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This paper generalizes the existing minimal mathematical model of a given generic touristic site by including a distributed time-delay to reflect the whole past history of the number of tourists in their influence on the environment and capital flow. A stability and bifurcation analysis is carried out on the coexisting equilibria of the model, with special emphasis on the positive equilibrium. Considering general delay kernels and choosing the average time-delay as bifurcation parameter, a Hopf bifurcation analysis is undertaken in the neighborhood of the positive equilibrium. This leads to the theoretical characterization of the critical values of the average time delay which are responsible for the occurrence of oscillatory behavior in the system. Extensive numerical simulations are also presented, where the influence of the investment rate and competition parameter on the qualitative behavior of the system in a neighborhood of the positive equilibrium is also discussed.

In this paper, we study an epidemic model describing two strains with non-monotone incidence rates. The model consists of six ordinary differential equations illustrating the interaction between the susceptible, the exposed, the infected and the removed individuals. The system of equations has four equilibrium points, disease-free equilibrium, endemic equilibrium with respect to strain 1, endemic equilibrium with respect to strain 2, and the last endemic equilibrium with respect to both strains. The global stability analysis of the equilibrium points was carried out through the use of suitable Lyapunov functions. Two basic reproduction numbers R 0 1 and R 0 2 are found; we have shown that if both are less than one, the disease dies out. It was established that the global stability of each endemic equilibrium depends on both basic reproduction numbers and also on the strain inhibitory effect reproduction number(s) Rm and/or Rk. It was also shown that any strain with highest basic reproduction number will automatically dominate the other strain. Numerical simulations were carried out to support the analytic results and to show the effect of different problem parameters on the infection spread.

In this paper, we study to find the numerical solution of fractional Schistosomiasis disease by using a numerical method. Fractional Schistosomiasis disease model is used to symbolize a parasitic disease caused by trematode flukes of the genus Schistosoma. The physical behaviour of results obtained by using q-homotopy analyses transform method (q-HATM) in terms of plots for different fractional-order is captured. The results obtained by using considered method is more effective and easy to apply in order to examine the nature of multi-dimensional differential equations of fractional order arising in biological disease.

In this paper, the Hyers–Ulam stability of linear Caputo–Fabrizio fractional differential equations with Mittag–Leffler kernel is studied using the Laplace transform method (via the Wright function). Existence, uniqueness and generalized Hyers–Ulam–Rassias stability results for nonlinear problems are established.

In this paper, we derive extensions of classical Lyapunov and Chetaev instability theorems and LaSalle’s invariance principle to the case of Atangana–Baleanu derivative in the Caputo sence. Moreover, we get some results to estimates fractional derivatives of quadratic and Volterra–type Lyapunov functions when γ ∈ (0, 1). Finally, we present rigorous proofs about the complete classification for global dynamics of an HIV/AIDS epidemic model with Atangana–Baleanu Caputo derivative (Chaos, Solitons and Fractals 126 (2019) 41–49).

Time-delays and fractional-order play a vital role in modeling biological systems with memory. In this paper, we propose a novel delay differential model with fractional-order for tumor immune system with external treatments. Non-negativity of the solution of such model has been investigated. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to two differenttumor time-delays τ1 and τ2. The occurrence of Hopf bifurcation is captured when any of the time-delay passes through critical value τ 1 * , or τ 2 * . Theoretical results are validated numerically by solving the governing system, using the modified Adams–Bashforth–Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model.

The paper proposes to apply the K-Fisher (KF) index and the complexity-entropy curves based on k-entropy into studying the time series. The KF index is a good extension of the traditional Shannon-Fisher (SF) index. In the complexity-entropy curves, the results show that the normalized k-entropy and its corresponding complexity has the same extreme value k = 0.6 , interestingly, the KF index is the smallest when k = 0.6 , which suggests that the time series with k = 0.6 might be the most stable and verifies that the KF index is an effective tool to measure the stability of sequence. Furthermore, we conclude that this new complexity-entropy curves and the KF index could clearly distinguish financial stock indices and find that the stability of US stock indices are higher than the Australian and Chinese stock indices.

