Binary differential equations with symmetries
Miriam Manoel Patríicia Tempesta
Discrete & Continuous Dynamical Systems - A 2019, 39(4): 1957-1974 doi: 10.3934/dcds.2019082

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, $ a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0, $ for $ a, b, c $ smooth real functions defined on an open set of $ \mathbb{R}^2 $. Generically, solutions of a BDE are given as leaves of a pair of foliations, and the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts methods from invariant theory of compact Lie groups to obtain an algorithm to compute general expressions of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group $ {{\bf{O}}(2)} $.

keywords: Binary differential equation symmetry group representation equivariant quadratic differential form compact Lie group
Divergent diagrams of folds and simultaneous conjugacy of involutions
Solange Mancini Miriam Manoel Marco Antonio Teixeira
Discrete & Continuous Dynamical Systems - A 2005, 12(4): 657-674 doi: 10.3934/dcds.2005.12.657
In this work we show that the smooth classification of divergent diagrams of folds $(f_1, \ldots, f_s) : (\mathbb R^n,0) \to (\mathbb R^n \times \cdots \times \mathbb R^n,0)$ can be reduced to the classification of the $s$-tuples $(\varphi_1, \ldots, \varphi_s)$ of associated involutions. We apply the result to obtain normal forms when $s \leq n$ and $\{\varphi_1, \ldots, \varphi_s\}$ is a transversal set of linear involutions. A complete description is given when $s=2$ and $n\geq 2$. We also present a brief discussion on applications of our results to the study of discontinuous vector fields and discrete reversible dynamical systems.
keywords: singularities involution discontinuous vector fields reversible diffeomorphisms. Divergent diagram of folds normal form

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