Natural invariant measures

Fractals and Choas

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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 AffiliationUniversity of Paderborn, Germany
Author Email
Author Postal MailInstitute for Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
Notes
The pictures have been rendered in collaboration with Martin Rumpf (rumpf@math.uni-duisburg.de) using the software platform GRAPE.
Keywordsnatural invariant measure, SRB-measure, transfer operator, set oriented method
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