:2526–2530, 2014. https://doi.org/10.1007/s11071-016-3170-x; Kengne in Commun Nonlinear Sci Numer Simul 52:62–76, 2017. https://doi.org/10.1016/j.cnsns.2017.04.017; Mobayen et al. in Int J Syst Sci 49:1–15, 2018. https://doi.org/10.1080/00207721.2017.1410251; Zhang et al. in Int J Non-linear Mech 106:1–12, 2018. https://doi.org/10.1016/j.ijnonlinmec.2018.08.012; Pham et al. in Chaos Solitons Fractals 120:213–221, 2019. https://doi.org/10.1016/j.chaos.2019.02.003), the model considered in this work is the only one in which such type of complex dynamics has already been found. A suitable analog simulator (electronics circuit) is designed and used to support the theoretical analysis.

On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes

Very recently, a hepatitis B epidemic model with saturated incidence rate has been proposed and analyzed in Khan et al. (Chaos Solitons Fractals 124:1–9, 2019). The local asymptotic stability of the disease endemic equilibrium (DEE) point of model has been established theoretically but its global asymptotic stability has not been studied. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of the DEE point of the model. More precisely, we prove that if the DEE point exists, then it is not only locally asymptotically stable but also globally asymptotically stable. Furthermore, we present an alternative proof for the global stability of the disease-free equilibrium point. The main result is that we obtain the complete global stability of the hepatitis B virus model. Besides, we construct nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties of the continuous model. These properties include the positivity of solutions and the stability of the model. Dynamical properties of the proposed schemes are rigorously investigated by mathematical analyses and numerical simulations. Finally, numerical simulations are performed to confirm the validity of the theoretical results as well as the advantages of the NSFD schemes. Numerical simulations also indicate that the NSFD schemes are appropriate and effective to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the classical fourth-order Runge–Kutta scheme cannot preserve the essential properties of the continuous model. Consequently, they can generate numerical approximations which are completely different from the solutions of the model.

On the effect of projections on the Billingsley dimensions

We are interested in the behavior of Billingsley dimensions under projections onto a lower dimensional linear subspace. The results in this paper establish the connections with various dimensions of subsets [Formula: see text] of [Formula: see text] and their projections, and generalize many known results about Hausdorff and packing dimensions of projections of [Formula: see text]. In particular, we improve, through these results, one of the main theorems of Selmi et al. in [Multifractal variation for projections of measures, Chaos Solitons Fractals 91 (2016) 414–420] and treat an unsolved case in their work.