Ergodic sensitivity analysis of one-dimensional chaotic maps

Śliwiak, Adam A.; Chandramoorthy, Nisha; Wang, Qiqi

doi:10.1016/j.taml.2020.01.058

Cited by 17 publications

(26 citation statements)

References 21 publications

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“…The authors of [24] showed such splitting exists and is differentiable for a general uniformly hyperbolic systems and rigorously proved the convergence of all the components of S3. In addition, various numerical examples clearly support the computational efficiency of this method applied to low-dimensional systems [25].…”

Section: Introductionmentioning

confidence: 63%

“…The computation of these two quantities is the actual price for the regularization of the original Lebesgue integrals. The latter is known in the literature as the SRB density gradient [25,29,24]. It reflects a relative measure change along an unstable manifold and its value is thus independent from its corresponding quotient measure.…”

Section: Computation Of the Unstable Contributionmentioning

confidence: 99%

“…Indeed, Lebesgue integration by parts requires knowledge of the measure derivative itself [26]. In case of one-dimensional chaos, this iterative relation directly follows from the measure preservation property involving the Frobenius-Perron operator [25]. If the unstable manifold is geometrically more complex, it is convenient to apply the measure-based parameterization of the unstable subspace, directly relating the directional derivative of measure with the coordinate chart, and subsequently apply the chain rule on smooth manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension

Sliwiak,

Wang

2021

Preprint

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Accurate approximations of the change of system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the sensitivity of chaotic dynamical systems, which enables a better understanding of chaotic phenomena. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting regularizing Ruelle's original expression together with measure-based parameterization of the expanding subspace. We use these tools to rigorously derive trajectory-following recursive relations that exponentially converge, and construct a memory-efficient Monte Carlo scheme for derivatives of the output statistics. Thanks to the regularization and lack of simplifying assumptions on the behavior of the system, our method is immune to the common problems of other popular systems such as the exploding tangent solutions and unphysicality of shadowing directions. We provide a ready-to-use algorithm, analyze its complexity, and demonstrate several numerical examples of sensitivity computation of physically-inspired low-dimensional systems.

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Section: Introductionmentioning

confidence: 63%

Section: Computation Of the Unstable Contributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension

Sliwiak,

Wang

2021

Preprint

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“…For certain chaotic 1D processes, its chaotic cycles are very basic as well as completely predictable. After some data is collected, certain strategies can be used to approximate individual starting states [20]- [23].…”

Section: Introductionmentioning

confidence: 99%

Image Encryption Enabling Chaotic Ergodicity with Logistic and Sine Map

Khan¹,

Khan²,

Alghamdi³

et al. 2021

IJACSA

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Chaotic systems with complicated characteristics of ergodicity, impredictability as well as sensitivity to beginning stages are commonly utilized in the world of cryptography. A 2D logistic-adjusted-sine (LS) map is implemented in this article. Performance assessments reveal superior ergodicity as well as unpredictable and even a broader spectrum of chaotics than numerous previous chaotic maps. This research also develops a 2D-LS-based image encryption system and proposed LS-IES. The notion of diffusion as well as confusion is properly complied with enabled encryption functions. Research outcomes as well as security analyses demonstrate that LS-IES can swiftly encrypting different parameters in various images with a great resistance towards security threats.

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“…If the integrand involves highly-oscillatory derivatives, then the Monte Carlo integration might be prohibitively expensive due to a large variance of the sample [31]. In case of derivatives of functions evaluated at a future time (see examples of such integrands in [10,29,15,2]), the direct use of any integration scheme might be impossible due to the butterfly effect. Indeed, the application of the chain rule results in a product of the system's Jacobian matrices whose norms increase exponentially in time.…”

mentioning

confidence: 99%

A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds

Sliwiak,

Wang

2021

Preprint

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SRB measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectorydriven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We apply the concept of measure preservation together with the chain rule on smooth manifolds. Due to the recursive one-step nature of our derivations, the proposed procedure is memory-efficient and can be naturally integrated with existing Monte Carlo schemes widely used in computational chaotic dynamics. We numerically show the exponential convergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.

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Ergodic sensitivity analysis of one-dimensional chaotic maps

Cited by 17 publications

References 21 publications

Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension

Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension

Image Encryption Enabling Chaotic Ergodicity with Logistic and Sine Map

A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds

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