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.

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.

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.

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.

scholarly journals On the Existence of Eigenmodes of Linear Quasi-Periodic Differential Equations and their Relation to the MHD Continuum

1982 ◽  
Vol 37 (8) ◽  
pp. 830-839 ◽  
Author(s):  
A. Salat
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Infinite Sequence ◽  
Periodic Coefficient ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Magnetic Surfaces ◽  
Toroidal Geometry

The existence of quasi-periodic eigensolutions of a linear second order ordinary differential equation with quasi-periodic coefficient f{ω1t, ω2t) is investigated numerically and graphically. For sufficiently incommensurate frequencies ω1, ω2, a doubly indexed infinite sequence of eigenvalues and eigenmodes is obtained.The equation considered is a model for the magneto-hydrodynamic “continuum” in general toroidal geometry. The result suggests that continuum modes exist at least on sufficiently ir-rational magnetic surfaces

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

Self-Excited Oscillations in Dynamical Systems Possessing Retarded Actions

1942 ◽  
Vol 9 (2) ◽  
pp. A65-A71 ◽  
Author(s):  
Nicholas Minorsky
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Linear Equation ◽  
Finite Order ◽  
Characteristic Equation ◽  
High Derivative ◽  
Time Lags ◽  
Constant Coefficients ◽  
The Subject

Abstract There exists a variety of dynamical systems, possessing retarded actions, which are not entirely describable in terms of differential equations of a finite order. The differential equations of such systems are sometimes designated as hysterodifferential equations. An important particular case of such equations, encountered in practice, is when the original differential equation for unretarded quantities is a linear equation with constant coefficients and the time lags are constant. The characteristic equation, corresponding to the hysterodifferential equation for retarded quantities in such a case, has a series of subsequent high-derivative terms which generally converge. It is possible to develop a simple graphical interpretation for this equation. Such systems with retarded actions are capable of self-excitation. Self-excited oscillations of this character are generally undesirable in practice and it is to this phase of the subject that the present paper is devoted.

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 579-583
Author(s):  
Muhammad Abdullahi ◽  
Bashir Sule ◽  
Mustapha Isyaku
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Order Ordinary Differential Equation ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

1968 ◽  
Vol 20 ◽  
pp. 720-726
Author(s):  
T. G. Hallam ◽  
V. Komkov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Compact Set ◽  
Stability Of Solutions ◽  
Closed Set ◽  
The Stability ◽  
Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

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