Continuous-Time Difference Equations and Distributed Chaos Modelling

Olena Romanenko

doi:doi:10.11648/j.ml.20220801.12

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Continuous-Time Difference Equations and Distributed Chaos Modelling

Olena Romanenko

Published in Mathematics Letters (Volume 8, Issue 1)

Received: 13 February 2022 Accepted: 29 March 2022 Published: 20 April 2022

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Abstract

The article aims to attract the attention of researchers, experts and those interested in nonlinear dynamics and chaos theory to the not well known field of continuous-time difference equations, in the hopes of opening new doors into the study of chaotic system. Deterministic chaos and related notions are used in an increasing number of scientific works. There are a lot of problems associated with the mathematical aspects of the fine structure of chaos. Just as discrete-time difference equations have proven to be excellent models of temporal (discrete) chaos, so continuous-time difference equations provide new elegant mechanisms for onset and inside reconstructions of spatio-temporal (distributed) chaos. Distributed chaos is usually described by boundary value problems for partial differential equations. A number of these boundary value problems can be reduced to continuous-time difference equations, which enable one to build new chaos scenarios arising from the properties of the equations. Whereas the emergence of deterministic chaos is usually attributed to the complex structure of attractors, these new scenarios are based on a highly complex structure of spatially extended “points” of the attractor. Examples of reducible boundary value problems are set forth in the article, but the main focus is on a very elementary overview of the principal features of solutions of the simplest nonlinear continuous-time difference equations: loss of continuity, asymptotic periodicity, gradient catastrophe, fractal geometry, space-filling property, going beyond the horizon of predictability, self-stochasticity (deterministic solutions are asymptotically described by random processes), formation of hierarchical structures (down to arbitrarily small scales). Here we have a wonderful example of how very complex phenomena can be described with very simple equations. The use of continuous-time difference equations in the study of reducible and close-to-reducible boundary value problems migh help to advance in understanding possible mathematical mechanisms for distributed chaos.

Published in	Mathematics Letters (Volume 8, Issue 1)
DOI	10.11648/j.ml.20220801.12
Page(s)	11-21
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Continuous-Time Difference Equation, Boundary Value Problem, Deterministic Chaos

References

[1]	Sharkovsky, A. N., Maistrenko, Yu. L., & Romanenko, E. Yu. (1993). Difference Equations and Their Applications, Ser. Math. and Appl., 250, Dordrecht: Kluwer Academic。
[2]	Sharkovsky, A. N., & Romanenko, E. Yu. (1992). Ideal turbulence: Attractors of deterministic systems may lie in the space of random fields. Int. Journal of Bifurcation and Chaos, 2 (1), 31-36.
[3]	Romanenko, E. Yu. (1998). On attractors of continuous time difference equations. Computers Math. with Appl., 36 (10-12), 377-390.
[4]	Romanenko, E. Yu. (2003). Dynamical systems induced by continuous time difference equations and long-time behavior of solutions. Journal of Difference Equations and Appl., 9 (3-4), 263-280.
[5]	Romanenko, E. Yu. (2010). Randomness in deterministic continuous time difference equations. Ibid., 16 (2-3), 243-268.
[6]	Sharkovsky, A. N., & Romanenko, E. Yu. (2010). Difference Equations with Continuous Time: Theory and Applications. Discrete Dynamics and Difference Equations (Proc. ICDEA-2007), World Scientific, 104-119.
[7]	Romanenko, O. Yu, (2014). Difference Equations With Continuous Argument. Proc. of the Inst. of Math. Ukraine, 100, 346 (in Russian).
[8]	Sharkovsky, A. N., & Romanenko, E. Yu. (2005). Turbulence, ideal. Encyclopedia of Nonlinear Science (ed. Alwyn Scott), New York and London: Routledge, 955-957. (http://www.routledge-ny.com/nonlinsci).
[9]	Romanenko, E. Yu., & Sharkovskii, A. N. (2007). Dynamical systems and simulation of turbulence, Ukrainian Mathematical Journal, 59 (2), 229 - 242.
[10]	Sharkovsky, A. N., & Romanenko, E. Yu. (2020). Ideal turbulence: fractal and stochastic attractors in idealized models of mathematical physics. Proc. of the Inst. of Math. Ukraine, 106, 179p. (in Russian).
[11]	Witt, A. A. (1936) On the theory of the violin string. Journal. Tech. Physics, 6 (9), 1459-1479 (in Russian).
[12]	Sharkovsky, A. N. (1979). Oscillations described by autonomous difference and differential-difference equations. Proc. of Int. Conf. Nonliniar Osccilations (Prague, 1979), v. 2, Prague: Academia, 1073-1078.
[13]	Nagumo, J., & Shimura, M. (1961). Self-oscillation in a transmission line with a tunnel diode. Proc. IEEE., 49, 1281-1291.
[14]	Cooke, K. L., & Krumme, D. (1968). Differential-difference equations and nonlinear initial-boundary-value problems for linear hyperbolic partial differential equations. Journal of Math. Anal. Appl., 24, 372-387.
[15]	Romanenko, E. Yu., & Sharkovsky, A. N. (1999). From boundary value problems to difference equations: A method of investigation of chaotic vibrations. Intern. Journal of Bifurcation and Chaos, 9 (7), 1285-1306.
[16]	Sharkovsky, A. N. (2004). Difference equations and boundary value problems. New Progress in Difference Equations (Proc. ICDEA-2001), London: Taylor and Francis, 3-22.
[17]	Sharkovsky, A. N., & Romanenko, E. Yu. (2004). Difference equations and dynamical systems generated by some classes of boundary value problems. Proc. the Steklov Inst. of Math. Russia, 244, 264-279.
[18]	Fedorenko, V. V., Romanenko, E. Yu., & Sharkovsky, A. N. (2007). Trajectories of intervals in one-dimensional dynamical systems. Journal of Difference Equations and Appl., 13 (8-9), 821-828.

