On the Ultimate Dynamics of the Four-Dimensional Rössler System

2014 ◽  
Vol 24 (11) ◽  
pp. 1450149 ◽  
Author(s):  
Konstantin E. Starkov
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Invariant Sets ◽  
Limit Sets ◽  
Localization Method ◽  
Dimensional Plane ◽  
Rossler System ◽  
Lasalle Theorem ◽  
Rössler System ◽  
Positive Half

In this paper, we construct the polytope which contains all compact ω-limit sets of the four-dimensional Rössler system which is a generalization of the hyperchaotic Rössler system for the case of positive parameters. Further, we find a few three-dimensional planes containing all compact ω-limit sets for bounded positive half-trajectories located in some subdomains in the half-space z > 0. Besides, we analyze one case in which all compact ω-limit sets in the half-space z > 0 are contained in one three-dimensional plane. Our approach is based on a combination of the LaSalle theorem and the extreme-based localization method of compact invariant sets.

A Cancer Model for the Angiogenic Switch and Immunotherapy: Tumor Eradication in Analysis of Ultimate Dynamics

2020 ◽  
Vol 30 (10) ◽  
pp. 2050150
Author(s):  
Konstantin E. Starkov
Keyword(s):  
Cell Population ◽  
Effector Cell ◽  
Invariant Sets ◽  
Model Parameters ◽  
Cancer Model ◽  
Limit Sets ◽  
Localization Method ◽  
Angiogenic Switch ◽  
Lasalle Theorem ◽  
Tumor Eradication

In this paper, we study ultimate dynamics and derive tumor eradication conditions for the angiogenic switch model developed by Viger et al. This model describes the behavior and interactions between host ([Formula: see text]); effector ([Formula: see text]); tumor ([Formula: see text]); endothelial ([Formula: see text]) cell populations. Our approach is based on using the localization method of compact invariant sets and the LaSalle theorem. The ultimate upper bound for each cell population and ultimate lower bound for the effector cell population are found. These bounds describe a location of all bounded dynamics. We construct the domain bounded in [Formula: see text]- and [Formula: see text]-variables which contains the attracting set of the system. Further, we derive conditions imposed on the model parameters for the location of omega-limit sets in the plane [Formula: see text] (the case of a localized tumor). Next, we present conditions imposed on the model and treatment parameters for the location of omega-limit sets in the plane [Formula: see text] (the case of global tumor eradication). Various types of dynamics including the chaotic attractor and convergence dynamics are described. Numerical simulation illustrating tumor eradication theorems is fulfilled as well.

Evolutionary Algorithm Based Solution of Rössler Chaotic System Using Bernstein Polynomials

2020 ◽  
Vol 17 (7) ◽  
pp. 2932-2939
Author(s):  
Rania A. Alharbey ◽  
Kiran Sultan
Keyword(s):  
Chaotic System ◽  
Fitness Function ◽  
Bernstein Polynomials ◽  
Three Dimensional ◽  
Accurate Estimation ◽  
Interior Point Algorithm ◽  
Research Attention ◽  
Rossler System ◽  
Rössler System ◽  
Error Dynamics

Chaotic systems have gained enormous research attention since the pioneering work of Lorenz. Rössler system stands among the extensively studied classical chaotic models. This paper proposes a technique based on Bernstein Polynomial Basis Function to convert the three-dimensional Rössler system of Ordinary Differential Equations (ODEs) into an error minimization problem. First, the properties of Bernstein Polynomials are applied to derive the fitness function of Rössler chaotic system. Second, in order to obtain the values of unknown Bernstein coefficients to optimize the fitness function, the problem is solved using two versatile algorithms from the family of Evolutionary Algorithms (EAs), Genetic Algorithm (GA) hybridized with Interior Point Algorithm (IPA) and Differential Algorithm (DE). For validity of the proposed technique, simulation results are provided which verify the global stability of error dynamics and provide accurate estimation of the desired parameters.

scholarly journals Scattering of three-dimensional plane waves in a self-reinforced half-space lying over a triclinic half-space

2018 ◽  
Vol 15 (3) ◽  
pp. 759-774
Author(s):  
Shishir Gupta ◽  
Abhijit Pramanik ◽  
Smita ◽  
Snehamoy Pramanik
Keyword(s):  
Half Space ◽  
Plane Waves ◽  
Three Dimensional ◽  
Dimensional Plane

Calculating Positive Invariant Sets: A Quantifier Elimination Approach

2019 ◽  
Vol 14 (7) ◽  
Author(s):  
Klaus Röbenack ◽  
Rick Voßwinkel ◽  
Hendrik Richter
Keyword(s):  
Algebraic Geometry ◽  
Quantifier Elimination ◽  
Real Algebraic Geometry ◽  
Invariant Sets ◽  
Rossler System ◽  
Rössler System ◽  
Initial Choice

A Lyapunov-based approach for calculating positive invariant sets in an automatic manner is presented. This is done using real algebraic geometry techniques, which are summed up under the term quantifier elimination (QE). Using available tools, the approach presented yields an algorithmizable procedure whose conservatism only depends on the initial choice for the Lyapunov candidate function. The performance of the approach is illustrated on a variant of the Rössler system and on the Lorenz-Haken system.

scholarly journals Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

2021 ◽  
Vol 11 (15) ◽  
pp. 6955
Author(s):  
Andrzej Rysak ◽  
Magdalena Gregorczyk
Keyword(s):  
Optimal Parameter ◽  
Differential Transform Method ◽  
Bifurcation Diagrams ◽  
Transform Method ◽  
Rossler System ◽  
Differential Transform ◽  
Rössler System ◽  
Fractional System ◽  
Parameter Values

This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

Guided Surface Waves on an Elastic Half Space

1971 ◽  
Vol 38 (4) ◽  
pp. 899-905 ◽  
Author(s):  
L. B. Freund
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Surface Waves ◽  
Half Space ◽  
Body Wave ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Normal Displacement ◽  
Rigid Barrier ◽  
Wave Disturbances

Three-dimensional wave propagation in an elastic half space is considered. The half space is traction free on half its boundary, while the remaining part of the boundary is free of shear traction and is constrained against normal displacement by a smooth, rigid barrier. A time-harmonic surface wave, traveling on the traction free part of the surface, is obliquely incident on the edge of the barrier. The amplitude and the phase of the resulting reflected surface wave are determined by means of Laplace transform methods and the Wiener-Hopf technique. Wave propagation in an elastic half space in contact with two rigid, smooth barriers is then considered. The barriers are arranged so that a strip on the surface of uniform width is traction free, which forms a wave guide for surface waves. Results of the surface wave reflection problem are then used to geometrically construct dispersion relations for the propagation of unattenuated guided surface waves in the guiding structure. The rate of decay of body wave disturbances, localized near the edges of the guide, is discussed.

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