Measure of invertible dynamical maps under convex combinations of noninvertible dynamical maps

Discrete Dynamical Systems: Maps

Many physical systems displaying chaotic behavior are accurately described by mathematical models derived from well-understood physical principles. For example, the fundamental equations of fluid dynamics, namely the Navier–Stokes equations, are obtained from elementary mechanical and thermodynamical considerations. The simplest laser models are built from Maxwell’s laws of electromagnetism and...

Continuous iteration of dynamical maps

K-Dynamical Self Organizing Maps

Neural maps are a very popular class of unsupervised neural networks that project high-dimensional data of the input space onto a neuron position in a low-dimensional output space grid. It is desirable that the projection effectively preserves the structure of the data. In this paper we present a hybrid model called K-Dynamical Self Organizing Maps (KDSOM ) consisting of K Self Organizing Maps ...

Torus breakdown in noninvertible maps.

We propose a criterion for the destruction of a two-dimensional torus through the formation of an infinite set of cusp points on the closed invariant curves defining the resonance torus. This mechanism is specific to noninvertible maps. The cusp points arise when the tangent to the torus at the point of intersection with the critical curve L(0) coincides with the eigendirection corresponding to...

Homoclinic tangles for noninvertible maps

Not all naturally occurring iterated systems have the property that time is reversible. In which case, the dynamics must be modelled by a noninvertible map rather than by a di eomorphism. While one-dimensional theory describes dynamics of noninvertible maps, higher-dimensional theory has focussed on di eomorphisms, though there are many higher-dimensional examples of natural systems modelled by...