Structural stability of Lorenz attractors

Structural Stability and Hyperbolic Attractors

A necessary condition for structural stability is presented that in the two dimensional case means that the system has a finite number of topological attractors. Introduction. One of the basic questions in Dynamical System Theory is the characterization and the study of the properties of structurally stable systems (vector fields or diffeomorphisms). The standing conjecture about characterizati...

Computing geometric Lorenz attractors with arbitrary precision

The Lorenz attractor was introduced in 1963 by E. N. Lorenz as one of the first examples of strange attractors. However Lorenz’ research was mainly based on (non-rigourous) numerical simulations and, until recently, the proof of the existence of the Lorenz attractor remained elusive. To address that problem some authors introduced geometric Lorenz models and proved that geometric Lorenz models ...

Nonsymmetric Lorenz Attractors from a Homoclinic Bifurcation

We consider a bifurcation of a flow in three dimensions from a double homoclinic connection to a fixed point satisfying a resonance condition between the eigenvalues. For correctly chosen parameters in the unfolding, we prove that there is a transitive attractor of Lorenz type. In particular we show the existence of a bifurcation to an attractor of Lorenz type which is semiorientable, i.e., ori...

Lorenz attractors in flows and maps

Evolving Chaos: Identifying New Attractors of the Generalised Lorenz Family

In a recent paper, we presented an intelligent evolutionary search technique through genetic programming (GP) for finding new analytical expressions of nonlinear dynamical systems, similar to the classical Lorenz attractor’s which also exhibit chaotic behaviour in the phase space. In this paper, we extend our previous finding to explore yet another gallery of new chaotic attractors which are de...