Mathematics

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Shiv Nadar University

Institute for Innovations & Inventions with Mathematics and IT

The objective of this paper is to investigate analytically the non-linear structure and extent of complexity and chaos in public and private sector Bank Stocks as well as the indices: NIFTY and Bank NIFTY in India. In this paper, analysis of the time series was performed using the Wolf algorithm along with optimization using FET and BASGEN algorithms for numerical evaluation of the largest positive Lyapunov exponent. © 2022 Author(s).

World University root node

AIP Conference Proceedings

Investigation of chaos in Indian bank stocks, NIFTY and bank NIFTY indices

Asymptotic stability analysis applied to stabilize unstable fixed points and to control chaotic motions in two and three-dimensional discrete dynamical systems. A new set of parameter values obtained which stabilizes an unstable fixed point and control the chaotic evolution to regularity. The output of the considered model and that of the adjustable system continuously compared by a typical feedback and the difference used by the adaptation mechanism to modify the parameters. Suitable numerical simulation which are used thoroughly discussed and parameter values are adjusted. The findings are significant and interesting. This strategy has some advantages over many other chaos control methods in discrete systems but, however it can be applied within some limitations. © 2021, International Journal of Mathematical, Engineering and Management Sciences. All Rights Reserved

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International Journal of Mathematical, Engineering and Management Sciences

Asymptotic Stability Analysis Applied in Two and Three-Dimensional Discrete Systems to Control Chaos

Evolutionary dynamics of a two-gene model for chemical reactions, corresponding to gene expression and regulation, has been studied in detail. Bifurcation analysis has been carried out to understand behavior of steady state solutions leading to chaotic evolution for different cases. Numerical simulations have been performed and measurable quantities like Lyapunov exponents, topological entropies and correlation dimensions have been calculated for certain sets of parameter values. These measures explain complexity and chaotic nature of evolution. © 2019 Forum-Editrice Universitaria Udinese SRL. All Rights Reserved.

Italian Journal of Pure and Applied Mathematics

Dynamics of two-gene andrecut-Kauffman system: Chaos and complexity

Evolutionary dynamics of Lozi map have been studied and stability criteria of its fixed points are discussed in some detail. Bifurcation diagrams of this map are drawn by varying both of its parameters while assigning fixed value to other parameter. Various attractors of Lozi map appeared in diverse and interesting pattern during evolution with different pair of values of parameters and for different initial conditions. To examine complexity of the map during evolution, certain chaotic measures such as calculations of Lyapunov exponents, (LCEs), and topological entropies have been done and represented graphically. Finally, asymptotic stability method has been applied to stabilize an unstable orbit and chaotic evolution of the Lozi map has been controlled. © 2016, Jangjeon Mathematical Society. All rights reserved.

Advanced Studies in Contemporary Mathematics (Kyungshang)

Complexities in Lozi map and control of chaos

Complex evolutionary behavior of an age-structured predator-prey three-dimensional system has been investigated analytically as well as numerically. Natural populations, whose generations are non-overlapping, can be described by model of difference equations that explains how the populations evolve in discrete time steps. In this paper stability criteria of fixed points of 3-dimentional discrete model are discussed. Bifurcation diagrams of this map are drawn by varying one parameter while fixing value of other parameters. To examine complexity of the map during evolution, certain chaotic measures such as calculations of Lyapunov characteristic exponents (LCEs), topological entropies and correlation dimension have been done and represented graphically. © 2015 L. M. Saha and Neha Kumra.

Applied Mathematical Sciences

Complexities in age structured predator-prey system

We discuss the role of convective zones in the treatment of coupled pulsation and convection. In particular, we have investigated behaviors of the convective depth,n, and of the ratio between the convective and total luminosities, γc. In deep convection, additional fixed points appear, the values of which increase with the depth of zone, and become larger than unity. A diagram of stability criteria derived from the linearized model has been drawn, which shows stable regions fitting each criterion. Then, various flow patterns have been obtained by means of numerical simulations for sets of γc and n, and the structure of the mode has been discussed. The results of simulations are shown in tables and figures with a fixed set of parameters other than the set of γc and n, which show that deep convective zones cause chaotic solutions and long-period oscillations. Some suggestions are made based on these findings in stellar pulsation within the convective zone. © 2012 Astronomical Society of Japan.

Publications of the Astronomical Society of Japan

Nonlinear behaviors of pulsating stars with convective zones

Regular and chaotic oscillations in a modified discrete two dimensional coupled predator-prey model, proposed recently, is re-investigated by observing the bifurcation diagram, calculating Lyapunov exponents and correlation dimensions for various orbits. The map evolves from one cycle to three cycles followed by period doubling and then to a chaotic regime and show bistability as its parameter λ varies, 0 < ≤ 1.211. Initial population size of the species and the value their coupling coefficients play crucial role for subsequent evolutionary phenomena of the system. Regular and chaotic evolutions depend completely on the coefficient ,λ, and on the initial prey and predator population. Lyapunov exponents and correlation dimensions provide true measure of chaos and also, to identify the chaotic orbits. Some recently discovered indicators, FLI, SALI and DLI have also been used to understand the nature of orbits.Mathematical analysis have been carried out to find stable and unstable fixed points and to obtain numerical value of the parameter λ for sensitive state of the system to initial conditions i.e. emergence of chaos. Bistability condition have been discussed for variation of λ values. Graphical representation of Lyapunov exponents and correlation dimensions provide better understanding of the nature of orbits which are further justified by the use of FLI, SALI and DLI.

(SoNS) School of Natural Sciences

Chaotic evaluations in a modified coupledLogistic type predator-prey model

The state of chaos exhibited in nonlinear system may appear through various roots. However, for mostly one dimensional system, it appears through period doubling roots. Phenomena of bifurcations, drawn by varying certain parameter of the system, explain clearly regular as well as chaotic evolution of the system. As a measure of chaos, the most suitable tools to be considered are: Lyapunov characteristic exponents (LCE) and topological entropies. Though the plots of both of the LCE's and topological entropies are similar, but both have certain limitations. LCE would not work for systems having relativistic considerations but the topological entropy can work. In fact, topological entropy can nicely provide the measure of complexity of the system in the sense that the more complexity in the system means more topological entropy it will have. A chaotic system exhibits a chaotic set, strange attractor, having fractal property. Calculation of correlation dimension is required to obtain the dimension of such attractor. The work presented here explains the appearance of chaos through bifurcation in some nonlinear one dimensional discrete system. The stability of the steady state, (i.e. fixed points), have been examined for each system and then plots of Lyapunov exponents and topological entropies for their evolution have been obtained. Then, the calculations have been extended to find the correlation dimensions chaotic sets, chaotic attractors, seen for different systems. Graphical results reveal some interesting information.