This paper introduces a novel version of variable-order (VO) space-time mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels. An optimization scheme is proposed for solving this new class of VO fractional problems.The presented method is based on the hybrid of the generalized Lucas polynomials together with their operational matrices of VO fractional derivatives (which are obtained for the first time in the presented study), the collocation technique and the Lagrange multipliers scheme. The presented method transforms obtaining the solution of such problems into obtaining the solution of systems of algebraic equations. Two numerical examples are provided to show the validity and accuracy of the presented approach.

The response of the stochastic system hardly provides useful information for researching its characteristic. Therefore, the paper is aimed at utilizing a deterministic tool, namely, the most probable response, to explore the stochastic nonlinear system. Firstly, one defines the most probable response of the stochastic system. Then, its analytical solution is derived by incorporating the extremum theory into the associated Fokker–Planck equation. Finally, two examples are given to illustrate respectively the implication of this method. Meanwhile, it can conclude that the large the intensity of multiplicative noise is, the more obvious the impact is on the response of nonlinear system. However, no matter how the intensity of additive noise changes, the most probable response does not change. The numerical method verifies the validity of analytical solution.

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, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application.

This paper investigates the almost sure exponential stability of Milstein-type schemes for stochastic delay differential equations (SDDEs) using the discrete semimartingale convergence theorem. It is shown that the Milstein scheme can preserve the almost sure stability of the exact solution under a linear growth condition on the drift. If the drift defies the linear growth condition, but satisfies a one-sided Lipschitz condition, we show backward Milstein scheme can share the almost sure exponential stability. Moreover, the exponential decay rates of the two classes of Milstein schemes are also investigated. Numerical experiments are performed to confirm our theoretic findings.

In this paper, a two-strain mathematical model is proposed to explore the effect of the incubation period of malaria parasites on the evolution of malaria. Firstly, ignoring the effect of incubation period in the mosquito, the basic reproduction number is calculated by the next generation matrix method. We proved that the disease is extinct and this model has a global asymptotical stabile disease-free equilibrium if the basic reproduction numbers of strains-1 and 2 are less than unity. Otherwise the disease is persistent and the strain-i dominant equilibrium is globally asymptotically stable if the reproduction number of stain-i is larger than unity and the reproduction number of stain-j is less than unity. Further, the similar results are obtained for the model with incubation period. In addition, we also consider the competitive exclusion phenomenon in two models. The sensitivity analysis shows that ignoring the incubation period of malaria will overestimate the risk of malaria transmission. Finally, some numerical simulations are performed to illustrate the main theoretical results.

Infectious diseases have been ranked in the top ten causes of death by WHO in 2016 and despite of availability of various types of vaccines and antibiotics, a huge population is still dying by infectious diseases every day. This may happen due to, several reasons like, resistance of pathogens to antibiotics, improper hygiene and various types of difficulties in vaccination. It has also inspired mathematical modelers to develop dynamical systems predicting the infections in long run. During their spread in a particular population, infectious diseases show various kind of delays which essentially affects the dynamics. In this paper, a susceptible - vaccinated- exposed - infectious - removed (SVEIR) epidemic model is developed with vaccination and two discrete time delays. The first time delay has been incorporated for the time period used to cure the infectious population and another time delay denote the temporary immunity period. The existence of solution and its boundedness have been established. The local stability of disease free equilibrium in respect of both the delays have been discussed explicitly and we have found threshold values of both delay parameters for the local stability of disease free equilibrium. We have also established the local stability of interior equilibrium following the existence of Hopf-bifurcation. Theoretical result shows that considered model system undergoes a Hopf-bifurcation around the interior equilibrium when the time delay due to time period used to cure the infectious population crosses a threshold value. We have also discussed the direction and stability of delay induced Hopf bifurcation using normal form theory and centre manifold theorem. In presence of delay, by constructing a Lyapunov function, local asymptotic stability of the positive equilibrium point is discussed. The length of delay has been estimated to preserve the stability using Nyquist criterion. With the suitable choices of the parameters, some numerical simulations have been presented in the support of our analytical results.

