Chaotic behavior of the Riesz transforms for Hermite expansions

The intention of this research is to investigate the aspect of non-linearity and chaotic behavior of the Cyprus stock market. For this purpose, we use non-linearity and chaos theory. We perform BDS, Hinich-Bispectral tests and compute Lyapunov exponent of the Cyprus General index. The results show that existence of non-linear dependence and chaotic features as the maximum Lyapunov exponent was found to be positive. This study is important because chaos and efficient market hypothesis are mutually exclusive aspects. The efficient market hypothesis which requires returns to be independent and identically distributed (i.i.d.) cannot be accepted.

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The Tangled Tale of Phase Space

10.1093/oso/9780198805847.003.0006 ◽

2018 ◽

Author(s):

David D. Nolte

Keyword(s):

Phase Space ◽

Chaotic Behavior ◽

Three Body Problem ◽

Henri Poincaré ◽

History Of ◽

The Stability ◽

Three Body ◽

Space Phase ◽

Central Paradigm ◽

System Phase

This chapter presents the history of the development of the concept of phase space. Phase space is the central visualization tool used today to study complex systems. The chapter describes the origins of phase space with the work of Joseph Liouville and Carl Jacobi that was later refined by Ludwig Boltzmann and Rudolf Clausius in their attempts to define and explain the subtle concept of entropy. The turning point in the history of phase space was when Henri Poincaré used phase space to solve the three-body problem, uncovering chaotic behavior in his quest to answer questions on the stability of the solar system. Phase space was established as the central paradigm of statistical mechanics by JW Gibbs and Paul Ehrenfest.

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Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501979 ◽

2019 ◽

Vol 29 (14) ◽

pp. 1950197 ◽

Cited By ~ 2

Author(s):

P. D. Kamdem Kuate ◽

Qiang Lai ◽

Hilaire Fotsin

Keyword(s):

Chaotic System ◽

Chaotic Systems ◽

Lorenz System ◽

Chaotic Behavior ◽

Piecewise Linear ◽

Period Doubling ◽

Adaptive Synchronization ◽

The Novel ◽

Algebraic Equations ◽

The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

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