scholarly journals A High-Order Melnikov Method for Heteroclinic Orbits in Planar Vector Fields and Heteroclinic Persisting Perturbations

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yi Zhong
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Heteroclinic Orbit ◽  
Melnikov Method ◽  
High Order ◽  
Heteroclinic Orbits ◽  
Van Der Pol ◽  
Planar Vector Fields ◽  
Planar Vector

This work extends the high-order Melnikov method established by FJ Chen and QD Wang to heteroclinic orbits, and it is used to prove, under a certain class of perturbations, the heteroclinic orbit in a planar vector field that remains unbroken. Perturbations which have this property together form the heteroclinic persisting space. The Van der Pol system is analysed as an application.

scholarly journals Orbital Reversibility of Planar Vector Fields

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 14
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Center Problem ◽  
Normal Form Theory ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Form Theory ◽  
Nilpotent Center

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible.

THE SEPARATRIX VALUES OF A PLANAR HOMOCLINIC LOOP

2009 ◽  
Vol 19 (07) ◽  
pp. 2233-2247 ◽  
Author(s):  
LIQIN ZHAO ◽  
XUEXING WANG
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Homoclinic Bifurcation ◽  
Homoclinic Loop ◽  
Regular Map ◽  
Practical Applications ◽  
The Third ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
The Stability

It is well known that the stability of a homoclinic loop for planar vector fields is closely related to the cyclicity of this homoclinic loop. For a planar homoclinic loop consisting of a hyperbolic saddle, the loop values are crucial to the stability. The loop values are divided into two classes: saddle values and separatrix values. The saddle values are related to Dulac map near the saddle, and the separatrix values are related to the regular map near the homoclinic loop. The alternation of these quantities determines the stability of the homoclinic loop. So, it is important to investigate the separatrix values in both theory and for practical applications. For a given planar vector field, we can try to calculate the saddle values by means of dual Liapunov constants or by finding elementary invariants developed by Liu and Li [1990]. The first separatrix value was obtained by Dulac. The second separatrix value was given by Han and Zhu [2007] and by Hu and Feng [2001] independently. The third separatrix value was obtained by Luo and Li [2005] by means of Tkachev's method. In this paper, we shall establish the formulae for the third and fourth separatrix values. As applications, we will give an example with the homoclinic bifurcation of order 9 and prove that the cyclicity of homoclinic loop together with double homoclinic loops is 57.

scholarly journals Orbital Hypernormal Forms

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1500
Author(s):  
Antonio Algaba ◽  
Estanislao Gamero ◽  
Cristóbal García
Keyword(s):  
Vector Field ◽  
Autonomous System ◽  
Vector Fields ◽  
Computational Effort ◽  
Linear Operators ◽  
State Variables ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Identity Transformations ◽  
Reduced Time

In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector field of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector fields, with emphasis on a case of the Takens–Bogdanov singularity.

Uniqueness and non-uniqueness of normal forms for vector fields

1988 ◽  
Vol 108 (1-2) ◽  
pp. 27-33 ◽  
Author(s):  
Alberto Baider ◽  
Richard Churchill
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Resonance Problem ◽  
Main Result Theorem ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Result Theorem ◽  
The Relationship

SynopsisThe use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.

A note on analytic integrability of planar vector fields

2012 ◽  
Vol 23 (5) ◽  
pp. 555-562 ◽  
Author(s):  
A. ALGABA ◽  
C. GARCÍA ◽  
M. REYES
Keyword(s):  
Vector Field ◽  
Integrable Systems ◽  
Analytic Functions ◽  
Vector Fields ◽  
First Integral ◽  
Analytic Integrability ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Analytic First Integral ◽  
Image Position

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systemswhereh,K, Ψ and ξ are analytic functions defined in the neighbourhood ofOwithK(O) ≠ 0 or Ψ(O) ≠ 0 andn≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

scholarly journals A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
New Technique ◽  
Center Problem ◽  
New Normal ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
A New Technique

AbstractIn this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.

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