Amongst the several definitions of fractional operator (memory operator), these days the Caputo–Fabrizio operator (CF) without singular kernel is a useful extension of the classical Caputo derivative operator as the latter includes a singular mathematical expression also known as the kernel in its definition, leading to some difficulties in finding solutions to the corresponding differential equation, whereas the kernel of the former (CF) is non-singular. In this paper, the idea of CF derivative has been applied to solve a Brinkman nanoliquid (suspension of nanomaterials in a base fluid) problem. More exactly, the present article studies the effect of copper nanoparticles on the MHD free convection transient motion of C6H9NaO7 Brinkman nanoliquid lies over a vertical surface with time-dependent velocity, temperature, and concentration. Formulation of the problem in the fractional form (CF) is done and then closed-form solution is obtained. Graphs are drawn for different embedded parameters and variations in Nusselt number, skin friction and Sherwood number are shown in tabular form. C6H9NaO7 is a food product taken from seaweed or brown algae. C6H9NaO7 is used in the food industry for making gel-like foods such as pimento stuffing in prepared cocktail olives. In a liquid, C6H9NaO7 also acts as a thickener. C6H9NaO7 Brinkman nanoliquid in this direction (food technology) using CF derivative has not been investigated yet. This study will further provide new directions to carry this research using Atangana–Baleanu fractional derivatives for other non-Newtonian C6H9NaO7 nanoliquids used in different foods.

In this paper a fractional-like Hukuhara-type derivative is introduced for a set of equations. Connections between the new defined notion and the classical Hukuhara derivative are established and, in addition, some applications to the theory of set-valued differential equations are discussed. Namely, for a family of equations with fractional-like Hukuhara-type derivatives of the set of states: (a) a version of the comparison principle is proposed; (b) local existence conditions are derived; (c) an estimate of the deviation of the approximate solution from the exact one is given. With this research we extend the theory of fractional-like differential equations to the set-valued case.

In this paper, a fractional-order higher-dimensional multicavity chaotic map is investigated in the Caputo discrete delta’s sense. The numerical formula of discrete fractional-order chaotic map is deduced by utilizing the discrete fractional calculus (DFC). Taking a two-demensional model as an example, the dynamical analysis of the fractional-order multicavity chaotic map is carried out in detail by means of attractors, bifurcation diagrams, permutation entropy complexity and distribution characteristics. Moreover, there is a comparison between the fractional-order system and its integer-order counterpart for their behaviors. It shows that the fractional-order system has richer dynamical behaviors, higher complexity and more uniform distribution characteristics, which means that the fractional-order system has better engineering application.

This paper explores the idea of Bertrand duopoly in a context of bounded rationality, where two content providers (CP) endeavor to maximize their own profit. Each CP varies his price and fixes the level of his credibility of content. We suppose that one CP adjusts his price strategy according to a kind of smoothed information which average his marginal profit in the previous period and the one in a delayed period, while another CP makes its price strategy as the best response, which is a function the opponent action in the current period. Through a detailed analysis, we calculate explicitly the equilibrium points of the dynamical system, and we show the stability of its Nash equilibrium under some conditions. Numerical results reveal that while varying the model parameters (speed of adjustment and side payment), complex dynamic behaviors would occur, such as chaos. In addition, an appropriate method of chaos controlling is applied to force the system to go back to its stabilization behavior.

