scholarly journals Comment on “Period function and normalizers of vector fields in Rn with n−1 first integrals” by D. Peralta-Salas [J. Differential Equations 244 (6) (2008) 1287–1303]

2009 ◽  
Vol 246 (9) ◽  
pp. 3750-3753 ◽  
Author(s):  
G.E. Prince
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
First Integrals ◽  
Period Function

VECTOR FIELDS, VARIATIONAL EQUATIONS AND COMMUTATORS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250190
Author(s):  
WILLI-HANS STEEB ◽  
YORICK HARDY ◽  
IGOR TANSKI
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Fixed Points ◽  
Lie Algebra ◽  
Autonomous System ◽  
Vector Fields ◽  
Autonomous Systems ◽  
First Integrals ◽  
Variational Equations ◽  
First Order

We study autonomous systems of first order ordinary differential equations, their corresponding vector fields and the autonomous system corresponding to the vector field of the commutator of two such autonomous systems. These vector fields form a Lie algebra. From the variational equations of these autonomous systems, we form new vector fields consisting of the sum of the two vector fields. We show that these new vector fields also form a Lie algebra. Results about fixed points, first integrals and the divergence of the vector fields are also presented.

scholarly journals SYMMETRIES AND FIRST INTEGRALS FOR NON-VARIATIONAL EQUATIONS

2007 ◽  
Vol 04 (07) ◽  
pp. 1217-1230
Author(s):  
DIEGO CATALANO FERRAIOLI ◽  
PAOLA MORANDO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
First Integrals ◽  
Partial Differential ◽  
Variational Equations ◽  
First Order ◽  
The Ideal

For a class of exterior ideals, we present a method associating first integrals of the characteristic distributions to symmetries of the ideal. The method is applied, under some assumptions, to the study of first integrals of ordinary differential equations and first order partial differential equations as well as to the determination of first integrals for integrable distributions of vector fields.

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

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