Invariant manifolds of an autonomous ordinary differential equation from its generalized normal forms

2003 ◽  
Vol 13 (4) ◽  
pp. 1188-1204 ◽  
Author(s):  
Jesús Palacián
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Invariant Manifolds ◽  
Normal Forms ◽  
Autonomous Ordinary Differential Equation

ON THE STRUCTURE OF THE SOLUTIONS OF A TWO-PARAMETER FAMILY OF THREE-DIMENSIONAL ORDINARY DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 13 (05) ◽  
pp. 1287-1298 ◽  
Author(s):  
SERKAN T. IMPRAM ◽  
RUSSELL JOHNSON ◽  
RAFFAELLA PAVANI
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Wide Range ◽  
Two Parameters ◽  
Two Parameter ◽  
Autonomous Ordinary Differential Equation

We analyze the global structure of the solutions of a three-dimensional, autonomous ordinary differential equation which depends on two parameters. We use graphical, heuristic, and rigorous arguments to show that as the parameters vary, a wide range of dynamical behavior is displayed.

scholarly journals On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Alvaro Alvarez-Parrilla ◽  
Martín Eduardo Frías-Armenta ◽  
Elifalet López-González ◽  
Carlos Yee-Romero
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Normal Forms ◽  
Complex Field ◽  
System Of Differential Equations ◽  
Matrix Algebras ◽  
A New Technique ◽  
Previous Paragraph ◽  
Autonomous Ordinary Differential Equation

A new technique for solving a certain class of systems of autonomous ordinary differential equations over𝕂nis introduced (𝕂being the real or complex field). The technique is based on two observations: (1), if𝕂nhas the structure of certain normed, associative, commutative, and with a unit, algebras𝔸over𝕂, then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations overℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.

scholarly journals On the third order boundary value problems with asymmetric nonlinearity

2011 ◽  
Vol 16 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Sergey Smirnov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Third Order ◽  
Number Of Solutions ◽  
The Third ◽  
Asymmetric Nonlinearity ◽  
Autonomous Ordinary Differential Equation

The author considers two point third order boundary value problem with asymmetric nonlinearity. The structure and oscillatory properties of solutions of the third order nonlinear autonomous ordinary differential equation are discussed. Results on the estimation of the number of solutions to boundary value problem are provided. An illustrative example is given.

scholarly journals On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

2015 ◽  
Vol 29 (1) ◽  
pp. 7-17
Author(s):  
Roman Srzednicki
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Conley Index ◽  
Computer Assisted ◽  
Small Step ◽  
Index Pair ◽  
Time Periodic ◽  
Autonomous Ordinary Differential Equation

AbstractWe announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.

Some modified Michaelis–Menten equations having stable closed trajectories

1988 ◽  
Vol 109 (3-4) ◽  
pp. 341-359 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Global Convergence ◽  
Critical Point ◽  
Closed Trajectory ◽  
Biochemical Reactions ◽  
3 Dimensional ◽  
Convergence Of Solutions ◽  
Autonomous Ordinary Differential Equation

A 3-dimensional autonomous ordinary differential equation is studied which models certain cellular biochemical reactions. Extended Poincaré-Bendixson theory is used to obtain algebraic conditions on the parameters which are sufficient for the existence of at least one stable closed trajectory. Similar conditions are also obtained for the absence of chaos and for the global convergence of solutions to a critical point.

Sign in / Sign up

Export Citation Format

Share Document