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In this paper, we consider a time-fractional reaction-diffusion system with the same nonlinearities of the Newton-Leipnik chaotic system. Through analytical tools and numerical results, we derive sufficient conditions for the asymptotic stability of the proposed model and show the existence of chaos. We also propose a nonlinear synchronization controller for a pair of systems and establish the local and global asymptotic convergence of the trajectories by means of fractional stability theory and the Lyapunov method.

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We consider a reaction-diffusion system modeling the spread of an epidemic disease within a population divided into the susceptible and infective classes. We first consider the question of the uniform boundedness of the solutions for which we give a positive answer. Then we deal with the asymptotic behavior of the solutions where in particular we are interested in reasonable conditions leading to the extinction of the infection disease as the time goes to infinity.

Communicated by S. GeorgievThe aim of this work is to study the global existence of solutions to a triangular system of reaction-diffusion equations, which describes epidemiological or chemical situations. On the basis of the construction of a suitable Lyapunov functional, we show that for any initial data, classical global solutions exist even when the nonlinearities are of exponential growth.for some constants˛> 0, where ' is any non-negative continuously differentiable function on OE0, C1/ satisfyingThe reaction-diffusion system (1)-(3) arise in epidemiology; it describes the propagation of an infectious disease within a population divided into infected and susceptible classes of individuals ([1-3] and references therein).Our system has the same structure as the important models of Gierer-Meinhardt [4], Selkov [5], Schnakenberg [6], Edblom-Orban-Epstein [7], or the Brusselator model [8].In the particular case where ƒ D D 0, one can see that the system (1)-(3) corresponds to a modification of the problem of R. H. Martin for which has dealt with in a number of papers (see for instance [9][10][11][12][13]).In [14][15][16][17] are discussed systems of reaction diffusion equations with nonlinearities of exponential growth. In [14,15], Inamdar and Kulkarni considered exponential autocatalysis for a class of reaction-diffusion systems. This alternative form of autocatalysis revealed the existence of multiplicity and oscillatory behaviour important in practice. We note that in the case ƒ > 0, it is not obvious to obtain the global existence of solutions via the classical methods when the nonlinearity has an exponential growth for example.Our aim in this work was to prove that positive solutions exist globally in time for any initial bounded data. Our results improve the recent one of [18] by removing the constraint on ku 0 k 1 .Ã .Using (H.1) and applying the maximum principle to the second equation of the system (4), we obtain w.t, x/ 0 for all .t, x/ 2 .0, T / ,