Leon Brenig - Academia.edu (original) (raw)
Papers by Leon Brenig
Computer Physics Communications, 1999
We present the QPSI (Quasi-Polynomial Symmetries and Invariants) MAPLE package for the systematic... more We present the QPSI (Quasi-Polynomial Symmetries and Invariants) MAPLE package for the systematic determination of quasi-polynomial symmetries, invariants and invariant tensor fields for dynamical systems. A brief survey of the theoretical results obtained by the authors used in the package is given. @ 1999 Elsevier Science B.V.
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Algebraic tools are applied to find integrability properties of ODEs. Bilinear nonassociative alg... more 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.
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Ecological Modelling, 2005
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Phys Rev D, 2001
The dynamics of a universe dominated by a self-interacting nonminimally coupled scalar field are ... more The dynamics of a universe dominated by a self-interacting nonminimally coupled scalar field are considered. The structure of the phase space and complete phase portraits are given. New dynamical behaviors include superinflation (H˙>0), avoidance of big bang singularities through classical birth of the universe, and spontaneous entry into and exit from inflation. This model is promising for describing quintessence as a nonminimally coupled scalar field.
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Eprint Arxiv Gr Qc 0305011, May 1, 2003
We consider here the dynamics of some homogeneous and isotropic cosmological models with NNN inte... more We consider here the dynamics of some homogeneous and isotropic cosmological models with NNN interacting classical scalar fields non-minimally coupled to the spacetime curvature, as an attempt to generalize some recent results obtained for one and two scalar fields. We show that a Lyapunov function can be constructed under certain conditions for a large class of models, suggesting that chaotic behavior is ruled out for them. Typical solutions tend generically to the empty de Sitter (or Minkowski) fixed points, and the previous asymptotic results obtained for the one field model remain valid. In particular, we confirm that, for large times and a vanishing cosmological constant, even in the presence of the extra scalar fields, the universe tends to an infinite diluted matter dominated era.
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Physical Review A
ABSTRACT
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Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requir... more Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations keep the Heisenberg inequalities invariant and form a group. They are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schr\"odinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reduction.
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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
We developed computer simulations of an inelastic granular gas which show that the energy decay p... more We developed computer simulations of an inelastic granular gas which show that the energy decay proposed by Goldhirsch and Zanetti [Phys. Rev. Lett. 70, 1619 (1993)] has a limited validity. Moreover, we give an exact solution of the Liouville equation for the moments of the two-body homogeneous cooling distribution. The latter includes velocity correlations which raises questions about the derivation of kinetic equations for inelastic gases.
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Physics Letters A, 2005
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The second international conference on computing anticipatory systems, CASYS’98, 1999
We study changes of coordinates that allow the representation of the ordinary differential equati... more We study changes of coordinates that allow the representation of the 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, where we express the vector field of the dynamical system as a linear combination of products of powers of the
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Lecture Notes in Physics, 1978
A Master Equation describing a discrete Markov process which is a stochastic version of the Boltz... more A Master Equation describing a discrete Markov process which is a stochastic version of the Boltzmann equation is constructed. The relation of this formulation to the usual Langevin equation approach is discussed.
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Computer Algebra and Differential Equations, 1994
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Computer Physics Communications, 1999
We present the QPSI (Quasi-Polynomial Symmetries and Invariants) MAPLE package for the systematic... more We present the QPSI (Quasi-Polynomial Symmetries and Invariants) MAPLE package for the systematic determination of quasi-polynomial symmetries, invariants and invariant tensor fields for dynamical systems. A brief survey of the theoretical results obtained by the authors used in the package is given. @ 1999 Elsevier Science B.V.
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Algebraic tools are applied to find integrability properties of ODEs. Bilinear nonassociative alg... more 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.
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Ecological Modelling, 2005
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Phys Rev D, 2001
The dynamics of a universe dominated by a self-interacting nonminimally coupled scalar field are ... more The dynamics of a universe dominated by a self-interacting nonminimally coupled scalar field are considered. The structure of the phase space and complete phase portraits are given. New dynamical behaviors include superinflation (H˙>0), avoidance of big bang singularities through classical birth of the universe, and spontaneous entry into and exit from inflation. This model is promising for describing quintessence as a nonminimally coupled scalar field.
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Eprint Arxiv Gr Qc 0305011, May 1, 2003
We consider here the dynamics of some homogeneous and isotropic cosmological models with NNN inte... more We consider here the dynamics of some homogeneous and isotropic cosmological models with NNN interacting classical scalar fields non-minimally coupled to the spacetime curvature, as an attempt to generalize some recent results obtained for one and two scalar fields. We show that a Lyapunov function can be constructed under certain conditions for a large class of models, suggesting that chaotic behavior is ruled out for them. Typical solutions tend generically to the empty de Sitter (or Minkowski) fixed points, and the previous asymptotic results obtained for the one field model remain valid. In particular, we confirm that, for large times and a vanishing cosmological constant, even in the presence of the extra scalar fields, the universe tends to an infinite diluted matter dominated era.
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Physical Review A
ABSTRACT
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Bookmarks Related papers MentionsView impact
Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requir... more Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations keep the Heisenberg inequalities invariant and form a group. They are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schr\"odinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reduction.
Bookmarks Related papers MentionsView impact
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
We developed computer simulations of an inelastic granular gas which show that the energy decay p... more We developed computer simulations of an inelastic granular gas which show that the energy decay proposed by Goldhirsch and Zanetti [Phys. Rev. Lett. 70, 1619 (1993)] has a limited validity. Moreover, we give an exact solution of the Liouville equation for the moments of the two-body homogeneous cooling distribution. The latter includes velocity correlations which raises questions about the derivation of kinetic equations for inelastic gases.
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
Physics Letters A, 2005
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The second international conference on computing anticipatory systems, CASYS’98, 1999
We study changes of coordinates that allow the representation of the ordinary differential equati... more We study changes of coordinates that allow the representation of the 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, where we express the vector field of the dynamical system as a linear combination of products of powers of the
Bookmarks Related papers MentionsView impact
Lecture Notes in Physics, 1978
A Master Equation describing a discrete Markov process which is a stochastic version of the Boltz... more A Master Equation describing a discrete Markov process which is a stochastic version of the Boltzmann equation is constructed. The relation of this formulation to the usual Langevin equation approach is discussed.
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Computer Algebra and Differential Equations, 1994
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