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Olena Romanenko. (2022). Continuous-Time Difference Equations and Distributed Chaos Modelling. Mathematics Letters, 8(1), 11-21. https://doi.org/10.11648/j.ml.20220801.12

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ACS Style

Olena Romanenko. Continuous-Time Difference Equations and Distributed Chaos Modelling. Math. Lett. 2022, 8(1), 11-21. doi: 10.11648/j.ml.20220801.12

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AMA Style

Olena Romanenko. Continuous-Time Difference Equations and Distributed Chaos Modelling. Math Lett. 2022;8(1):11-21. doi: 10.11648/j.ml.20220801.12

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@article{10.11648/j.ml.20220801.12,
  author = {Olena Romanenko},
  title = {Continuous-Time Difference Equations and Distributed Chaos Modelling},
  journal = {Mathematics Letters},
  volume = {8},
  number = {1},
  pages = {11-21},
  doi = {10.11648/j.ml.20220801.12},
  url = {https://doi.org/10.11648/j.ml.20220801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20220801.12},
  abstract = {The article aims to attract the attention of researchers, experts and those interested in nonlinear dynamics and chaos theory to the not well known field of continuous-time difference equations, in the hopes of opening new doors into the study of chaotic system. Deterministic chaos and related notions are used in an increasing number of scientific works. There are a lot of problems associated with the mathematical aspects of the fine structure of chaos. Just as discrete-time difference equations have proven to be excellent models of temporal (discrete) chaos, so continuous-time difference equations provide new elegant mechanisms for onset and inside reconstructions of spatio-temporal (distributed) chaos. Distributed chaos is usually described by boundary value problems for partial differential equations. A number of these boundary value problems can be reduced to continuous-time difference equations, which enable one to build new chaos scenarios arising from the properties of the equations. Whereas the emergence of deterministic chaos is usually attributed to the complex structure of attractors, these new scenarios are based on a highly complex structure of spatially extended “points” of the attractor. Examples of reducible boundary value problems are set forth in the article, but the main focus is on a very elementary overview of the principal features of solutions of the simplest nonlinear continuous-time difference equations: loss of continuity, asymptotic periodicity, gradient catastrophe, fractal geometry, space-filling property, going beyond the horizon of predictability, self-stochasticity (deterministic solutions are asymptotically described by random processes), formation of hierarchical structures (down to arbitrarily small scales). Here we have a wonderful example of how very complex phenomena can be described with very simple equations. The use of continuous-time difference equations in the study of reducible and close-to-reducible boundary value problems migh help to advance in understanding possible mathematical mechanisms for distributed chaos.},
 year = {2022}
}

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TY  - JOUR
T1  - Continuous-Time Difference Equations and Distributed Chaos Modelling
AU  - Olena Romanenko
Y1  - 2022/04/20
PY  - 2022
N1  - https://doi.org/10.11648/j.ml.20220801.12
DO  - 10.11648/j.ml.20220801.12
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 11
EP  - 21
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20220801.12
AB  - The article aims to attract the attention of researchers, experts and those interested in nonlinear dynamics and chaos theory to the not well known field of continuous-time difference equations, in the hopes of opening new doors into the study of chaotic system. Deterministic chaos and related notions are used in an increasing number of scientific works. There are a lot of problems associated with the mathematical aspects of the fine structure of chaos. Just as discrete-time difference equations have proven to be excellent models of temporal (discrete) chaos, so continuous-time difference equations provide new elegant mechanisms for onset and inside reconstructions of spatio-temporal (distributed) chaos. Distributed chaos is usually described by boundary value problems for partial differential equations. A number of these boundary value problems can be reduced to continuous-time difference equations, which enable one to build new chaos scenarios arising from the properties of the equations. Whereas the emergence of deterministic chaos is usually attributed to the complex structure of attractors, these new scenarios are based on a highly complex structure of spatially extended “points” of the attractor. Examples of reducible boundary value problems are set forth in the article, but the main focus is on a very elementary overview of the principal features of solutions of the simplest nonlinear continuous-time difference equations: loss of continuity, asymptotic periodicity, gradient catastrophe, fractal geometry, space-filling property, going beyond the horizon of predictability, self-stochasticity (deterministic solutions are asymptotically described by random processes), formation of hierarchical structures (down to arbitrarily small scales). Here we have a wonderful example of how very complex phenomena can be described with very simple equations. The use of continuous-time difference equations in the study of reducible and close-to-reducible boundary value problems migh help to advance in understanding possible mathematical mechanisms for distributed chaos.
VL  - 8
IS  - 1
ER  -

