The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

André M. McDonald; Michaël A. van Wyk; Guanrong Chen

doi:10.3934/math.2021650

journal article Jan 01, 2021

The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

André M. McDonald Michaël A. van Wyk Guanrong Chen

AIMS Mathematics Vol. 6 No. 10 pp. 11200-11232 · American Institute of Mathematical Sciences (AIMS)

View at Publisher Save 10.3934/math.2021650

Abstract

<abstract>The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.</abstract>

Topics

No keywords indexed for this article. Browse by subject →

References

65

[1]

M. E. Borel, Les probabilités dénombrables et leurs applications arithmétiques, (French) [Countable probabilities and their arithmetic applications], Rendiconti del Circolo Matematico di Palermo, 27 (1909), 247-271. 10.1007/bf03019651

[2]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Mathematica Academiae Scientiarum Hungarica, 8 (1957), 477-493. 10.1007/bf02020331

[3]

S. M. Ulam, J. von Neumann, On combination of stochastic and deterministic processes, Bull. Amer. Math. Soc., 53 (1947), 1120.

[4]

Chaos, Fractals, and Noise

Andrzej Lasota, Michael C. Mackey

Applied Mathematical Sciences 10.1007/978-1-4612-4286-4

[5]

A. Boyarsky, P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, 1$^{\rm st}$ edition, Birkhäuser, Boston, 1997. 10.1007/978-1-4612-2024-4

[6]

O. W. Rechard, Invariant measures for many-one transformations, Duke Math. J., 23 (1956), 477-488. 10.1215/s0012-7094-56-02344-4

[7]

A. Lasota, J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, T. Am. Math. Soc., 186 (1973), 481-488. 10.1090/s0002-9947-1973-0335758-1

[8]

A. Boyarsky, M. Scarowsky, On a class of transformations which have unique absolutely continuous invariant measures, T. Am. Math. Soc., 255 (1979), 243-262. 10.1090/s0002-9947-1979-0542879-2

[9]

M. Jablonski, P. Góra, A. Boyarsky, A general existence theorem for absolutely continuous invariant measures on bounded and unbounded intervals, Nonlinear World, 3 (1996), 183-200.

[10]

G. D. Birkhoff, Proof of a recurrence theorem for strongly transitive systems, P. Natl Acad. Sci. USA, 17 (1931), 650. 10.1073/pnas.17.12.650

[11]

G. D. Birkhoff, Proof of the ergodic theorem, P. Natl Acad. Sci. USA, 17 (1931), 656-660. 10.1073/pnas.17.2.656

[12]

S. M. Ulam, Problems in Modern Mathematics, 1$^{\rm st}$ edition, John Wiley & Sons, New York, 1964.

[13]

Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture

Tien-Yien Li

Journal of Approximation Theory 10.1016/0021-9045(76)90037-x

[14]

J. Ding, A. Zhou, Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjecture to multi-dimensional transformations, Physica D, 92 (1996), 61-68. 10.1016/0167-2789(95)00292-8

[15]

S. Großman, S. Thomae, Invariant distributions and stationary correlation functions of one-dimensional discrete processes, Z. Naturforsch. A, 32 (1977), 1353-1363. 10.1515/zna-1977-1204

[16]

S. V. Ershov, G. G. Malinetskii, The solution of the inverse problem for the Perron-Frobenius equation, USSR Computational Mathematics and Mathematical Physics, 28 (1988), 136-141. 10.1016/0041-5553(88)90022-5

[17]

N. Friedman, A. Boyarsky, Construction of ergodic transformations, Adv. Math., 45 (1982), 213-254. 10.1016/s0001-8708(82)80004-2

[18]

P. Góra, A. Boyarsky, A matrix solution to the inverse Perron-Frobenius problem, P. Am. Math. Soc., 118 (1993), 409-414. 10.1090/s0002-9939-1993-1129877-8

[19]

A. Baranovsky, D. Daems, Design of one-dimensional chaotic maps with prescribed statistical properties, Int. J. Bifurcat. Chaos, 5 (1995), 1585-1598. 10.1142/s0218127495001198

[20]

F. K. Diakonos, D. Pingel, P. Schmelcher, A stochastic approach to the construction of one-dimensional chaotic maps with prescribed statistical properties, Phys. Lett. A, 264 (1999), 162-170. 10.1016/s0375-9601(99)00775-6

[21]

A. McDonald, M. van Wyk, Solution of the inverse Frobenius-Perron problem for semi-Markov chaotic maps via recursive Markov state disaggregation, In: Proceedings of the 25$^{\rm th}$ IEEE European Signal Processing Conference (EUSIPCO), (2017), 1604-1608. 10.23919/eusipco.2017.8081480

[22]

A. McDonald, A. van Wyk, Construction of semi-Markov ergodic maps with selectable spectral characteristics via the solution of the inverse eigenvalue problem, In: Proceedings of the 2017 Asia Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC), (2017), 987-993. 10.1109/apsipa.2017.8282167

[23]

A. M. McDonald, M. A. van Wyk, A novel approach to solving the generalized inverse Frobenius-Perron problem, In: Proceedings of the 2020 IEEE International Symposium on Circuits and Systems (ISCAS), (2020), 1-5. 10.1109/iscas45731.2020.9181115

[24]

X. Nie, D. Coca, A new approach to solving the inverse Frobenius-Perron problem, In: Proceedings of the 2013 IEEE European Control Conference (ECC), (2013), 2916-2920. 10.23919/ecc.2013.6669502

[25]

