ARTICLE | doi:10.20944/preprints202305.0775.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: chaotic attractor; hyperchaotic attractor; Lozi map; sheet-attractor; thread-attractor
Online: 11 May 2023 (03:48:45 CEST)
Since its original publication 1 in 1978, Lozi’s chaotic map has been thoroughly explored and continues to be. Hundreds of publications analyze its particular structure or apply its properties in many fields (electronic devices like memristor, A.I. with swarm intelligence, etc.). Several generalizations have been proposed, transforming the initial two-dimensional map into multidimensional one. However, they do not respect the original constraint that allows this map to be one of the few strictly hyperbolic: a constant Jacobian. In this paper we introduce a three-dimensional piece-wise linear extension respecting this constraint and we explore a special property never highlighted for chaotic mappings: the coexistence of thread-chaotic attractors (i.e., attractors which are formed by collection of lines) and sheet-chaotic attractors (i.e., attractors which are formed by collection of planes). This new 3-dimensional mapping can generate a large variety of chaotic and hyperchaotic attractors. We give five examples of such behavior in this article. In the first three examples, there is coexistence of thread and sheet-chaotic attractors. However, their shape are different and they are constituted by a different number of pieces. In the two last examples, the blow up of the attractors with respect to parameter a and b is highlighted.
ARTICLE | doi:10.20944/preprints202204.0167.v1
Subject: Computer Science And Mathematics, Data Structures, Algorithms And Complexity Keywords: time series interpolation; phase space reconstruction; takens’ theorem; interpolation; stochastic interpolation; genetic algorithm; time series data; preprocessing; strange attractor; attractor; attractor reconstruction
Online: 18 April 2022 (11:02:00 CEST)
We present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens’ theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population of different stochastically interpolated time series data, and secondly, to use a genetic algorithm to find the pieces in the population which generate the smoothest reconstructed phase space trajectory. A smooth trajectory curve is hereby found to have a low variance of second derivatives along the curve. For simplicity, we refer to the developed method as PhaSpaSto-interpolation, which is an abbreviation for phase-space-trajectory-smoothing stochastic interpolation. The proposed approach is tested and validated with a univariate time series of the Lorenz system, five non-model data sets and tested against a cubic spline interpolation and a linear linear interpolation. We find that the criterion for smoothness guarantees low errors on known model and non-model data. Finally, we interpolate the discussed non-model data sets, and show the corresponding improved phase space portraits. The proposed method is useful for interpolating low-sampled time series data sets for, e.g., machine learning, regression analysis, or time series prediction approaches.
ARTICLE | doi:10.20944/preprints202002.0350.v1
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: Swift-Hohenberg equation; Random-pullback attractor; Non-autonomous random dynamical system
Online: 24 February 2020 (12:30:08 CET)
In this paper, we study the existence of the random -pullback attractor of a non-autonomous local modiﬁed stochastic Swift-Hohenberg equation with multiplicative noise in stratonovich sense. It is shown that a random -pullback attractor exists in when its external force has exponential growth. Due to the stochastic term, the estimate are delicate, we overcome this difficulty by using the Ornstein-Uhlenbeck(O-U) transformation and its properties.
ARTICLE | doi:10.20944/preprints201808.0251.v1
Subject: Social Sciences, Behavior Sciences Keywords: potts network; attractor neural networks; auto-associative memory; cortex; semantic memory
Online: 14 August 2018 (12:33:52 CEST)
A statistical analysis of semantic memory should reflect the complex, multifactorial structure of the relations among its items. Still, a dominant paradigm in the study of semantic memory has been the idea that the mental representation of concepts is structured along a simple branching tree spanned by superordinate and subordinate categories. We propose a generative model of item representation with correlations that overcomes the limitations of a tree structure. The items are generated through "factors" that represent semantic features or real-world attributes. The correlation between items has its source in the extent to which items share such factors and the strength of such factors: if many factors are balanced, correlations are overall low; whereas if a few factors dominate, they become strong. Our model allows for correlations that are neither trivial nor hierarchical, but may reproduce the general spectrum of correlations present in a data-set of nouns. We provide an estimate of the number of concepts that can be stored and retrieved by a large-scale cortical network, the Potts network, which is perhaps approximately 107 with human cortical parameters. When this storage capacity is exceeded, however, retrieval fails completely only for balanced factors; above a critical degree of imbalance, a phase transition leads to a regime where the network still extracts considerable information about the cued item, even if not recovering its detailed representation: partial categorization seems to emerge spontaneously as a consequence of the dominance of particular factors, rather than being imposed ad hoc. We argue this to be a relevant model of semantic memory resilience in Tulving’s remember/know paradigms.
