Is the largest Lyapunov exponent preserved in embedded dynamics?

Dechert, W. Davis; Gençay, Ramazan

doi:10.1016/s0375-9601(00)00657-5

Cited by 16 publications

(7 citation statements)

References 20 publications

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“…Therefore, the multiplicative ergodic theorem implies that the Lyapunov exponents are invariant numbers representing ‘globally’ the complexity of the dynamical system under study, independently of the initial condition considered. In this sense it is also important to point out that Gen$\def\rotf#1{\hbox to\wd#1{\hskip\wd#1\rotstart{-1 1 scale} \box#1\hss}\rotfinish} \newbox\rotfigbox \newdimen\rotdimen \def\vspec#1{\special{ps:#1}} \def\rotstart#1{\vspec{gsave currentpoint currentpoint translate #1 neg exch neg exch translate}} \def\rotfinish{\vspec{currentpoint grestore moveto}} \def\rotlt#1{\rotdimen=\ht#1\advance\rotdimen by\dp#1 \hbox to\rotdimen{\vbox to\wd#1{\vskip\wd#1\rotstart{270 rotate} \copy#1\vss}\hss}\rotfinish} \def\revcedil#1{\setbox1=\hbox{\char24}\setbox0\hbox{#1} {\ooalign{{\kern0pt\hbox to\wd0{\hfil\rotf{1}\hfil}}\crcr\noalign{\kern-0.65em\nointerlineskip}\unhbox0}}} \revcedil{c}$ ay and Dechert (1992, 1996) and Dechert and Gen$\def\rotf#1{\hbox to\wd#1{\hskip\wd#1\rotstart{-1 1 scale} \box#1\hss}\rotfinish} \newbox\rotfigbox \newdimen\rotdimen \def\vspec#1{\special{ps:#1}} \def\rotstart#1{\vspec{gsave currentpoint currentpoint translate #1 neg exch neg exch translate}} \def\rotfinish{\vspec{currentpoint grestore moveto}} \def\rotlt#1{\rotdimen=\ht#1\advance\rotdimen by\dp#1 \hbox to\rotdimen{\vbox to\wd#1{\vskip\wd#1\rotstart{270 rotate} \copy#1\vss}\hss}\rotfinish} \def\revcedil#1{\setbox1=\hbox{\char24}\setbox0\hbox{#1} {\ooalign{{\kern0pt\hbox to\wd0{\hfil\rotf{1}\hfil}}\crcr\noalign{\kern-0.65em\nointerlineskip}\unhbox0}}} \revcedil{c}$ ay (2000) have studied the topological invariance of the Lyapunov exponent estimator from the observed dynamics.…”

Section: A Statistical Framework For Testing Chaotic Dynamics Via mentioning

confidence: 99%

Testing chaotic dynamics via Lyapunov exponents

Rodríguez

Sosvilla‐Rivero

Félix

2005

J of Applied Econometrics

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SUMMARYWe propose a new test to detect chaotic dynamics, based on the stability of the largest Lyapunov exponent from different sample sizes. This test is applied to the data used in the single-blind controlled competition tests for non-linearity and chaos that were generated by Barnett et al. (1997), as well as to several other chaotic series. The results suggest that the new test is particularly effective when compared to other stochastic alternatives (both linear and non-linear). For large sample sizes the power of the test is one, although for small sample sizes it diminishes occasionally.

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Section: A Statistical Framework For Testing Chaotic Dynamics Via mentioning

confidence: 99%

Testing chaotic dynamics via Lyapunov exponents

Rodríguez

Sosvilla‐Rivero

Félix

2005

J of Applied Econometrics

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“…These additional Lyapunov exponents have been labeled as spurious Lyapunov exponents [e.g. Sauer et al, 1998, Dechert andGençay, 2000]. They pose significant challenges in data-driven identification of true Lyapunov exponents.…”

Section: Paradigm II : Delay Coordinates An Effective and Numerically...mentioning

confidence: 99%

Learning theory for dynamical systems

Berry¹,

Das²

2022

Preprint

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The task of modelling and forecasting a dynamical system is one of the oldest problems, and it remains challenging. Broadly, this task has two subtasks -extracting the full dynamical information from a partial observation; and then explicitly learning the dynamics from this information. We present a mathematical framework in which the dynamical information is represented in the form of an embedding. The framework combines the two subtasks using the language of spaces, maps, and commutations. The framework also unifies two of the most common learning paradigms -delaycoordinates and reservoir computing. We use this framework as a platform for two other investigations of the reconstructed system -its dynamical stability; and the growth of error under iterations. We show that these questions are deeply tied to more fundamental properties of the underlying systemthe behavior of matrix cocycles over the base dynamics, its non-uniform hyperbolic behavior, and its decay of correlations. Thus, our framework bridges the gap between universally observed behavior of dynamics modelling; and the spectral, differential and ergodic properties intrinsic to the dynamics.

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“…(However, since the mdimensional system h has a larger dimension than the n-dimensional system f , the number of smooth Lyapunov exponents that are spurious is m n. This issue is discussed in detail in Dechert and Gencay [10]- [11] and Gencay and…”

Section: Estimation Ofmentioning

confidence: 99%

Market structure and the stability and volatility of electricity prices

Bask

Widerberg

2009

Energy Economics

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By using a novel approach in this paper, ;2 -analysis, we have found that electricity prices most of the time have increased in stability and decreased in volatility when the Nordic power market has expanded and the degree of competition has increased. That electricity prices at Nord Pool have been generated by a stochastic dynamic system that most often has become more stable during the step-wise integration of the Nordic power market means that this market is less sensitive to shocks after the integration process than it was before this process. This is good news.JEL codes: C14; C22; D43.

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Is the largest Lyapunov exponent preserved in embedded dynamics?

Cited by 16 publications

References 20 publications

Testing chaotic dynamics via Lyapunov exponents

Testing chaotic dynamics via Lyapunov exponents

Learning theory for dynamical systems

Market structure and the stability and volatility of electricity prices

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