Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

1995 ◽  
Vol 3 (2) ◽  
pp. 181-194 ◽  
Author(s):  
Walter M. Miller
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures

Using Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds

1997 ◽  
Vol 07 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
Rabbijah Guder ◽  
Edwin Kreuzer
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Nonlinear Dynamical Systems ◽  
Powerful Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Long Term Behavior ◽  
Mapping Theory

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.

scholarly journals Invariant Measures for Set-Valued Dynamical Systems

1999 ◽  
Vol 351 (3) ◽  
pp. 1203-1225 ◽  
Author(s):  
Walter Miller ◽  
Ethan Akin
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures

scholarly journals Invariant measures for the flow of a first order partial differential equation

1985 ◽  
Vol 5 (3) ◽  
pp. 437-443 ◽  
Author(s):  
R. Rudnicki
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Invariant Measures ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order

AbstractWe prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.

Basin of attraction of cycles of discretizations of dynamical systems with SRB invariant measures

1996 ◽  
Vol 84 (3-4) ◽  
pp. 713-733 ◽  
Author(s):  
Phil Diamond ◽  
Anthony Klemm ◽  
Peter Kloeden ◽  
Aleksej Pokrovskii
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Basin Of Attraction
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