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 Postal Mail||Institute for Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany|
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|