Viruses have different mechanisms in causing a disease in an organism, which largely depend on the viral species. The recent advancement, through coupling data analysis and mathematical modeling, has allowed the identification and characterization of the nature of the virus. In the present paper, the homotopy analysis method is applied to provide an approximate solution of the basic HIV viral dynamic model describing the viral dynamics in a susceptible population. The proposed method allows for the solution of the governing system of differential equations to be calculated in the form of an infinite series with components which can be easily calculated. The homotopy analysis method utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter. By using the homotopy series solutions, firstly, several β − curves using an appropriate ratio are plotted to demonstrate the regions of convergence and the optimum value of ℏ, then the residual and absolute errors are obtained for different values of these regions. Secondly, the residual error functions are applied to show the accuracy of the applied homotopy analysis method. Also, the convergence theorem of homotopy analysis method for the HIV viral dynamic model is proved. Mathematica software is used for the calculations and numerical results. The results obtained show the effectiveness and strength of the homotopy analysis method.

The development and evolution of economic system is the important support for economic analyses and researches. The application of nonlinear science, which is represented by chaos theory and has undergone major changes, makes people understand the ideological pivots and theoretical perspectives of the economic system that also includes the recognition of industrial economic system evolution. As a mathematical tool, fractional calculus has gradually penetrated into the economic field from the purely mathematical category. Based on the shift Chebyshev-tau idea, this paper solved a series of fractional diffusion equations with boundary conditions; using the tau method, the chaotic economic evolution is transformed into an algebraic equation system, and the equation’s approximate solution is obtained by combining these boundary conditions. On the basis of discovering chaotic characteristics in economic system evolution, the law of economic development and evolution is recognized by applying the perspective analysis of chaos theory. The results show that the spatial fractional calculus diffusion equation can effectively reveal the essence of economic system evolution and provide a new reference for the analysis mode of economic theories.

The fuzzy systems with interval approach use an infinite valued parameter in the range of [0,1] as a confidence degree of belief. This parameter makes more complicity but plays the main role in creating the fuzzy solution of the fuzzy systems. In solving process of the model, the Atangana–Baleanu derivative in the fractional case of differential equations has a memory to use all the previous information. Therefore this is as a key point and advantage of using this derivative to reduce the complicity of numerical results in comparison with other known derivatives. In this paper, first, the ABC fractional derivative on fuzzy set-valued functions in parametric interval form is defined. Then it is applied for proving the existence and uniqueness of the solution of fuzzy fractional differential equation with ABC fractional derivative. In general, it is shown that the last interval model is as a coupled system of nonlinear equations. To solve the final system an efficient numerical method called ABC-PI is used. For more illustration, several examples are solved numerically and analyzed by the figures.

This paper investigates the global stability of discrete-time pathogen infection model with pathogen-to-cell and cell-to-cell transmissions and humoral immunity. We consider both latently and actively infected cells. The model incorporates three types of intracellular time delays. We use nonstandard finite difference method to discretize the continuous-time model. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number R 0 and the humoral immune response activation number R 1 . We have proven that if R 0 ≤ 1 , then the pathogen-free equilibrium Q0 is globally asymptotically stable, if R 1 ≤ 1 R 0 , then the persistent pathogen equilibrium without immune response Q* is globally asymptotically stable, and if R 1 > 1 , then the persistent pathogen equilibrium with immune response Q ¯ is globally asymptotically stable. We illustrate our theoretical results by using numerical simulations. The effect of time delay on the pathogen dynamics is also studied. We have shown that the time delay has similar effect as the drug therapy. This gives some impression to develop new class of treatment to increase the delay period and then suppress the pathogen replication.

In this paper, we mainly investigate chaos control of a piecewise linear circuit system. According to the characteristic of this system, we modify Hwang’s linear continuous controller and obtain a more simple controller consisting of two parts, by which we find from theory the extent of control parameter when chaotic motion is controlled to equilibrium manifold, equilibrium point, periodic orbit or limit cycle. Numerical simulation also verifies the method is effective.

