Stability of Invariant Measures

Boyarsky, Abraham; Góra, Paweł

doi:10.1007/978-1-4612-2024-4_11

Cited by 41 publications

(71 citation statements)

References 0 publications

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“…Condition (iii) is a technical restriction, which is sometimes called expanding in the literature [28], although this terminology is usually reserved for stronger conditions [29]. It will be important in lemma 3.7 later.…”

Section: Discrete-time Random Walksmentioning

confidence: 99%

Limiting stochastic processes of shift-periodic dynamical systems

Stadlmann

Erban

2019

R. Soc. open sci.

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A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences xn+1 = F(xn) generated by such maps display rich dynamical behaviour. The integer parts falsefalse⌊xnfalsefalse⌋ give a discrete-time random walk for a suitable initial distribution of x0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

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Section: Discrete-time Random Walksmentioning

confidence: 99%

Limiting stochastic processes of shift-periodic dynamical systems

Stadlmann

Erban

2019

R. Soc. open sci.

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“…Let

be a partition of R into intervals, and

if i ≠ j . Assuming that S is piecewise monotonic and expanding [18] ,

where S i is the monotonic restriction of S on the interval R i .…”

Section: Description Of the Inverse Problemmentioning

confidence: 99%

“…Let f n be a piecewise constant function over the partition R such that

. Its image under transformation P S f n is also a piecewise constant function over ℜ [18] such that

and the Frobenius–Perron operator can be represented by a finite-dimensional matrix

where

is the Frobenius–Perron matrix induced by S with entries given by

…”

Section: A Matrix Representation Of the Transfer Operatormentioning

confidence: 99%

“…In such cases, in the absence of individual point trajectories, it is desirable to determine the underlying dynamical system that generated the observed density functions. Given a non-singular transformation, the evolution of an initial density function under the action of the transformation is described by the Frobenius–Perron operator associated with the transformation [18] . The fixed point of such an operator represents the invariant density under the transformation.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A matrix-based approach to solving the inverse Frobenius–Perron problem using sequences of density functions of stochastically perturbed dynamical systems

Nie¹,

Coca²

2018

Communications in Nonlinear Science and Numerical Simulation

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HighlightsThe paper introduces new method of reconstructing an unknown one-dimensional transformation that is subject to constantly applied stochastic perturbations based on temporal sequences of probability density functions.The main assumption is that the one-dimensional transformation that generated the densities is piecewise-linear, semi-Markov.A matrix approximation of the transfer operator associated with the stochastically perturbed transformation, which forms the basis for the reconstruction algorithm, is introduced.A practical algorithm to estimate the matrix-representation of the Frobenius-Perron operator associated with the unperturbed transformation and reconstruct the onedimensional map is proposed.The algorithm is extended to nonlinear continuous maps.Numerical simulation examples are provided to demonstrate the performance of the approach and to compare it with that of an existing algorithm.

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“…By Proposition 2 (ii), ( I , ℬ I , γ m , τ m ) is a “dynamical system” in the sense of [ 7 , Definition 3.1.3].…”

Section: Introductionmentioning

confidence: 99%