The dynamic system method and the traps

1998 ◽  
Vol 30 (01) ◽  
pp. 137-151
Author(s):  
Odile Brandière
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Dynamic System ◽  
Stochastic Algorithms ◽  
System Method ◽  
Dynamical System Method ◽  
Unstable Direction ◽  
Differential Equation Method

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.

The dynamic system method and the traps

1998 ◽  
Vol 30 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Odile Brandière
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Dynamic System ◽  
Stochastic Algorithms ◽  
System Method ◽  
Dynamical System Method ◽  
Unstable Direction ◽  
Differential Equation Method

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.

Discussion on the Complicated Topological Dynamic System of Ordinary Differential Equation (ODE)

2014 ◽  
Vol 696 ◽  
pp. 30-37
Author(s):  
Yun Xia Wang
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Dynamic System ◽  
Closed Form ◽  
Topological Dynamic ◽  
Autonomous Differential Equations

The dynamical system of ODEs is about closed-form researches into the field of ODEs from the perspective of dynamical systems. This paper, starting with the research of path in autonomous differential equations and discussion on Poincaré’s viewpoints, probes into the complicated topological dynamic system of ODEs.

Synchronization of chaotic jerk systems

2020 ◽  
Vol 34 (20) ◽  
pp. 2050189
Author(s):  
Z. Wang ◽  
S. Panahi ◽  
A. J. M. Khalaf ◽  
S. Jafari ◽  
I. Hussain
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Collective Behavior ◽  
Chaotic Systems ◽  
Dynamical Network ◽  
General Systems

Chaotic jerk oscillators belong to the simplest chaotic systems. These systems try to model the behavior of dynamical systems efficiently. Jerk oscillators can be known as the most general systems in science, especially physics. It has been proved that every dynamical system expressed with an ordinary differential equation is able to describe as a jerky system in particular conditions. One of its main topics is investigating the collective behavior of chaotic jerk oscillators in the dynamical network. In this paper, the synchronizability of the identical network of jerk oscillators is examined in three different coupling configurations, which are velocity, acceleration, and jerk coupling, and the results are compared with each other.

The evolution equation of non-linear waves and its exact solutions by subsidiary ordinary differential equation method

2020 ◽  
Vol 34 (34) ◽  
pp. 2050390
Author(s):  
Xiaojun Yin ◽  
Liangui Yang ◽  
Quansheng Liu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Rossby Waves ◽  
Burgers Equation ◽  
Adomian Decomposition Method ◽  
Equation Method ◽  
Adomian Decomposition ◽  
Non Linear ◽  
Differential Equation Method ◽  
Equatorial Rossby Waves

In this work, we investigate the dynamics of the equatorial Rossby waves by including the complete Coriolis force, external source and dissipation. The amplitude evolution of equatorial Rossby waves is described as an extended non-linear mKdV–Burgers equation from a potential vorticity equation and it is unlike the standard mKdV–Burgers equation. Built on the obtained model, the corresponding physical phenomena related to the non-linear Rossby waves are analyzed. Also, the subsidiary ordinary differential equation method is employed to solve the solitary solution of the mKdV equation. By analyzing the solution, we find that the horizontal component of Coriolis parameter works on the amplitude of the Rossby waves. Meanwhile, we use the Adomian decomposition method to obtain the approximate soliton solution of the model.

A note on the generation of random dynamical systems from fractional stochastic delay differential equations

2015 ◽  
Vol 15 (03) ◽  
pp. 1550018 ◽  
Author(s):  
Luu Hoang Duc ◽  
Björn Schmalfuß ◽  
Stefan Siegmund
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Dynamical Systems ◽  
Delay Differential Equations ◽  
Regularity Conditions ◽  
Random Dynamical System ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Stochastic Delay Differential Equation ◽  
Delay Differential

In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Hölder space which is separable.

scholarly journals Separation Method of Semi-Fixed Variables Together with Dynamical System Method for Solving Nonlinear Time-Fractional PDEs with Higher-Order Terms

2021 ◽  
Author(s):  
Weiguo Rui
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
Exact Solutions ◽  
Fractional Order ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
System Method ◽  
Higher Order Terms ◽  
Fractional Models ◽  
Dynamical System Method

Abstract It is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. In this paper, a new approach which combined with the separation method of semi-fixed variables and dynamical system method is introduced. As example, a time-fractional reaction-diffusion equation with higher-order terms is studied under the different kinds of fractional-order differential operators. In different parametric regions, phase portraits of systems which derived from the reaction-diffusion equation are presented. Existence and dynamic properties of solutions of this nonlinear time-fractional models are investigated. In some special parametric conditions, some exact solutions of this time-fractional models are obtained. The dynamical properties of some exact solutions are discussed and the graphs of them are illustrated.PACS: 02.30.Jr; 02.30.Oz; 02.70.-c; 02.70.Mv; 02.90.+p; 04.20.Jb; 05.10.-a

scholarly journals A canonical form of the equation of motion of linear dynamical systems

2018 ◽  
Vol 474 (2211) ◽  
pp. 20170809
Author(s):  
Daniel T. Kawano ◽  
Rubens Goncalves Salsa ◽  
Fai Ma ◽  
Matthias Morzfeld
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Linear System ◽  
Canonical Form ◽  
Equation Of Motion ◽  
Positive Definite ◽  
Order Ordinary Differential Equation ◽  
Linear Dynamical Systems ◽  
Almost All

The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.

scholarly journals Anisotropic power-law inflation for a model of two scalar and two vector fields

2021 ◽  
Vol 81 (6) ◽  
Author(s):  
Tuan Q. Do ◽  
W. F. Kao
Keyword(s):  
Dynamical System ◽  
Power Law ◽  
Vector Fields ◽  
Bianchi Type ◽  
Scalar Fields ◽  
Field Extension ◽  
Type I ◽  
Anisotropic Solution ◽  
System Method ◽  
Dynamical System Method

AbstractInspired by an interesting counterexample to the cosmic no-hair conjecture found in a supergravity-motivated model recently, we propose a multi-field extension, in which two scalar fields are allowed to non-minimally couple to two vector fields, respectively. This model is shown to admit an exact Bianchi type I power-law solution. Furthermore, stability analysis based on the dynamical system method is performed to show that this anisotropic solution is indeed stable and attractive if both scalar fields are canonical. Nevertheless, if one of the two scalar fields is phantom then the corresponding anisotropic power-law inflation turns unstable as expected.

Sign in / Sign up

Export Citation Format

Share Document