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2013
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# Basil Grammaticos

Jussieu 75251 Paris Cedex 05, 75251, Paris, France
IMNC, Universite Paris 7

## Publications:

 Grammaticos B., Willox R., Satsuma J. Revisiting the Human and Nature Dynamics Model 2020, vol. 25, no. 2, pp.  178-198 Abstract We present a simple model for describing the dynamics of the interaction between a homogeneous population or society, and the natural resources and reserves that the society needs for its survival. The model is formulated in terms of ordinary differential equations, which are subsequently discretised, the discrete system providing a natural integrator for the continuous one. An ultradiscrete, generalised cellular automaton-like, model is also derived. The dynamics of our simple, three-component, model are particularly rich exhibiting either a route to a steady state or an oscillating, limit cycle-type regime or to a collapse. While these dynamical behaviours depend strongly on the choice of the details of the model, the important conclusion is that a collapse or near collapse, leading to the disappearance of the population or to a complete transfiguration of its societal model, is indeed possible. Keywords: population dynamics, dynamical systems, collapse, resources and reserves, discretisation, generalised cellular automaton Citation: Grammaticos B., Willox R., Satsuma J.,  Revisiting the Human and Nature Dynamics Model, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 178-198 DOI:10.1134/S1560354720020045
 Grammaticos B., Ramani A., Guha P. Second-degree Painlevé Equations and Their Contiguity Relations 2014, vol. 19, no. 1, pp.  37-47 Abstract We study second-order, second-degree systems related to the Painlevé equations which possess one and two parameters. In every case we show that by introducing a quantity related to the canonical Hamiltonian variables it is possible to derive such a second-degree equation. We investigate also the contiguity relations of the solutions of these higher-degree equations. In most cases these relations have the form of correspondences, which would make them non-integrable in general. However, as we show, in our case these contiguity relations are indeed integrable mappings, with a single ambiguity in their evolution (due to the sign of a square root). Keywords: Painlevé equations, contiguity relations, second-degree differential equations, Hamiltonian formalism Citation: Grammaticos B., Ramani A., Guha P.,  Second-degree Painlevé Equations and Their Contiguity Relations, Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 37-47 DOI:10.1134/S1560354714010031
 Nagoya H., Grammaticos B., Ramani A. Quantum Painlevé Equations: from Continuous to Discrete and Back 2008, vol. 13, no. 5, pp.  417-423 Abstract We present a cascade of quantum Painlevé equations consisting in successive contiguity relations, whereupon starting form a continuous equations we obtain a discrete one, and continuous limits of the latter. We start from the quantum Painlevé V and in the process derive the quantum form of continuous PIII which was missing in previous studies. Keywords: discrete systems, quantization, Painlevé equations Citation: Nagoya H., Grammaticos B., Ramani A.,  Quantum Painlevé Equations: from Continuous to Discrete and Back, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 417-423 DOI:10.1134/S1560354708050031
 Grammaticos B., Ramani A. Generating discrete Painlevé equations from affine Weyl groups 2005, vol. 10, no. 2, pp.  145-152 Abstract We show how, starting from the geometrical description of discrete Painlevé equations in terms of affine Weyl groups, one can generate new second-order systems. We use this approach to introduce a new definition of the discrete Painlevé equations which eschews the reference to continuous systems. Keywords: discrete Painlevé equations, integrability, Weyl groups, Bäcklund transformations Citation: Grammaticos B., Ramani A.,  Generating discrete Painlevé equations from affine Weyl groups , Regular and Chaotic Dynamics, 2005, vol. 10, no. 2, pp. 145-152 DOI: 10.1070/RD2005v010n02ABEH000308
 Tamizhmani K. M., Grammaticos B., Carstea A. S., Ramani A. The $q$-discrete Painlevé IV equations and their properties 2004, vol. 9, no. 1, pp.  13-20 Abstract We present a detailed study of the properties of two q-discrete Painlevé IV equations: singularity structure, bilinear forms, auto-Bäcklund/Schlesinger transformations, rational solutions as well as special solutions obtained through second- or first-order linear equations. Citation: Tamizhmani K. M., Grammaticos B., Carstea A. S., Ramani A.,  The $q$-discrete Painlevé IV equations and their properties, Regular and Chaotic Dynamics, 2004, vol. 9, no. 1, pp. 13-20 DOI:10.1070/RD2004v009n01ABEH000260
 Lafortune S., Carstea A. S., Ramani A., Grammaticos B., Ohta Y. Integrable Third-Order Mappings and their Growth Properties 2001, vol. 6, no. 4, pp.  443-448 Abstract We study the degree growth of the iterates of the initial conditions for a class of third-order integrable mappings which result from the coupling of a discrete Painlevé equation to an homographic mapping. We show that the degree grows like $n^3$. In the special cases where the mapping satisfies the singularity confinement requirement we find a slower, quadratic growth. Finally we present a method for the construction of integrable $N$th-order mappings with degree growth $n^N$. Citation: Lafortune S., Carstea A. S., Ramani A., Grammaticos B., Ohta Y.,  Integrable Third-Order Mappings and their Growth Properties, Regular and Chaotic Dynamics, 2001, vol. 6, no. 4, pp. 443-448 DOI:10.1070/RD2001v006n04ABEH000188
 Kruskal M. D., Tamizhmani K. M., Grammaticos B., Ramani A. Asymmetric Discrete Painleve Equations 2000, vol. 5, no. 3, pp.  273-280 Abstract We investigate the possible integrable nonautonomous forms of a given class of mappings involving more than one dependent variable. These integrable discrete systems define "asymmetric" Painlevé equations. Our main tool of investigation is the application of the singularity confinement discrete integrability criterion. A new way of implementing it, first proposed for the singularity analysis of continuous systems, is also introduced. Citation: Kruskal M. D., Tamizhmani K. M., Grammaticos B., Ramani A.,  Asymmetric Discrete Painleve Equations, Regular and Chaotic Dynamics, 2000, vol. 5, no. 3, pp. 273-280 DOI:10.1070/RD2000v005n03ABEH000149
 Grammaticos B., Ramani A. The Hunting for the Discrete Painleve Equations 2000, vol. 5, no. 1, pp.  53-66 Abstract We review our findings on discrete Painleve equations with emphasis on the two direct methods we have proposed for their derivation: deautonomisation through singularity confinement and geometrical approach based on affine Weyl groups. The question of integrability of discrete Painleve equations is also addressed in terms of the existence of a Lax pair or of a description in the frame of the Grand Scheme. A list of discrete Painleve equations, as complete as possible but still not exhaustive, is also presented. Citation: Grammaticos B., Ramani A.,  The Hunting for the Discrete Painleve Equations, Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 53-66 DOI:10.1070/RD2000v005n01ABEH000123