This paper formulates and analyzes a virus dynamics model with impairment of B-cell functions. The model includes two modes of viral transmission, cell-free and cell-to-cell. The cell-free and cell-cell incidence rates are modeled by general nonlinear functions. We integrate both latently and actively infected cells as well as three time delays into the model. Nonnegativity and boundedness properties of the solutions are proven to show the well-posedness of the model. The model admits two equilibria which are determined by the basic reproduction number R 0 G . The global stability of each equilibrium is proven by utilizing Lyapunov function and applying LaSalle’s invariance principle. The theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions on the virus dynamics is studied. We have shown that if the functions of B-cell are impaired, then the concentration of viruses is increased in the plasma.

In this paper, an analytical exact solution for the fractional differential equation of the oscillator in a resisting medium was obtained successfully via the natural transform method. The fractional derivatives were described in the Caputo sense. The results illustrated the power, efficiency, simplicity, and reliability of the proposed method.

In the current paper, a new approach is applied to solve time fractional advection-diffusion equation. The utilized fractional derivative operator is the Atangana–Baleanu (AB) derivative in Caputo sense. The mentioned fractional derivative involves the Mittag–Leffler function as the kernel that is both non-singular and non-local. A new operational matrix of AB fractional integration is obtained for the Bernstein polynomials (Bps). By applying the aforesaid matrix, the considered problems are reduced to a system of equations. The approximate solution is derived by solving the yielded system. Also, the error bound is studied. The obtained results show that the applied scheme is simple and powerful tool in finding numerical solutions of fractional equations.

This paper introduces a novel class of nonlinear optimal control problems generated by dynamical systems involved with variable-order fractional derivatives in the Atangana–Baleanu–Caputo sense. A computational method based on the Chebyshev cardinal functions and their operational matrix of variable-order fractional derivative (which is generated for the first time in the present study) is proposed for the numerical solution of this class of problems. The presented method is based on transformation of the main problem to solving system of nonlinear algebraic equations. To do this, the state and control variables are expanded in terms of the Chebyshev cardinal functions with unknown coefficients, then the cardinal property of these basis functions together with their operational matrix are employed to generate a constrained extremum problem, which is solved by the Lagrange multipliers method. The applicability and accuracy of the established method are investigated through some numerical examples. The reported results confirm that the established scheme is highly accurate in providing acceptable results.

In this paper we consider multi variable orders differential equations (MVODEs) with non-local and no-singular kernel. The derivative is described in Atangana and Baleanu sense of variable order. We use the fifth-kind Chebyshev polynomials as basic functions to obtain operational matrices. We transfer the original equations to a system of algebraic equations using operational matrices and collocation method. The convergence analysis of the presented method is discussed. Few examples are presented to show the efficiency of the presented method.

This paper involves complex valued versions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation. Under various suitable assumptions the results are established in the setting of complex valued double controlled metric space. Thereafter, by making consequent use of the fixed point method, short and simple proofs are obtained for solutions of Riemann-Liouville integral, complex valued Atangana-Baleanu integral operator and non-linear Telegraph equation.

The regional economic income multiplication plan is the target set by many economic development zones. The multiplication plan time is an important research hotspot of economic development planning. The paper uses fractional-order accumulation to generate GMλ(1, 1)-gray prediction model to predict the per capita income level of a region. The data is derived from the per capita income of the region in 2009–2013, and predicts the per capita income capacity. Through research and calculation, it is found that using the fractional calculus λ value in the error controllable range can make the prediction result more reasonable and more accurate. The final case study demonstrates that the region’s per capita income doubling plan can be completed by 2020.

In this paper, we suggest a new integro-differential equation by utilizing from the concept of fractal-fractional derivative and integral newly introduced by Atangana. We offer that under which conditions the solution of the suggested equation is exist and unique benefitting from Banach fixed-point theorem. Also, we construct a new numerical scheme for the numerical solution of our problem and we give numerical simulation and illustrations for different values of fractional order α and fractal order β. That’s why, this study will guide for researchers significantly in theory and applications.

