Randomness in deterministic difference equations

Romanenko, E. Yu.

doi:10.1080/10236190902766843

Cited by 8 publications

(7 citation statements)

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“…The map g again has an attracting cycle, but of period 3; let it be denoted by {γ 1 , γ 2 , γ 3 }. 5 The basin of this cycle is formally written in the same 5 The numerical values of β1, β2 and γ1, γ2, γ3 can be found as being the zeros of form as previously: I \ D(g), but now the separator has a much more complicated structure: D(g) is a Cantor-like set. For every point z ∈ D(g), z ≠ − α 0 , α 0 , its neighborhood expands in the limit to the interval [λ, λ+λ 2 ].…”

Section: Periodicity In the Asymptotic Dynamics Of Solutionsmentioning

confidence: 99%

“…In its general sense, the self-stochasticity concept means that the asymptotic dynamics of a deterministic chaotic system can be described in probability terms [2,5]. As regards (23), self-stochasticity consists in the existence of solutions that "go" beyond the predictability horizon but whose behavior is asymptotically accurately described by some random processes.…”

Section: Self-stochasticity Propertymentioning

confidence: 99%

“…Over the past decades, the asymptotic dynamics of continuous-time difference equations was being the focus of Sharkovsky's scientific team (Institute of Mathematics, National Academy of Sciences of Ukraine). Although the main ideas of their research were published about 30 years ago [1][2][3], certain fundamental steps were made relatively recently [4][5][6], in particular it was clearly shown that such equations are very suitable for numerical and substantive analytical discussion of chaotic dynamics. Now it can already be stated that the primary theory of continuous-time difference equations is completed [7].…”

Section: Introductionmentioning

confidence: 99%

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Continuous-Time Difference Equations and Distributed Chaos Modelling

Romanenko¹

2022

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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 continuoustime 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.

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Section: Periodicity In the Asymptotic Dynamics Of Solutionsmentioning

confidence: 99%

Section: Self-stochasticity Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continuous-Time Difference Equations and Distributed Chaos Modelling

Romanenko¹

2022

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“…The map F has a smooth ergodic invariant measure concentrated on {I 1 , I 2 }, which, in addition, has the intermixing property. Therefore, reasoning from the method developed in [5,8], we can safely state that the long-term behavior of a chaotic solution of the BVP (8), (9) is described with a certain random process, defined in terms of the invariant measure of F . More specifically, the averaged finite-dimensional distributions of the solution are close to the corresponding proper (non-averaged) finite-dimensional distributions of the random process when time is large.…”

Section: Solutions Of the Bvp: Constructing And Propertiesmentioning

confidence: 99%

“…For instance, if f (z) = z(1 − z) and ln 3 < a ≤ ln 4 in (2), (3), then all the nonstationary solutions approach (in Hausdorff metric for graphs) to upper semicontinuous functions and exhibit the cascade process of emergence of self-similar spatial-temporal structures; moreover, if the map z → e a z(1 − z) has a smooth invariant measure, then the cascade process results in the self-stochasticity phenomenon -the behavior of solutions becomes unpredictable with time but it can be described with a certain random process (see [7,8] for a fuller treatment).…”

mentioning

confidence: 99%