scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

scholarly journals ON POLYNOMIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON A CIRCLE WITHOUT SINGULAR POINTS

2020 ◽  
Vol 12 (4) ◽  
pp. 33-40
Author(s):  
V.Sh. Roitenberg ◽  
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Vector Fields ◽  
Second Order ◽  
Singular Points ◽  
Free Term ◽  
Small Perturbations ◽  
Cylindrical Phase ◽  
Autonomous Differential Equations ◽  
The Right

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550114 ◽  
Author(s):  
Shuang Chen ◽  
Zhengdong Du
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Planar System ◽  
The Stability ◽  
First Time

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

scholarly journals Singular points and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 179-185
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Vector Fields ◽  
Singular Points ◽  
Variable Parameters ◽  
Definition Of

This paper gives the definition of Lie rotated vector fields in the plane and the conditions of movement of singular points on Lie rotated vector fields with variable parameters.

Bifurcation of Homoclinic Orbits to a Saddle-Center in Reversible Systems

2003 ◽  
Vol 13 (09) ◽  
pp. 2603-2622 ◽  
Author(s):  
J. Klaus ◽  
J. Knobloch
Keyword(s):  
Fixed Point ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Critical Parameter ◽  
Center Manifold ◽  
Homoclinic Orbits ◽  
Reversible Systems ◽  
Lin's Method ◽  
Two Parameter ◽  
Lin’S Method

We consider two-parameter families of reversible vector fields having (at the critical parameter value) a homoclinic orbit to a nonhyperbolic fixed point. The nonhyperbolicity is due to a pair of purely imaginary eigenvalues. We give a complete description of the bifurcating one-homoclinic orbits to the center manifold. For that purpose we adapt Lin's method.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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