Chaotic dynamics in Parkfield region, California and characteristics earthquakes

Based on about 75000 earthquakes in the California region detected through Parkfield network during the years 1969-1987, the occurrence of chaos was examined by two different approaches, namely, strange at tractor dimension and the Lyapunov exponent. The strange at tractor dimension was found as 6.3 in this region suggesting atleast 7 parameters for earthquake predictability. Small positive Lyapunov exponent of 0.045 provided further evidence for deterministic chaos in the region which showed strong dependence on the initial conditions. Implications of chaotic dynamics on characteristic Parkfield earthquakes has been discussed. The strange at tractor dimension in the region could be representative for the Transform type of plate boundary which is lower than that reported for continent collision type of plate boundary which is lower than that reported for continent collision type of plate boundary near Hindukush northwest Himalayan region.

Determination of fractal dimension of chaotic at tractor for Maximum temperature over Madras

The dimensions of attractors of daily maximum temperature (during March-May) recorded by the two observatories of Madras, viz., Nungarnbakkarn and Meenambakkarn are estimated from phase space trajectories by the method of deterministic chaos, The dimensions provide the basic information on the minimum number of parameters required to understand the complex dynamical system and also the upper bound (degrees of freedom) of such parameters that are sufficient to model the system, The fractal dimension for the weather event, viz. maximum temperature over Madras is between 3.5 and 3.9 suggesting 4 parameters are necessary to model the system and a maximum of 19 parameters are sufficient.

Recent Trends of Controlling Chaotic Resonance and Future Perspectives

Stochastic resonance is a phenomenon in which the effects of additive noise strengthen the signal response against weak input signals in non-linear systems with a specific barrier or threshold. Recently, several studies on stochastic resonance have been conducted considering various engineering applications. In addition to additive stochastic noise, deterministic chaos causes a phenomenon similar to the stochastic resonance, which is known as chaotic resonance. The signal response of the chaotic resonance is maximized around the attractor-merging bifurcation for the emergence of chaos-chaos intermittency. Previous studies have shown that the sensitivity of chaotic resonance is higher than that of stochastic resonance. However, the engineering applications of chaotic resonance are limited. There are two possible reasons for this. First, the stochastic noise required to induce stochastic resonance can be easily controlled from outside of the stochastic resonance system. Conversely, in chaotic resonance, the attractor-merging bifurcation must be induced via the adjustment of internal system parameters. In many cases, achieving this adjustment from outside the system is difficult, particularly in biological systems. Second, chaotic resonance degrades owing to the influence of noise, which is generally inevitable in real-world systems. Herein, we introduce the findings of previous studies concerning chaotic resonance over the past decade and summarize the recent findings and conceivable approaches for the reduced region of orbit feedback method to address the aforementioned difficulties.

Strange attractor in earthquake swarms near Valsad (Gujarat), India

Valsad district in south Gujarat near the western coast of the peninsular India experienced earthquake swarms since early February 1986. Seismic monitoring through a network of micro earthquake seismographs showed a well concentrated seismic activity over an area of 7 × 10 km2 with the depth of foci extending from 1 to 15 km. A total number of 21,830 earthquakes were recorded during March 1986 to June 1988. The daily frequency of earthquakes for this period was utilized to examine deterministic chaos through evaluation of dimension of strange attractor and Lyapunov exponent. The low dimension of 2.1 for the strange attractor and positive value of the largest Lyapunov exponent suggest chaotic dynamics in Valsad earthquake swarms with at least 3 parameters for earthquake predictability. The results indicate differences in the characteristics of deterministic chaos in intraplate and interplate regions of India.

Deterministic Chaos Detection and Simplicial Local Predictions Applied to Strawberry Production Time Series

In this work, we attempted to find a non-linear dependency in the time series of strawberry production in Huelva (Spain) using a procedure based on metric tests measuring chaos. This study aims to develop a novel method for yield prediction. To do this, we study the system’s sensitivity to initial conditions (exponential growth of the errors) using the maximal Lyapunov exponent. To check the soundness of its computation on non-stationary and not excessively long time series, we employed the method of over-embedding, apart from repeating the computation with parts of the transformed time series. We determine the existence of deterministic chaos, and we conclude that non-linear techniques from chaos theory are better suited to describe the data than linear techniques such as the ARIMA (autoregressive integrated moving average) or SARIMA (seasonal autoregressive moving average) models. We proceed to predict short-term strawberry production using Lorenz’s Analog Method.