X. Nie, D. Coca, Reconstruction of one-dimensional chaotic maps from sequences of probability density functions, Nonlinear Dynam., 80 (2015), 1373-1390. 10.1007/s11071-015-1949-9

[26]

X. Nie, D. Coca, A matrix-based approach to solving the inverse Frobenius-Perron problem using sequences of density functions of stochastically perturbed dynamical systems, Commun. Nonlinear Sci., 54 (2018), 248-266. 10.1016/j.cnsns.2017.05.011

[27]

X. Nie, J. Luo, D. Coca, M. Birkin, J. Chen, Identification of stochastically perturbed autonomous systems from temporal sequences of probability density functions, J. Nonlinear Sci., 28 (2018), 1467-1487. 10.1007/s00332-018-9455-0

[28]

X. Nie, D. Coca, J. Luo, M. Birkin, Solving the inverse Frobenius-Perron problem using stationary densities of dynamical systems with input perturbations, Commun. Nonlinear Sci., 90 (2020), 105302. 10.1016/j.cnsns.2020.105302

[29]

F. K. Diakonos, P. Schmelcher, On the construction of one-dimensional iterative maps from the invariant density: The dynamical route to the beta distribution, Phys. Lett. A, 211 (1996), 199-203. 10.1016/0375-9601(95)00971-x

[30]

D. Pingel, P. Schmelcher, F. K. Diakonos, Theory and examples of the inverse Frobenius-Perron problem for complete chaotic maps, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 357-366. 10.1063/1.166413

[31]

N. Santitissadeekorn, Transport analysis and motion estimation of dynamical systems of time-series data [dissertation], Clarkson University, New York, 2008.

[32]

A. G. Lozowski, M. Lysetskiy, J. M. Zurada, Signal processing with temporal sequences in olfactory systems, IEEE T. Neural Networks, 15 (2004), 1268-1275. 10.1109/tnn.2004.832730

[33]

A. Lasota, P. Rusek, An application of ergodic theory to the determination of the efficiency of cogged drilling bits, Archiwum Górnictwa, 3 (1974), 281-295.

[34]

A. M. McDonald, M. A. van Wyk, Efficient generation of random signals with prescribed probability distribution and spectral bandwidth via ergodic transformations, In: Proceedings of the 26$^{\rm th}$ IEEE European Signal Processing Conference (EUSIPCO), (2018), 331-335. 10.23919/eusipco.2018.8553152

[35]

H. G. Guzman, Wideband chaotic signal analysis and processing [dissertation], University of Texas, El Paso, 2007.

[36]

X. Nie, J. Wang, O. Kiselychnyk, J. Chen, Modelling of one-dimensional noisy dynamical systems with a Frobenius-Perron solution, In: Proceedings of the 22$^{\rm nd}$ IEEE International Conference on Automation and Computing (ICAC), (2016), 219-224. 10.1109/iconac.2016.7604922

[37]

R. M. Lueptow, A. Akonur, T. Shinbrot, PIV for granular flows, Exp. Fluids, 28 (2000), 183-186. 10.1007/s003480050023

[38]

J. Wu, E. S. Tzanakakis, Deconstructing stem cell population heterogeneity: Single-cell analysis and modeling approaches, Biotechnol. Adv., 31 (2013), 1047-1062. 10.1016/j.biotechadv.2013.09.001

[39]

E. Kreyszig, Introductory Functional Analysis with Applications, 1$^{\rm st}$ edition, Wiley, New York, 1978.

[40]

A. Villani, On Lusin's condition for the inverse function, Rendiconti del Circolo Matematico di Palermo, 33 (1984), 331-335. 10.1007/bf02844496

[41]

A. Boyarsky, G. Haddad, All invariant densities of piecewise linear Markov maps are piecewise constant, Adv. Appl. Math., 2 (1981), 284-289. 10.1016/0196-8858(81)90008-7

[42]

O. Perron, Zur theorie der matrices, Math. Ann., 64 (1907), 248-263. 10.1007/bf01449896

[43]

G. Frobenius, Über matrizen aus nicht negativen elementen, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 26 (1912), 456-477.

[44]

M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities, 1$^{\rm st}$ edition, Allyn and Bacon, Boston, 1964.

[45]

P. R. Halmos, Lectures on Ergodic Theory, 1$^{\rm st}$ edition, Chelsea Publishing Company, New York, 1956.

[46]

S. M. Ulam, A Collection of Mathematical Problems, 1$^{\rm st}$ edition, Interscience Publishers, New York, 1960.

[47]

M. A. van Wyk, W-H. Steeb, Chaos in Electronics, 1$^{\rm st}$ edition, Springer Science and Business Media, Dordrecht, 1997. 10.1007/978-94-015-8921-5

[48]

L. O. Chua, Y. Yao, Q. Yang, Generating randomness from chaos and constructing chaos with desired randomness, Int. J. Circ. Theor. Appl., 18 (1990), 215-240. 10.1002/cta.4490180302

[49]

A. Berlinet, G. Biau, A chaotic non-uniform random variate generator, University Montpellier II, (1999), 1-22, Report No.: 99-06.

[50]

N. Wei, Solutions of the inverse Frobenius-Perron problem [thesis], Concordia University, Montréal, 2015.

Showing 50 of 65 references

Metrics

7

Citations

65

References

Details

Published: Jan 01, 2021
Vol/Issue: 6(10)
Pages: 11200-11232

Authors

Cite This Article

André M. McDonald, Michaël A. van Wyk, Guanrong Chen (2021). The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation. AIMS Mathematics, 6(10), 11200-11232. https://doi.org/10.3934/math.2021650

The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

You May Also Like