ARTICLE | doi:10.20944/preprints202112.0303.v1
Subject: Computer Science And Mathematics, Computational Mathematics Keywords: Discrete fractional-order system; Caputo delta fractional difference; Hidden attractor; Dihedral symmetry D3
Online: 20 December 2021 (10:20:56 CET)
In this paper the D 3 dihedral logistic map of fractional order is introduced. The map 1 presents a dihedral symmetry D 3 . It is numerically shown that the construction and interpretation 2 of the bifurcation diagram versus the fractional order require special attention. The system stability 3 is determined and the problem of hidden attractors is analyzed. Also, analytical and numerical 4 results show that the chaotic attractor of integer order, with D 3 symmetries, looses its symmetry 5 in the fractional-order variant.
ARTICLE | doi:10.20944/preprints201807.0357.v1
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: stochastic processes; Langevin equation; Fokker-Planck equation; information length; Fisher information; relaxation; chaos; attractor
Online: 19 July 2018 (11:30:59 CEST)
A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic processes and geometry. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time and is called the information length. By using this concept, we investigate classical and quantum systems and demonstrate the utility of the information length as a unique Lagrangian diagnostic to quantify the information change as a system continuously evolves in time and to map out attractor structure. We further elucidate quantum effects (uncertainty relation) and the dual role of the width of PDF in quantum systems.
ARTICLE | doi:10.20944/preprints202111.0528.v2
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: Conformable calculus; Fractional-order financial system; ESDDFD and NSFD methods; Hyperchaotic attractor; Market confidence; Ethics risk
Online: 6 December 2021 (12:48:05 CET)
Four discrete models using the exact spectral derivative discretization finite difference (ESDDFD) method are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index , the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is in part an artifact of the CEFD method as it is absent from the ESDDFD models.
ARTICLE | doi:10.20944/preprints202004.0265.v1
Subject: Medicine And Pharmacology, Epidemiology And Infectious Diseases Keywords: epidemic; COVID-19; contamination kinetics; immunological response; dynamical systems; reproduction rate; critical state; attractor; stable cycle; chaos
Online: 16 April 2020 (08:20:50 CEST)
In the context of the COVID-19 epidemic, and on the basis of the Theory of Dynamical Systems, we propose a simple model for the expansion of contagious diseases, with a particular focus on viral respiratory tracts. The infection develops through contacts between contagious and exposed people, with a rate proportional to contact duration and turnover, inversely proportional to the efficiency of protection measures, and balanced by the average immunological response. The obvious initial exponential increase is readily hindered by the size reduction of the exposed population. The system converges towards a stable attractor whose value is expressed in terms of the ratio C/D of contamination vs decay factors. Decreasing this ratio below a critical value leads to a tipping point beyond which the epidemic is over. By contrast, significant values of C/D may bring the system through a bifurcating hierarchy of stable cycles up to a chaotic behaviour.
ARTICLE | doi:10.20944/preprints202303.0503.v1
Subject: Physical Sciences, Other Keywords: dynamical systems; self-organization; temporal periodicity; attractor topology; phase transitions; synchronization patterns; physical vacuum; elementary particles; unification of forces
Online: 29 March 2023 (07:08:35 CEST)
The phenomenon of dissipative self-organization is studied on the example of time-crystal networks. Particular attention was given to transient processes, attractor topologies, phase transitions, and asymptotic stability. New concepts were introduced, including topological phases, spinorial states, and bond flavors. Concepts such as ground states, chemical potentials, elastic forces, temperature, and statistical distributions were endowed with a new meaning associated with asymptotic stability. Phenomena usually attributed exclusively to quantum physics have been shown to occur in this essentially classical environment. They coexist with, and in some cases, such as charge quantization, are related to the phenomenon of time dilation. The approach was applied to model vacuum self-organization. We have shown that under the ac-tion of competing forces, such as gravity and antigravity, a cascade of phase transitions can transform an unorganized vacuum into phases in which interactions, fields and waves resemble electromagnetic, weak, and strong, and their elements can be used as building blocks for prototype particles. In addition, some interrelation field parameters and probabilities of particle transmutations were calculated, which are not predicted by the standard model. The results are consistent with the experiment. The presented material and methodology may be of interest for studying self-organization in different environments.