ON EQUILIBRIUM POINTS AND BIFURCATIONS IN FIXED EQUI-TRIANGULAR ARRANGEMENT OF THREE RANKINE VORTEX FILAMENTS

This paper revisits Chua's electrical circuit in the context of the Caputo derivative. We introduce the Caputo derivative into the modeling of the electrical circuit. The solutions of the new model are proposed using numerical discretizations. The discretizations use the numerical scheme of the Riemann-Liouville integral. We have determined the equilibrium points and study their local stability. The existence of the chaotic behaviors with the used fractional-order has been characterized by the determination of the maximal Lyapunov exponent value. The variations of the parameters of the model into the Chua's electrical circuit have been quantified using the bifurcation concept. We also propose adaptive controls under which the master and the slave fractional Chua's electrical circuits go in the same way. The graphical representations have supported all the main results of the paper.

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Dynamics of a delayed competitive system affected by toxic substances with imprecise biological parameters

Filomat ◽

10.2298/fil1716271p ◽

2017 ◽

Vol 31 (16) ◽

pp. 5271-5293

Author(s):

A.K. Pal ◽

P. Dolai ◽

G.P. Samanta

Keyword(s):

Equilibrium Points ◽

Stable Limit Cycle ◽

Limit Cycle Oscillation ◽

Toxic Substances ◽

Stable Limit ◽

Biological Parameters ◽

Delay Model ◽

Competitive System ◽

Interval Numbers ◽

Cycle Oscillation

In this paper we have studied the dynamical behaviours of a delayed two-species competitive system affected by toxicant with imprecise biological parameters. We have proposed a method to handle these imprecise parameters by using parametric form of interval numbers. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate our analytical findings.

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Cascades and reconnection in interacting vortex filaments

Physical Review Fluids ◽

10.1103/physrevfluids.6.074701 ◽

2021 ◽

Vol 6 (7) ◽

Author(s):

Rodolfo Ostilla-Mónico ◽

Ryan McKeown ◽

Michael P. Brenner ◽

Shmuel M. Rubinstein ◽

Alain Pumir

Keyword(s):

Vortex Filaments

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Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

Entropy ◽

10.3390/e23070876 ◽

2021 ◽

Vol 23 (7) ◽

pp. 876

Author(s):

Wieslaw Marszalek ◽

Jan Sadecki ◽

Maciej Walczak

Keyword(s):

Equilibrium Points ◽

Autonomous Systems ◽

Cytosolic Calcium ◽

Calcium Oscillations ◽

Dynamical Model ◽

Bifurcation Diagrams ◽

Step Size ◽

Oscillatory Systems ◽

Two Parameter ◽

Parameter Bifurcation

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation.

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Complex dynamical behaviors of a novel exponential hyper–chaotic system and its application in fast synchronization and color image encryption

Science Progress ◽

10.1177/00368504211003388 ◽

2021 ◽

Vol 104 (1) ◽

pp. 003685042110033

Author(s):

Javad Mostafaee ◽

Saleh Mobayen ◽

Behrouz Vaseghi ◽

Mohammad Vahedi ◽

Afef Fekih

Keyword(s):

Image Encryption ◽

Chaotic System ◽

Sliding Mode ◽

Color Image ◽

Equilibrium Points ◽

Lyapunov Stability Theory ◽

System Stability ◽

Color Image Encryption ◽

Terminal Sliding Mode ◽

Finite Time Stability

This paper proposes a novel exponential hyper–chaotic system with complex dynamic behaviors. It also analyzes the chaotic attractor, bifurcation diagram, equilibrium points, Poincare map, Kaplan–Yorke dimension, and Lyapunov exponent behaviors. A fast terminal sliding mode control scheme is then designed to ensure the fast synchronization and stability of the new exponential hyper–chaotic system. Stability analysis was performed using the Lyapunov stability theory. One of the main features of the proposed controller is the finite time stability of the terminal sliding surface designed with high–order power function of error and derivative of error. The approach was implemented for image cryptosystem. Color image encryption was carried out to confirm the performance of the new hyper–chaotic system. For image encryption, the DNA encryption-based RGB algorithm was used. Performance assessment of the proposed approach confirmed the ability of the proposed hyper–chaotic system to increase the security of image encryption.

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Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽

10.3390/sym13050785 ◽

2021 ◽

Vol 13 (5) ◽

pp. 785

Author(s):

Hasan S. Panigoro ◽

Agus Suryanto ◽

Wuryansari Muharini Kusumawinahyu ◽

Isnani Darti

Keyword(s):

Hopf Bifurcation ◽

Fractional Derivative ◽

Equilibrium Point ◽

Power Law ◽

Equilibrium Points ◽

Essential Difference ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

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Pareto-Optimal Equilibrium Points in non-Cooperative Multi-Objective Optimization Problems

Expert Systems with Applications ◽

10.1016/j.eswa.2021.114995 ◽

2021 ◽

pp. 114995

Author(s):

Mohammadali Saniee Monfared ◽

Sayyed Ehsan Monabbati ◽

Atefeh Rajabi Kafshgar

Keyword(s):

Optimization Problems ◽

Equilibrium Points ◽

Pareto Optimal ◽

Multi Objective Optimization ◽

Multi Objective

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Some Special Types of Orbits around Jupiter

Aerospace ◽

10.3390/aerospace8070183 ◽

2021 ◽

Vol 8 (7) ◽

pp. 183

Author(s):

Yongjie Liu ◽

Yu Jiang ◽

Hengnian Li ◽

Hui Zhang

Keyword(s):

Gravity Model ◽

Small Perturbation ◽

Equilibrium Points ◽

Major Axis ◽

Semi Major Axis ◽

Ground Track ◽

The Earth ◽

Degenerate Equilibrium ◽

The Mean ◽

Element Theory

This paper intends to show some special types of orbits around Jupiter based on the mean element theory, including stationary orbits, sun-synchronous orbits, orbits at the critical inclination, and repeating ground track orbits. A gravity model concerning only the perturbations of J2 and J4 terms is used here. Compared with special orbits around the Earth, the orbit dynamics differ greatly: (1) There do not exist longitude drifts on stationary orbits due to non-spherical gravity since only J2 and J4 terms are taken into account in the gravity model. All points on stationary orbits are degenerate equilibrium points. Moreover, the satellite will oscillate in the radial and North-South directions after a sufficiently small perturbation of stationary orbits. (2) The inclinations of sun-synchronous orbits are always bigger than 90 degrees, but smaller than those for satellites around the Earth. (3) The critical inclinations are no-longer independent of the semi-major axis and eccentricity of the orbits. The results show that if the eccentricity is small, the critical inclinations will decrease as the altitudes of orbits increase; if the eccentricity is larger, the critical inclinations will increase as the altitudes of orbits increase. (4) The inclinations of repeating ground track orbits are monotonically increasing rapidly with respect to the altitudes of orbits.

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