Acoustic structure and information content of trumpets in female Asian elephants (Elephas maximus)

Most studies on elephant vocal communication have focused on the low-frequency rumble, with less effort on other vocalization types such as the most characteristic elephant call, the trumpet. Yet, a better and more complete understanding of the elephant vocal system requires investigating other vocalization types and their functioning in more detail as well. We recorded adult female Asian elephants (Elephas maximus) at a private facility in Nepal and analyzed 206 trumpets from six individuals regarding their frequency, temporal and contour shape, and related acoustic parameters of the fundamental frequency. We also tested for information content regarding individuality and context. Finally, we recorded the occurrence of non-linear phenomena such as bifurcation, biphonation, subharmonics and deterministic chaos. We documented a mean fundamental frequency ± SD of 474 ± 70 Hz and a mean duration ± SD of 1.38 ± 1.46 s (Nindiv. = 6, Ncalls = 206). Our study reveals that the contour of the fundamental frequency of trumpets encodes information about individuality, but we found no evidence for trumpet subtypes in greeting versus disturbance contexts. Non-linear phenomena prevailed and varied in abundance among individuals, suggesting that irregularities in trumpets might enhance the potential for individual recognition. We propose that trumpets in adult female Asian elephants serve to convey an individual’s identity as well as to signal arousal and excitement to conspecifics.

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Aims: The aim of this study was to develop the mathematical models of the linear elasticity theory of biomaterials by taking into account their fractal structure. This study further aimed to construct a variational formulation of the problem, obtain the main relationships of the finite element method to calculate the rheological characteristics of a biomaterial with a fractal structure, and develop application software for calculating the components of the stress-strain state of biomaterials while considering their fractal structure. The obtained results were analyzed. Background: The development of adequate mathematical models of the linear elasticity theory for biomaterials with a fractal structure is an urgent scientific task. Finding its solution will make it possible to analyze the rheological behavior of biomaterials exposed to external loads by taking into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Objective: The objective of this study was the deformation process of biomaterials with a fractal structure under external load. Methods: The equations of the linear elasticity theory for the construction of the mathematical models of the deformation process of biomaterials under external load were used. Mathematical apparatus of integro-differentiation of fractional order to take into account the fractal structure of the biomaterial was used. A variational formulation of the linear elasticity problem while taking into account the fractal structure of the biomaterial was formulated. The finite element method with a piecewise linear basis for finding an approximate solution to the problem was used. Results: The main relations of the linear elasticity problem, which takes into account the fractal structure of the biomaterial, were obtained. A variational formulation of the problem was constructed. The main relations of the finite-element calculation of the linear elasticity problem of a biomaterial with a fractal structure using a piecewise-linear basis are found. The main components of the stress-strain state of the biomaterial exposed to external loads are found. Conclusion: Using the mathematical apparatus of integro-differentiation of fractional order in the construction of the mathematical models of the deformation process of biomaterials with a fractal structure makes it possible to take into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Also, this approach makes it possible to determine the residual stresses in the biomaterial, which play an important role in the appearance of stresses during repeated loads.

Solving the problem of mathematical models overparameterization for some nonlinear oscillating systems

This study proposes a numerical-analytical method that allows us to simplify the model, which is obtained on the basis of the single observable variable of an object under the study, and which may be overparameterized. As a model, we consider a system of ordinary differential equations with polynomial right-hand sides. To solve this problem, the so-called differential model is used, that is, a system in which unknown variables are replaced by derivatives of the observed variable, and which is derived on the basis of a system under the study so that the observed variables of these systems coincide. The method of simplification of a system under the study is based on the fact that using a numerical method, a simpler differential model can be obtained. Next, an analytical transition from a simplified differential model to a simplified original system is performed. In this case, the time series error remains within given limits even for systems with deterministic chaos, despite their high sensitivity to the initial conditions.

Development of a hash algorithm based on cellular automata and chaos theory

Information security, reliability of data transfer are today an important component of the globalization of information technology. Therefore, the proposed work is devoted to highlighting the results of the design and development of a hacking-resistant algorithm to ensure the integrity of information transfer via digital technology and computer engineering. To solve such problems, cryptographic hashing functions are used. In particular, elements of deterministic Chaos were introduced into the developed cyclic hashing algorithm. The investigation analyzes in detail the strengths and weaknesses of known hashing algorithms. They are shown to have disadvantages. The main ones are a large number of matches (Hamming (x, y) and the presence of a weak avalanche effect, which lead to a significant decrease in the reliability of the algorithm for hacking. The designed hashing algorithm uses an iterative Merkley-Damgard structure, augmented by the input message to a length multiple of 512 bits. Processing in blocks of 128-bit uses cellular automata with mixed rules of 30, 105 and 90, 150 and takes into account the dependence of the generation of the initial vector on the incoming message. This allows half of the 10,000 pairs of arbitrary messages to have an inverse Hamming distance of 0 to 2. The proposed algorithm is four times slower than the well-known family of "secure hash algorithms." However, computation speed is not a critical requirement for a hash function. Decreasing the sensitivity to the avalanche effect allows the generation time to be approximately halved. Optimization of the algorithm, as well as its testing was carried out using new technologies of the Java programming language (version 15). Suggestions and recommendations for improving this approach to data hashing are given also

Justifying Born’s Rule Pα = |Ψα|2 Using Deterministic Chaos, Decoherence, and the de Broglie-Bohm Quantum Theory

In this work, we derive Born’s rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to an environment made of “qubits” (i.e., Bohmian pointers), we show that entanglement together with deterministic chaos leads to a fast relaxation from any statistical distribution ρ(x) of finding a particle at point x to the Born probability law |Ψ(x)|2. Our model is discussed in the context of Boltzmann’s kinetic theory, and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.