Dynamical process of complex systems and fractional differential equations

Hara, Hiroaki; Tamura, Yoshiyasu

doi:10.2478/s11534-013-0224-2

Cited by 4 publications

(5 citation statements)

References 32 publications

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“…In order to establish the fundamental base for magnetic relaxation, we applied the stretched exponential law to the data. In complex systems, such as aggregates of single‐domain, pseudo single‐domain, and multi‐domain grains, the relaxation behavior can be explained by stretched exponential law with a transformation from τ to τ * [ Hara and Tamura , ]. The relaxation rate of the power law describing the stretched exponential law has been shown to reflect the distribution of relaxation time in complex systems [ Plonka , , ].…”

Section: Discussionmentioning

confidence: 99%

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Stretched exponential relaxation of viscous remanence and magnetic dating of erratic boulders

et al. 2016

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Viscous remanence continuously increases with the duration of reorientation of rocks, and the remanence gets partially overprinted in rocks parallel to the Earth's magnetic field. This overprinted viscous remanence is unblocked at a certain temperature that enables the estimation of the time required for the rock to acquire the magnetism, by assuming the exponential law of Néel's single‐domain theory. However, previous results of dating the rocks by the exponential law have shown older ages than radiometric or cosmogenic exposure ages. Néel's exponential decay law is applicable to a system whose magnetic grains have an identical relaxation time. However, in real systems, the expected behavior is not usually observed because relaxation times vary for individual grains. Moreover, the variation of viscous remanence with the logarithmic law for a distribution of relaxation times is predicted to be concave downward. Here we found that the stretched exponential law, exp{−(t/τ)1 − n} with 0 ≤ n < 1, explains previously published data on viscous decay. The observed stretched exponential relaxation can be interpreted in terms of the global relaxation of a system containing many relaxing species, each of which decays exponentially in time with a specific fixed relaxation rate. Using this law, we derived an extended time‐temperature relationship of magnetite involving the Néel's exponential decay law with n = 0 and a system containing many relaxing times with variable n. The extended time‐temperature relationship shows that the age of a coral tsunami boulder with high anomalous unblocking temperatures can be fitted with an assigned radiometric age.

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Section: Discussionmentioning

confidence: 99%

“…For n = 0, equation (8) is reduced to a simple Néel's exponential relaxation [Ngai and Strom, 1988;Ulrich et al, 2003]. The dynamics of slow relaxation is expressed by a longer relaxation time instead of simple relaxation time [Ngai, 1998;Hara and Tamura, 2013].…”

Section: 1002/2016jb013281mentioning

confidence: 99%

Stretched exponential relaxation of viscous remanence and magnetic dating of erratic boulders

et al. 2016

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“…The use of this type of models is appropiate since diffusion processes have been associated with dielectric charging in MEMS [5]. Furthermore, there is a link in [6] between fractional systems and the typically observed multiexponential or stretched-time exponential type responses [2], [7] usually found in dielectric charging.…”

Section: Introductionmentioning

confidence: 99%

Characterization of Dielectric Charging in MEMS Using Diffusive Representation

Atienza

Gorreta

Pons-Nin

et al. 2017

IEEE Trans. Ind. Electron.

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Abstract-This paper introduces Diffusive Representation as a novel approach to characterize the dynamics of charge trapped in dielectric layers of microelectromechanical systems (MEMS). Diffusive Representation provides a computationally efficient method to achieve an arbitrary order state-space model of the charging dynamics. This approach is particularly well suited to analyze the dynamics of the dielectric charge under nontrivial controls, as in the case of sliding mode controllers. The diffusive symbol of the experimental structure has been obtained from open-loop measurements, in which Pseudo Random Binary Sequences (PRBS) are applied to the device. The obtained model exhibits good agreement with experimental data and also allows to model the behaviour of the charge dynamics under excitation with arbitrary binary signals.

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“…Note that the constitutive relation of the GM and GKV models become identical to a system of overdamped Langevin equations when random inputs are considered [20,22], and that imposing a scaling rule on the GM and GKV models is analogous to considering a system of scaled overdamped Langevin equations, which is known to exhibit power-law time-dependent relaxation behavior [56]. Thus, the interpretation of the differentiation order a as the internal scaling of the characteristic time scales may be valid for a range of complex systems described by scaled overdamped Langevin equations [57,58], whose information geometrical characterization has been recently discussed in [59].…”

Section: The Interpretation Of the Differentiation Ordermentioning

confidence: 99%

Physically meaningful parameters of viscoelastic models based on the general functional form of the transfer functions

Suda,

Nagahama

2023

Phys. Scr.

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Viscoelastic models aim to characterize material properties by extracting the parameters that best describe observed deformations. However, due to the equivalence of various model structures, the conventional model parameters as constants assigned to each model element cannot be uniquely determined for a given deformation and, therefore, are unfit to represent material properties. To resolve this problem, we formulate the framework of viscoelastic models based on their transfer functions. We explicitly derive the functional form of the transfer functions of viscoelastic models and show that a set of parameters is uniquely determined for each equivalent set of viscoelastic models. The newly identified parameters correspond to the instantaneous modulus and the distribution of the characteristic time scales of the system, and based on this perspective, we show that the differentiation order of a fractional viscoelastic element, spring-pot, is determined solely by the recursive nature of the characteristic time scales of the system. Furthermore, we show that all viscoelastic models are equivalent to four canonical model structures.

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Dynamical process of complex systems and fractional differential equations

Cited by 4 publications

References 32 publications

Stretched exponential relaxation of viscous remanence and magnetic dating of erratic boulders

Stretched exponential relaxation of viscous remanence and magnetic dating of erratic boulders

Characterization of Dielectric Charging in MEMS Using Diffusive Representation

Physically meaningful parameters of viscoelastic models based on the general functional form of the transfer functions

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