Léon Brenig

Algebraic recasting of nonlinear systems of ODEs into universal formats

It is sometimes desirable to produce for a nonlinear system of ODEs a new representation of simpler structural form, but it is well known that this goal may imply an increase in the dimension of the system. This is what happens if in this new representation the vector field has a lower degree of nonlinearity or a smaller number of nonlinear contributions. Until now both issues have been treated separately, rather unsystematically and, in some cases, at the expense of an excessive increase in the number of dimensions. We unify here the treatment of both issues in a common algebraic framework. This allows us to proceed algorithmically in terms of simple matrix operations.

Algebraic structures and invariant manifolds of differential systems

Algebraic tools are applied to find integrability properties of ODEs. Bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations, since all equations in this class are related to a canonical quadratic differential system: the Lotka–Volterra system. These algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given. Their subalgebras are associated to nonlinear invariant manifolds in the phase space. These manifolds are calculated explicitly. More general algebraic invariant surfaces are also obtained by combining a theorem of Walcher and the Lotka–Volterra canonical form. Applications are given for Lorenz model, Lotka, May–Leonard, and Rikitake systems.

Embedding recurrent neural networks into predator-prey models

We study changes of coordinates that allow the embedding of ordinary differential equations describing continuous-time recurrent neural networks into differential equations describing predator-prey models-also called Lotka-Volterra systems. We transform the equations for the neural network first into quasi-monomial form (Brenig, L. (1988). Complete factorization and analytic solutions of generalized Lotka-Volterra equations. Physics Letters A, 133(7-8), 378-382), where we express the vector field of the dynamical system as a linear combination of products of powers of the variables. In practice, this transformation is possible only if the activation function is the hyperbolic tangent or the logistic sigmoid. From this quasi-monomial form, we can directly transform the system further into Lotka-Volterra equations. The resulting Lotka-Volterra system is of higher dimension than the original system, but the behavior of its first variables is equivalent to the behavior of the original neural network. We expect that this transformation will permit the application of existing techniques for the analysis of Lotka-Volterra systems to recurrent neural networks. Furthermore, our results show that Lotka-Volterra systems are universal approximators of dynamical systems, just as are continuous-time neural networks.

From particle segregation to the granular clock

Recently several authors studied the segregation of particles for a system composed of mono-dispersed inelastic spheres contained in a box divided by a wall in the middle. The system exhibited a symmetry breaking leading to an overpopulation of particles in one side of the box. Here we study the segregation of a mixture of particles composed of inelastic hard spheres and fluidized by a vibrating wall. Our numerical simulations show a rich phenomenology: horizontal segregation and periodic behavior. We also propose an empirical system of ODEs representing the proportion of each type of particles and the segregation flux of particles. These equations reproduce the major features observed by the simulations.

Is quantum mechanics based on an invariance principle

Non-relativistic quantum mechanics for a free particle is shown to emerge from classical mechanics through an invariance principle under transformations that preserve the Heisenberg position?momentum inequality. These transformations are induced by isotropic space dilations. This invariance imposes a change in the laws of classical mechanics that exactly corresponds to the transition-to-quantum mechanics. The Schr?dinger equation appears jointly with a second nonlinear equation describing non-unitary processes. Unitary and non-unitary evolutions are exclusive and appear sequentially in time. The non-unitary equation admits solutions that seem to correspond to the collapse of the wavefunction.

Quasipolynomial generalization of Lotka-Volterra mappings

In recent years, it has been shown that Lotka-Volterra mappings constitute a valuable tool from both the theoretical and the applied points of view, with developments in very diverse fields such as physics, population dynamics, chemistry and economy. The purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism (originally conceived for the analysis of ordinary differential equations) can be extended into the mapping domain. The extension of the formalism into the discrete-time context is remarkable as far as the quasipolynomial methodology had never been shown to be applicable beyond the differential case. It will be demonstrated that Lotka-Volterra mappings play a central role in the quasipolynomial formalism for the discrete-time case. Moreover, the extension of the formalism into the discrete-time domain allows a significant generalization of Lotka-Volterra mappings as well as a whole transfer of algebraic methods into the discrete-time context. The result is a novel and more general conceptual framework for the understanding of Lotka-Volterra mappings as well as a new range of possibilities that become open not only for the theoretical analysis of Lotka-Volterra mappings and their generalizations, but also for the development of new applications.

Some global results on quasipolynomial discrete systems

Abstract The quasipolynomial (QP) generalization of Lotka–Volterra discrete-time systems is considered. Use of the QP formalism is made for the investigation of various global dynamical properties of QP discrete-time systems including permanence, attractivity, dissipativity and chaos. The results obtained generalize previously known criteria for discrete Lotka–Volterra models.

Truncated Levy distributions in an inelastic gas

We study a one-dimensional model for granular gases, the so-called inelastic Maxwell model. We show theoretically the existence of stationary solutions of the unforced case, that are characterized by an infinite average energy per particle. Moreover, we verify the quasi-stationarity of these states by performing numerical simulations with a finite number of particles, thereby highlighting truncated Levy distributions for the velocities.

Airbone particles dynamics: towards a theoretical approach

Wind-tunnel simulations and field experiments for dust transport by air show that deposition is predominant on the windward side of hills whereas sans deposition is mainly observed on the lee side. That and other aspects such as the existence of several deposition regimes depending on wind intensity call for theoretical explanations. We present some preliminary considerations leading to such a theoretical framework.

Structure and convergence of Poincaré-like normal forms

Abstract The general term of the Poincare normalizing series is explicitly constructed for non-resonant systems of ODEs in a large class of equations. In the resonant case, a non-local transformation is found, which exactly linearizes the ODEs and whose series expansion always converges in a finite domain. Examples are treated.