In this presented paper, we investigate the novel numerical scheme for the non-linear reaction-diffusion equation and non-linear integro reaction-diffusion equation equipped with Atangana Baleanu derivative in Caputo sense (ABC). A difference scheme with the help of Taylor series is applied to deal with fractional differential term in the time direction of differential equation. We applied a numerical method based on quasi wavelet for discretization of unknown function and their spatial derivatives. A formulation to deal with Dirichlet boundary condition is also included. To demonstrate the effectiveness and validity of our proposed method some numerical examples are also presented. We compare our obtained numerical results with the analytical results and we conclude that quasi wavelet method achieve accurate results and this method has a distinctive local property. On the other hand the method is easy to apply on higher order fractional partial differential equation and integro differential equation.

For k ≥ 2, the k-generalized Fibonacci sequence ( F n ( k ) ) n is defined by the initial values 0 , 0 , … , 0 , 1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we shall study for which x, k and t the expression x t F n + t ( k ) + ⋯ + x F n + 1 ( k ) + F n ( k ) belongs to ( F m ( k ) ) m for infinitely many integers n. This work generalizes [13, Theorem 2] which is related to the case t = 1 .

International audience; In this paper we analyze the behavior of a nonlinear system under impulse loadings. The system is composed of a master ''linear'' degree of freedom (dof) substructure which is attached to a slave ''nonlinear'' energy sink (NES) for the sake of the control. Melnikov integral is endowed in order to study the possibility of existence of chaos and transversal homoclinic orbits in the system. Then, the complexification method as an alternative to nonlinear normal modes is implemented to reveal the behavior of the system during the energy exchange between two oscillators. The non-smooth time transformation (NSTT) technique is implemented in order to enlighten the system behavior during its extremely nonlinear regime, meanwhile stable and unstable zones of the system during its quasilinear regime are highlighted.

The main purpose of this paper is to introduce a new class of Banach type fuzzy contractions and to present some fixed and common fixed point theorems for these mappings, as well as for the Edelstein fuzzy locally contractive mappings. Two examples are presented to show that our results are genuine generalizations of many known results.

The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurrance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domains. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes [1] .

In this paper we introduce and investigate a class of asymptotically nonexpansive mappings which properly extends the class of nonexpansive mappings. We proved general existence theorems for fixed and periodic points of these mappings in arbitrary intuitionistic fuzzy metric spaces and so we solved an open problem, related to periodic points.

We consider in this paper a family of modified hyperchaotic Rossler systems and investigate both problems of understanding hyperchaos–chaos–hyperchaos transition and computing the prediction time. These systems were obtained and numerically investigated by Nikolov and Clodong [Nikolov S, Clodong S. Occurrence of regular, chaotic and hyperchaotic behavior in a family of modified Rossler hyperchaotic systems. Chaos, Solitons & Fractals 2004;22:407–31]. Our studies confirm that transition hyperchaos–chaos–hyperchaos (i) depends on the change of the sign of the corresponding characteristic equation roots or (ii) can be obtained as a result of the absorption/repulsion of the repeller originally located out of the attractor by the growing attractor. It is also shown that the prediction time is a more reliable predictor of the evolution than the information dimension. We conclude that the prediction time in hyperchaotic regimes is at least one order of magnitude smaller than those in chaotic zones.

International audience; This paper is devoted to the study of the stability of limit cycles of a nonlinear delay differential equation with a distributed delay. The equation arises from a model of population dynamics describing the evolution of a pluripotent stem cells population. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We then investigate the stability of the limit cycles yielded by the bifurcation using the normal form theory and the center manifold theorem. We illustrate our results with some numerics.

In this paper we review some aspects of constant negative curvature models in connection with the problems of quantum chaos. Special emphasis is put on so-called arithmetic systems, which display strong nongeneric behaviour, both classically and quantically.