Vector fields, invariant varieties and linear systems

Admissible vector fields and quasi-invariant measures∗

Lorenz and Rössler Systems with Piecewise-Linear Vector Fields

Dynamical systems of class C [1] are described by the 3-order autonomous differential equation with nonlinearity given as a three-segment piecewise linear (PWL) function. Argument of this function is a linear combination of state variables. These systems form an extensive group of nonlinear systems with PWL vector fields and may produce rich set of chaotic attractors. The paper shows how this g...

Invariant Hypersurfaces for Positive Characteristic Vector Fields

We show that a generic vector field on an affine space of positive characteristic admits an invariant algebraic hypersurface. This is in sharp contrast with the characteristic zero case where Jouanolou’s Theorem says that a generic vector field on the complex plane does not admit any invariant algebraic curve.

Invariant varieties for polynomial dynamical systems

We study algebraic dynamical systems (and, more generally, σ-varieties) Φ : AC → AC given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of “clusters” from which one may easily read off possible compositional identities. Our main res...

Cyclic wavelet systems in prime dimensional linear vector spaces

Finite affine groups are given by groups of translations and di- lations on finite cyclic groups. For cyclic groups of prime order we develop a time-scale (wavelet) analysis and show that for a large class of non-zero window signals/vectors, the generated full cyclic wavelet system constitutes a frame whose canonical dual is a cyclic wavelet frame.