Self-Similar Random Process and Chaotic Behavior In Serrated Flow of High Entropy Alloys

Abstract

The statistical and dynamic analyses of the serrated-flow behavior in the nanoindentation of a high-entropy alloy, Al0.5CoCrCuFeNi, at various holding times and temperatures, are performed to reveal the hidden order associated with the seemingly-irregular intermittent flow. Two distinct types of dynamics are identified in the high-entropy alloy, which are based on the chaotic time-series, approximate entropy, fractal dimension, and Hurst exponent. The dynamic plastic behavior at both room temperature and 200 °C exhibits a positive Lyapunov exponent, suggesting that the underlying dynamics is chaotic. The fractal dimension of the indentation depth increases with the increase of temperature, and there is an inflection at the holding time of 10 s at the same temperature. A large fractal dimension suggests the concurrent nucleation of a large number of slip bands. In particular, for the indentation with the holding time of 10 s at room temperature, the slip process evolves as a self-similar random process with a weak negative correlation similar to a random walk.

Characteristic Time of Diffusive Mixing in Cube with Reflecting Edges

“…However, a representation described in Figure 1 allows getting over this difficulty. Diffusion models with fractional Brownian motion are considered as in physical [7] so in informatics applications [12]. Some last results for multivariate fractional Levi motion (but without reflecting from segment ends) are obtained in [13].…”

“…Next consideration closely connecting with a concept of a self-similar random process (see for example [6]) has a large application in modern physics [7][8][9][10].…”

Determining the chaotic behaviour of copper prices in the long-term using annual price data

“…SampEn is even less sensitivity to the length of data sets and provides more consistent results which being also reliable to be used in small data sets (Chen et al, 2006;Ferrario et al, 2006;Lake et al, 2002;Yentes et al, 2013). It is important to note that ApEn and SampEn have also been used as a metric to evidence the chaotic behaviour of time series that presents positive λ (Cencini and Ginelli, 2013;Chen et al, 2016;Gaspard et al, 1998;Kaplan et al, 1991;Pincus, 1995;Sanei, 2013;Schreiber and Kantz, 1995) The chaotic behaviour of the price sequence of several mineral commodities has been assessed previously, however these studies (Abdullah and Zeng, 2010;He et al, 2015;Panas, 2001;Panas and Ninni, 2000) have been limited to sample periods of 15 years or less, and used daily or monthly prices. It is important to verify the time-related behaviour of mineral commodity prices in the longterm by using annual data, due to the impact of economic decisions, technological and regulatory changes which can be observed over years (Bernanke, 2013;Slade, 2015;Yellen, 2013).…”

“…However, a method has been introduced by Rosenstein et al (1993) to overcome this limitation, allowing sufficiently accurate and precise calculations of λ in small datasets (Becks et al, 2005;Graham et al, 2007;Kantz, 1994;Navarro-Urrios et al, 2017). Previous studies have proven the suitability and reliability of small data sets with 50 observations in regards to their calculation of λ for assessing chaotic behaviour of complex dynamic systems (Becks et al, 2005;Blank, 1991;Chen et al, 2016;Gaspard et al, 1998;Navarro-Urrios et al, 2017;Raffalt et al, 2017;Reynolds et al, 2016;Sivakumar, 2000). Moreover, the same methodology has been successfully applied to study the complexity of various dynamic systems, including electrocardiograms, human gait recording, and laser droplet generation (Krese et al, 2010;Perc, 2005aPerc, , 2005b.…”

“…They occur in parallel sets with a gap between any two members of the set. The most useful tool for examining the sensitivity to initial conditions and detecting the presence of chaotic behaviour has been the widely documented largest Lyapunov exponent (λ) method (Becks et al, 2005;Chen et al, 2016;Gaspard et al, 1998;Gottwald, 2009;Kodba et al, 2005;Navarro-Urrios et al, 2017;Panas, 2001;Panas and Ninni, 2000;Perc, 2006;Reynolds et al, 2016;Rosenstein et al, 1993;Savi, 2005;Showalter and Hamilton, 2015;Wernecke et al, 2017;Zhong et al, 2017). According to a large body of literature,, large datasets are required to assess chaotic behaviour, but this is often not achievable due to cost, technical or temporal limitations (Abraham et al, 1986;Kantz, 1994;Rosenstein et al, 1993).…”