Survey: Lyapunov exponents for stochastic flows on manifolds

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

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Lyapunov Exponents of the Half-Line SHE

Journal of Statistical Physics ◽

10.1007/s10955-021-02772-8 ◽

2021 ◽

Vol 183 (3) ◽

Author(s):

Yier Lin

Keyword(s):

Lyapunov Exponents ◽

Half Line

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Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

Physical Review Fluids ◽

10.1103/physrevfluids.5.024602 ◽

2020 ◽

Vol 5 (2) ◽

Author(s):

Richard D. J. G. Ho ◽

Andres Armua ◽

Arjun Berera

Keyword(s):

Lyapunov Exponents ◽

Isotropic Turbulence ◽

Homogeneous And Isotropic Turbulence

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Hyperchaotic Image Encryption Based on Multiple Bit Permutation and Diffusion

Entropy ◽

10.3390/e23050510 ◽

2021 ◽

Vol 23 (5) ◽

pp. 510

Author(s):

Taiyong Li ◽

Duzhong Zhang

Keyword(s):

Big Data ◽

Lyapunov Exponents ◽

Image Encryption ◽

Hyperchaotic System ◽

Encryption Scheme ◽

Image Content ◽

Image Security ◽

Level Data ◽

Different Types ◽

And Diffusion

Image security is a hot topic in the era of Internet and big data. Hyperchaotic image encryption, which can effectively prevent unauthorized users from accessing image content, has become more and more popular in the community of image security. In general, such approaches conduct encryption on pixel-level, bit-level, DNA-level data or their combinations, lacking diversity of processed data levels and limiting security. This paper proposes a novel hyperchaotic image encryption scheme via multiple bit permutation and diffusion, namely MBPD, to cope with this issue. Specifically, a four-dimensional hyperchaotic system with three positive Lyapunov exponents is firstly proposed. Second, a hyperchaotic sequence is generated from the proposed hyperchaotic system for consequent encryption operations. Third, multiple bit permutation and diffusion (permutation and/or diffusion can be conducted with 1–8 or more bits) determined by the hyperchaotic sequence is designed. Finally, the proposed MBPD is applied to image encryption. We conduct extensive experiments on a couple of public test images to validate the proposed MBPD. The results verify that the MBPD can effectively resist different types of attacks and has better performance than the compared popular encryption methods.

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Chaos for communication

Nonlinear Dynamics ◽

10.1007/s11071-021-06644-4 ◽

2021 ◽

Author(s):

Murilo S. Baptista

Keyword(s):

Information Content ◽

Lyapunov Exponents ◽

Communication System ◽

Linear Superposition ◽

Chaotic Signals ◽

Physical Medium ◽

Periodic Signals ◽

Low Probability ◽

Chaotic Interference ◽

Source Signals

AbstractThis work shows that chaotic signals with different power spectrum and different positive Lyapunov exponents are robust to linear superposition, meaning that the superposition preserves the Lyapunov exponents and the information content of the source signals, even after being transmitted over non-ideal physical medium. This work tackles with great detail how chaotic signals and their information content are affected when travelling through medium that presents the non-ideal properties of multi-path propagation, noise and chaotic interference (linear superposition), and how this impacts on the proposed communication system. Physical media with other non-ideal properties (dispersion and interference with periodic signals) are also discussed. These wonderful properties that chaotic signals have allow me to propose a novel communication system based on chaos, where information composed from and to multiple users each operating with different base frequencies and that is carried by chaotic wavesignals, can be fully preserved after transmission in the open air wireless physical medium, and it can be trivially decoded with low probability of errors.

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