# Natural invariant measures

### Fractals and Choas

By Michael Dellnitz, Oliver Junge
Print

Shown are approximations to natural invariant measures ("SRB-measures") in the Lorenz system and in Chua's circuit. The underlying computational technique consists of (1) computing a covering of the underlying invariant set via a multilevel-subdivision algorithm and (2) discretization of the corresponding transfer-operator (Perron-Frobenius operator) using a Galerkin-approach. An invariant vector of the discretized operator yields an approximate invariant measure.

Natural invariant measure in Chua's circuit for the parameter values $$\alpha=18$$, $$\beta=33$$, $$m_0=-0.2$$ and $$m_1=0.01$$.

Natural invariant measures in the Lorenz system for the parameter values $$\beta = 0.4, 0.8, 1.2$$ and $$8/3$$ (from left to right, top to bottom). The other parameter values were fixed to $$\sigma = 10$$ and $$\rho = 28$$.

 Author Institutional Affiliation University of Paderborn, Germany Author Email Author Postal Mail Institute for Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany Notes The pictures have been rendered in collaboration with Martin Rumpf ([email protected]) using the software platform GRAPE. Keywords natural invariant measure, SRB-measure, transfer operator, set oriented method
Tags: