Non-hyperbolicity at large scales of a high-dimensional chaotic system

Wormell, Caroline L.

doi:10.1098/rspa.2021.0808

“…The chaotic hypothesis of Gallavotti and Cohen states that many high-dimensional physical systems are hyperbolic [10,26,63]. It seems that this hypothesis does not hold strictly, since Wormell and Gottwald recently gave a counter example [62,63]. Also, weather systems seem to have a significant amount of nonhyperbolic directions [14], but still seem to have linear responses.…”

Section: Literature Reviewmentioning

confidence: 99%

Backpropagation in hyperbolic chaos via adjoint shadowing

Ni

2024

Nonlinearity

0

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To generalise the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator S acting on covector fields. We show that S can be equivalently defined as: S is the adjoint of the linear shadowing operator S; S is given by a ‘split then propagate’ expansion formula; S ( ω ) is the only bounded inhomogeneous adjoint solution of ω. By (a), S adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), S also expresses the other part of the linear response, the unstable contribution. By (c), S can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395 690–709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions.

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“…Hyperbolic systems have structurally stable dynamics, and linear response, meaning that their statistics vary smoothly with parameter variations [45]. In practice, violations of hyperbolicity are commonly reported in the literature [46,47,43], whereas true hyperbolic systems are rare [48]. Thanks to the chaotic hypothesis [4,49,50], high-dimensional chaotic systems can be practically treated as hyperbolic systems, i.e.…”

Section: Stability Of Chaotic Systemsmentioning

confidence: 99%

Stability analysis of chaotic systems from data

Margazoglou

¹

,

Magri

²

2022

Preprint

0

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The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point, and compute the properties of the tangent space (i.e., the Jacobian). The main goal of this paper is to propose a method that learns the Jacobian, thus, the stability properties, from observables (data). First, we propose the Echo State Network (ESN) with the Recycle Validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the Echo State Network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN, and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution, and its tangent space with negligible numerical errors. In detail, (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (ii) the Lyapunov spectrum; (iii) the finite-time Lyapunov exponents; (iv) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations. Mathematics Subject Classification (2020) 68T07 · 34D08 · 37D45 · 37M22

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“…In summary, a tri-valued memristor not only enriches the types of nonlinear systems compared to a two-valued and continuous memristor, but also broadens the design of chaotic systems, increases the stability and robustness of the system and produces rich and diverse dynamic behaviors. At this stage, the research on tri-valued memristor chaos is relatively small and limited to the basis of four-dimensional chaotic systems, and the constructed four-dimensional chaotic systems have the problems of simple structure and low sequence randomness, whereas the highdimensional chaotic systems have more complex dynamical behaviors and a richer class of stochastic phenomena in comparison with the four-dimensional chaotic systems [8][9][10][11]. Therefore, it is challenging research to introduce tri-valued memristors into high-dimensional chaotic systems, and in this paper, we address the above problems by constructing high-dimensional chaotic systems using tri-valued memristors to produce richer dynamical behaviors and more stochastic sequences.…”

Section: Introductionmentioning

confidence: 99%

Research on Variable Parameter Color Image Encryption Based on Five-Dimensional Tri-Valued Memristor Chaotic System

Wang,

Ding

2024

Entropy

0

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To construct a chaotic system with complex characteristics and to improve the security of image data, a five-dimensional tri-valued memristor chaotic system with high complexity is innovatively constructed. Firstly, a pressure-controlled tri-valued memristor on Liu’s pseudo-four-wing chaotic system is introduced. Through analytical methods, such as Lyapunov exponential map, bifurcation map and attractor phase diagram, it is demonstrated that the new system has rich dynamical behaviors with periodic limit rings varying with the coupling parameter of the system, variable airfoil phenomenon as well as transient chaotic phenomenon of chaos-periodic depending on the system parameter and chaos-quasi-periodic depending on the memristor parameter. The system is simulated with dynamic circuits based on Simulink. Secondly, the differently structured synchronous controls of chaotic systems are realized using a nonlinear feedback control method. Finally, based on the newly constructed five-dimensional chaotic system, a variable parameter color image encryption scheme is proposed to iteratively generate varying chaotic pseudo-random sequences by varying the system parameters, which will be used for repetition-free disambiguation, additive modulo left-shift diffusion and DNA encryption for the three components of RGB of the color image after chunking. The simulation results are analyzed by histogram, information entropy, adjacent pixel correlation, etc., and the images are tested using differential attack, noise attack and geometric attack, as well as analyzing the PSNR and SSIM of the decrypted image quality. The results show that the encryption method has a certain degree of security and can be applied to medical, military and financial fields with more complex environmental requirements.

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Non-hyperbolicity at large scales of a high-dimensional chaotic system

Cited by 5 publications

References 60 publications

Backpropagation in hyperbolic chaos via adjoint shadowing

Backpropagation in hyperbolic chaos via adjoint shadowing

Stability analysis of chaotic systems from data

Research on Variable Parameter Color Image Encryption Based on Five-Dimensional Tri-Valued Memristor Chaotic System

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