Simulating a chaotic process

Viana, Ricardo L.; Barbosa, Jose Renato Ramos; Grebogi, Celso; Batista, Antônio M.

doi:10.1590/s0103-97332005000100010

Cited by 4 publications

(5 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The selection of augmenting the nonlinear volatility models with chaos theory is based on the following. Chaotic behavior means high sensitivity to initial conditions (Viana and Barbosa 2005). Small differences in initial conditions give important outcomes in chaotic behavior, since very small differentiation in initial conditions (Wernecke et al 2017) leads to enormous differences between expected and realized values in the long term.…”

Section: Introductionmentioning

confidence: 99%

The chaotic behavior among the oil prices, expectation of investors and stock returns: TAR-TR-GARCH copula and TAR-TR-TGARCH copula

Bildirici

2018

Pet. Sci.

View full text Add to dashboard Cite

This paper has two aims. The first one is to investigate the existence of chaotic structures in the oil prices, expectations of investors and stock returns by combining the Lyapunov exponent and Kolmogorov entropy, and the second one is to analyze the dependence behavior of oil prices, expectations of investors and stock returns from January 02, 1990, to June 06, 2017. Lyapunov exponents and Kolmogorov entropy determined that the oil price and the stock return series exhibited chaotic behavior. TAR-TR-GARCH and TAR-TR-TGARCH copula methods were applied to study the co-movement among the selected variables. The results showed significant evidence of nonlinear tail dependence between the volatility of the oil prices, the expectations of investors and the stock returns. Further, upper and lower tail dependence and comovement between the analyzed series could not be rejected. Moreover, the TAR-TR-GARCH and TAR-TR-TGARCH copula methods revealed that the volatility of oil price had crucial effects on the stock returns and on the expectations of investors in the long run.

show abstract

Section: Introductionmentioning

confidence: 99%

The chaotic behavior among the oil prices, expectation of investors and stock returns: TAR-TR-GARCH copula and TAR-TR-TGARCH copula

Bildirici

2018

Pet. Sci.

View full text Add to dashboard Cite

show abstract

“…The UDV is reflected and quantified by the fluctuations around zero of the finite-time exponent closest to zero (Davidchack & Lai 2000;Viana et al 2005). These fluctuations around the zero value of the finite-time exponents are then a good indicator of the non-hyperbolic nature of the orbit.…”

Section: Analysis Of the Resultsmentioning

confidence: 99%

Role of dark matter haloes on the predictability of computed orbits

Vallejo

Sanjuán

2016

A&A

View full text Add to dashboard Cite

Aims. The predictability of a system indicates how often a computed orbit is close to a real orbit of the system, independent of its stability or chaotic nature. We explore the effect of dark halo shapes on the predictability of computed orbits in a Milky Way mean field model. We also present the sources for the low predictability found in some orbits. Methods. We derived a predictability index from the distributions of the finite-time Lyapunov exponents. We computed those distributions and analysed the evolution of their shapes when the finite-time interval sizes are varied. The predictability index can be computed using the interval lengths corresponding to the timescales when the flow dynamics leaves the local regime and enters the global regime. Results. These analyses reveal that not all chaotic orbits have the same predictability and that the predictability of some orbits is more affected than others by the orientation and shape of the dark halo. We show that the lowest predictability may be linked to strong unstable dimension variability.

show abstract

“…If f were globally hyperbolic, then a shadowing trajectory exists for all time, which has zero mismatch, and the gradient descent algorithm will converge to such a trajectory if the observational noise is sufficiently small [28]. This cannot be said of a non-hyperbolic system [8,21,22,31]. Of course, if a non-hyperbolic system happens to be locally hyperbolic in the region of x and y, then some of the results pertaining to (globally) hyperbolic systems may apply, but this does not help when the system is non-hyperbolic in this region.…”

Section: Resultsmentioning

confidence: 99%

“…The author, for example, has applied GDI to a simple quasi-geostrophic model of the atmosphere [18] (dimension ≈ 1500) and an operational weather forecasting model at reduced resolution [15] (dimension ≈ 7.5 × 10 5 ). In both of these applications the model certainly has Lyapunov exponents near zero, and possibly local Lyapunov exponents that fluctuate about zero [8,21,22,31]. Three important issues that arise in the application of GDI are:…”

Section: Applications and Discussionmentioning

confidence: 99%

“…There are other problems. It has been shown that there exist non-hyperbolic systems for which shadowing trajectories cannot not be obtained [8,21,22,31]; these systems have local Lyapunov exponents that fluctuate about zero over time. On the other hand, Pilyugin and others have shown that shadowing is a generic property of homeomorphisms in the C 0 -topology [26].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Shadowing Pseudo-Orbits and Gradient Descent Noise Reduction

Judd

2007

J Nonlinear Sci

View full text Add to dashboard Cite

Shadowing trajectories are one of the most powerful ideas of modern dynamical systems theory, providing a tool for proving some central theorems and a means to assess the relevance of models and numerically computed trajectories of chaotic systems. Shadowing has also been seen to have a role in state estimation and forecasting of nonlinear systems. Shadowing trajectories are guaranteed to exist in hyperbolic systems, but this is not true of non-hyperbolic systems, indeed it can be shown there are systems that cannot have long shadowing trajectories. In this paper we consider what might be called shadowing pseudo-orbits. These are pseudo-orbits that remain close to a given pseudo-orbit, but have smaller mismatches between forecast state and verifying state. Shadowing pseudo-orbits play a useful role in the understanding and analysis of gradient descent noise reduction, state estimation and forecasting nonlinear systems, because their existence can be ensured for a wide class of non-hyperbolic systems. New theoretical results are presented that that extend classical shadowing theorems to shadowing pseudo-orbits. These new results provide some insight into the convergence behaviour of gradient descent noise reduction methods. The paper also discusses, in the light of the new results, some recent numerical results for an operational weather forecasting model when gradient descent noise reduction was employed.

show abstract

Simulating a chaotic process

Cited by 4 publications

References 31 publications

The chaotic behavior among the oil prices, expectation of investors and stock returns: TAR-TR-GARCH copula and TAR-TR-TGARCH copula

The chaotic behavior among the oil prices, expectation of investors and stock returns: TAR-TR-GARCH copula and TAR-TR-TGARCH copula

Role of dark matter haloes on the predictability of computed orbits

Shadowing Pseudo-Orbits and Gradient Descent Noise Reduction

Contact Info

Product

Resources

About