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Author Information

Olena Romanenko

Department of Dynamical Systems Theory and Fractal Analysis, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

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APA Style

Olena Romanenko. (2022). Continuous-Time Difference Equations and Distributed Chaos Modelling. Mathematics Letters, 8(1), 11-21. https://doi.org/10.11648/j.ml.20220801.12

Copy | Download

ACS Style

Olena Romanenko. Continuous-Time Difference Equations and Distributed Chaos Modelling. Math. Lett. 2022, 8(1), 11-21. doi: 10.11648/j.ml.20220801.12

Copy | Download

AMA Style

Olena Romanenko. Continuous-Time Difference Equations and Distributed Chaos Modelling. Math Lett. 2022;8(1):11-21. doi: 10.11648/j.ml.20220801.12

Copy | Download

@article{10.11648/j.ml.20220801.12,
  author = {Olena Romanenko},
  title = {Continuous-Time Difference Equations and Distributed Chaos Modelling},
  journal = {Mathematics Letters},
  volume = {8},
  number = {1},
  pages = {11-21},
  doi = {10.11648/j.ml.20220801.12},
  url = {https://doi.org/10.11648/j.ml.20220801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20220801.12},
  abstract = {The article aims to attract the attention of researchers, experts and those interested in nonlinear dynamics and chaos theory to the not well known field of continuous-time difference equations, in the hopes of opening new doors into the study of chaotic system. Deterministic chaos and related notions are used in an increasing number of scientific works. There are a lot of problems associated with the mathematical aspects of the fine structure of chaos. Just as discrete-time difference equations have proven to be excellent models of temporal (discrete) chaos, so continuous-time difference equations provide new elegant mechanisms for onset and inside reconstructions of spatio-temporal (distributed) chaos. Distributed chaos is usually described by boundary value problems for partial differential equations. A number of these boundary value problems can be reduced to continuous-time difference equations, which enable one to build new chaos scenarios arising from the properties of the equations. Whereas the emergence of deterministic chaos is usually attributed to the complex structure of attractors, these new scenarios are based on a highly complex structure of spatially extended “points” of the attractor. Examples of reducible boundary value problems are set forth in the article, but the main focus is on a very elementary overview of the principal features of solutions of the simplest nonlinear continuous-time difference equations: loss of continuity, asymptotic periodicity, gradient catastrophe, fractal geometry, space-filling property, going beyond the horizon of predictability, self-stochasticity (deterministic solutions are asymptotically described by random processes), formation of hierarchical structures (down to arbitrarily small scales). Here we have a wonderful example of how very complex phenomena can be described with very simple equations. The use of continuous-time difference equations in the study of reducible and close-to-reducible boundary value problems migh help to advance in understanding possible mathematical mechanisms for distributed chaos.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Continuous-Time Difference Equations and Distributed Chaos Modelling
AU  - Olena Romanenko
Y1  - 2022/04/20
PY  - 2022
N1  - https://doi.org/10.11648/j.ml.20220801.12
DO  - 10.11648/j.ml.20220801.12
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 11
EP  - 21
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20220801.12
AB  - The article aims to attract the attention of researchers, experts and those interested in nonlinear dynamics and chaos theory to the not well known field of continuous-time difference equations, in the hopes of opening new doors into the study of chaotic system. Deterministic chaos and related notions are used in an increasing number of scientific works. There are a lot of problems associated with the mathematical aspects of the fine structure of chaos. Just as discrete-time difference equations have proven to be excellent models of temporal (discrete) chaos, so continuous-time difference equations provide new elegant mechanisms for onset and inside reconstructions of spatio-temporal (distributed) chaos. Distributed chaos is usually described by boundary value problems for partial differential equations. A number of these boundary value problems can be reduced to continuous-time difference equations, which enable one to build new chaos scenarios arising from the properties of the equations. Whereas the emergence of deterministic chaos is usually attributed to the complex structure of attractors, these new scenarios are based on a highly complex structure of spatially extended “points” of the attractor. Examples of reducible boundary value problems are set forth in the article, but the main focus is on a very elementary overview of the principal features of solutions of the simplest nonlinear continuous-time difference equations: loss of continuity, asymptotic periodicity, gradient catastrophe, fractal geometry, space-filling property, going beyond the horizon of predictability, self-stochasticity (deterministic solutions are asymptotically described by random processes), formation of hierarchical structures (down to arbitrarily small scales). Here we have a wonderful example of how very complex phenomena can be described with very simple equations. The use of continuous-time difference equations in the study of reducible and close-to-reducible boundary value problems migh help to advance in understanding possible mathematical mechanisms for distributed chaos.
VL  - 8
IS  - 1
ER  -

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