SINGULAR POINTS AND LIE ROTATED VECTOR FIELDS

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.


Introduction.
Many engineering problems are usually run into a class of nonlinear equations that contain variable parameters.In order to study whole orbits or whole phase diagrams of vector fields that contain parameters, it is a complicated and interesting problem how the whole orbit or whole phase diagram change as parameter is changed.It is extremely complicated for general containing parameter vector fields to change in the plane, but for some special containing parameter rotated vector fields, their change has regular rule as parameter is changed.These are many results in this respects [3,4,5,6,7].
In Section 2, we present the basic definitions of Lie rotated vector fields.We define Lie rotated vector fields using one parameter group approach.In accordance with the strict definition of rotated vector field, the singular points of X(µ) must be kept fixed, but in this paper, the singular points of X(µ) can be moved as parameter µ is changed.In Section 3, we discuss the motion of singular points on Lie rotated vector fields.In the section, we require the singular points of X(µ) to be strictly moved as parameter µ is changed, and permit the moved singular points to disappear or decompose, which do not coincide with the singular points of original vector field.We give some conditions and properties corresponding to the vector field Y .In this paper, we give some examples to illustrate the concept and notion of Lie rotated vector fields.

Lie rotated vector fields. We consider vector fields on the plane
For the vector fields (2.1), we define If X and Y are vector fields, then [X, Y ] is a vector field which is operated by Lie bracket, i.e., where Z 1 and Z 2 are expressed as, respectively, where ∇ is gradient operator.Let the plane vector fields X(µ) = (X 1 (x, µ), X 2 (x, µ)) be defined by the following differential equations: where X 1 and X 2 are functions of x and parameter µ ∈ I ⊂ R, and the singular points are isolated.
Definition 2.1.Let the plane vector field X(µ) be determined by (2.5), where If vector field Y exists which is defined by the following differential equations: where Y 1 and Y 2 ∈ C 3 (R 2 , R).At all ordinary points of X(0), such that the following relation holds where X µ (0) is the derivative of the vector field X(µ) at µ = 0, then X(µ), µ ∈ I, is called Lie rotated vector fields.
Remark 2.2.If the vector field X(µ) is defined on D × I, where D ⊂ R 2 , such that X(0) satisfies relation (2.7) at all ordinary points of X(0) on D, then X(µ), µ ∈ I, is called Lie rotated vector fields on D. Lemma 2.3.Let ψ s be a one parameter transform group which is produced by C 1 vector field Y , s ∈ R, and let X be C 1 vector field.If s is fixed, and ϕ p (t) is an integral curve of X through the point p, ϕ p (0 Proof.The proof follows from [1] and [2].In fact, if ϕ p (t) is an integral curve of X through the point p, then and (2.9) Proof.Let the singular points of ψ s * X(µ), µ = 0, on the plane R 2 be p µ 1 ,...,p µ k and the singular points of X(0) on the plane R 2 are p 1 ,...,p m , ∀ε > 0, 0 < ε , and radius ε, such that S ε (p)∩S ε (q) = ∅, where p and q ∈ {p µ i }∪{p j } (1 ≤ i ≤ k, 1 ≤ j ≤ m), p = q.Let ψ s be a one parameter transform group which is produced by C 1 vector field Y , s ∈ R. By the limit definition of Lie bracket, we have (2.12) Next, we notice that X(µ) can be unfolded as Let s = µ, it follows from (2.12), (2.13), and (2.14) that , for given ε > 0, we sooner or later can find δ 1 = δ 1 (ε) > 0, such that when |µ| < δ 1 , we have and let ϑ(µ) be the crossing angle of ψ µ * X(µ) and the x 1 axis, for given ε > 0, we sooner or later can find δ 2 = δ 2 (ε), such that when |µ| < δ 2 , at the ordinary points of 0) is the crossing angle of X(0) and the x 1 axis), so (2.17) Take δ = min{δ 1 ,δ 2 }, then when |µ| < δ, ψ µ * X(µ) constitutes a rotated vector field.
Remark 2.6.In accordance with the strict definition of rotated vector field, the singular points must be kept fixed, but the singular points of ψ µ * X(µ) in Lemma 2.5 can be moved as parameter µ is changed.In the unmistakable circumstance, when |µ| < δ, we call ψ µ * X(µ) a rotated vector.
In the above lemma, δ needs not be a quite small positive number, i.e., 0 < δ 1 need not be set up.For the sake of distinctness, we cite an example to illustrate this equation.
3. The motion of singular points.Let X(µ) be a Lie rotated vector field, we require the singular points of X(µ) to be strictly moved as parameter µ is changed, and permit the singular points that have been moved disappear or decompose, but require the singular points that have been decomposed to be at most limited in number, which do not coincide with the singular points of the original vector field.
Proof.We only prove the circumstance of point p µ (the proof is completely alike as the circumstance of point p 0 ).
From Lemma 3.6, we know that there is no elliptic region which links with the singular point p 0 of X(0), the same do the singular point p µ of X(µ), and from Theorem 3.1, we know that the index of p µ of X(µ) is zero.Take p µ as circular center, make the circumference of a circle l with radius rather small, and let that hyperbolic region of point p µ which intersects with the circumference of a circle l has h.By the Bendixson's formula in §6 of Chapter 3 of [8], we can immediately find h = 2.
From Lemmas 3.6 and 3.7, we have the following theorem.Theorem 3.8.Let X(µ) be a Lie rotated vector field, X(0)| p 0 = 0, and let Y | p 0 = 0, then some singulars while can be moved as parameter µ is changed in X(µ) only contain two hyperbolic regions and their index is zero.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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