scholarly journals On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2057
Author(s):  
Alexander N. Pchelintsev
Keyword(s):  
Dynamical System ◽  
Numerical Method ◽  
High Precision ◽  
Fourth Order ◽  
Nonlinear Dynamical System ◽  
Search Procedure ◽  
Nonlinear Dynamical ◽  
Poisson Stability ◽  
Quadratic Nonlinearities ◽  
The Stability

This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm to study the stability of the found regime and confirm the type of attractor.

scholarly journals Dynamical System Analysis of Interacting Hessence Dark Energy inf(T)Gravity

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Jyotirmay Das Mandal ◽  
Ujjal Debnath
Keyword(s):  
Dynamical System ◽  
Autonomous System ◽  
System Analysis ◽  
Nonlinear Dynamical System ◽  
Computational Method ◽  
Dynamical Analysis ◽  
Dynamical System Theory ◽  
Nonlinear Dynamical ◽  
The Stability ◽  
The Universe

We have carried out dynamical system analysis of hessence field coupling with dark matter inf(T)gravity. We have analysed the critical points due to autonomous system. The resulting autonomous system is nonlinear. So, we have applied the theory of nonlinear dynamical system. We have noticed that very few papers are devoted to this kind of study. Maximum works in literature are done treating the dynamical system as done in linear dynamical analysis, which are unable to predict correct evolution. Our work is totally different from those kinds of works. We have used nonlinear dynamical system theory, developed till date, in our analysis. This approach gives totally different stable solutions, in contrast to what the linear analysis would have predicted. We have discussed the stability analysis in detail due to exponential potential through computational method in tabular form and analysed the evolution of the universe. Some plots are drawn to investigate the behaviour of the system(this plotting technique is different from usual phase plot and that devised by us). Interestingly, the analysis shows that the universe may resemble the “cosmological constant” like evolution (i.e.,ΛCDM model is a subset of the solution set). Also, all the fixed points of our model are able to avoid Big Rip singularity.

Jacobi stability analysis of Rikitake system

2016 ◽  
Vol 13 (07) ◽  
pp. 1650098 ◽  
Author(s):  
M. K. Gupta ◽  
C. K. Yadav
Keyword(s):  
Dynamical System ◽  
Differential Geometry ◽  
Stability Analysis ◽  
Nonlinear Differential Equations ◽  
Equilibrium Points ◽  
Nonlinear Dynamical System ◽  
Intrinsic Properties ◽  
Nonlinear Dynamical ◽  
Rikitake System ◽  
The Stability

We study the Rikitake system through the method of differential geometry, i.e. Kosambi–Cartan–Chern (KCC) theory for Jacobi stability analysis. For applying KCC theory we reformulate the Rikitake system as two second-order nonlinear differential equations. The five KCC invariants are obtained which express the intrinsic properties of nonlinear dynamical system. The deviation curvature tensor and its eigenvalues are obtained which determine the stability of the system. Jacobi stability of the equilibrium points is studied and obtain the conditions for stability. We study the dynamics of Rikitake system which shows the chaotic behaviour near the equilibrium points.

NOISE REDUCTION BY GRADIENT DESCENT

1993 ◽  
Vol 03 (01) ◽  
pp. 113-118 ◽  
Author(s):  
MIKE DAVIES
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Gradient Descent ◽  
Nonlinear Dynamical System ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Gradient Descent Method ◽  
Local Minima ◽  
Nonlinear Dynamical ◽  
The Stability

The problem of reducing noise in a time series from a nonlinear dynamical system can be formulated as a nonlinear minimisation process. This paper demonstrates that this can be easily solved using a steepest descent method without any of the stability problems that have been associated with using a Newton method [Hammel, 1990; Farmer & Sidorowich, 1991]. The optimisation function to be minimised is also shown not to contain any local minima if the trajectory is always hyperbolic. So that in this case this method will converge eventually to a purely deterministic trajectory. Finally this method is compared with a recently proposed algorithm [Schreiber & Grassberger, 1991], which can be viewed as an alternative gradient descent method.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

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