An Index Interpretation For the Number Of Limit Cycles Of A Vector Field

arXiv:math/0408037v2 [math.DS] 11 Aug 2004

An Index Interpretation For The Number Of Limit Cycles of a Vector Field
Ali Taghavi Institute for Advanced Studies in Basic Sciences
Zanjan 45195-159, Iran
February 1, 2008
Does the Hilbert 16th Problem have a P DE nature?The Hilbert 16th Problem asks for a uniform upper bound H(n) for the number of limit cycles of a polynomial vector ?eld X of degree n on the Plane. More ever it seems that ”limit cycles” are The only obstructions for solving the ”P DE” X.g = f , globally in the plane
The following observation about Lienard equation suggests to look at the Hilbert 16th problem as a P DE problem
Proposition. Let L be the Lienard polynomial Vector ?eld
x˙ = y ? F (x) y˙ = ?x
where F is an odd degree polynomial with F ′(0) = 0 ,L de?nes a linear operator on function space, L(f ) = L.f If All Limit Cycles of L be Hyperbolic Then The number of limit cycles of L is equal to ?indexL. Proof.The origin is The only singularity of vector ?eld L and let we have n limit cycles,γ1,γ2. . . γn which all surround the origin Let f satis?es the following conditions: its integral along all closed orbits is zero and f (0) = 0, actually such f is in the kernel of an operator de?ned on the function space to n + 1 dimension We prove there is a map g with L.g = f ,This shows that the Fredholm Index of L is equal to ?n,because the kernel is one dimensional space since around attractors the only ?rst integrals are constant maps ). Since the origin is a Hyperbolic singularity,we can de?ne g in a unique integral way in the interior of γ1, see below as a similar situation near hyperbolic
1

limit cycle ,g is uniform continuous in the interior of γ1 and has a unique

extension to the boundary, because the integral of f along γ1 is zero and

g(x) ? g(p(x) is near to zero where p(x) is a poincare map with respect to

some transverse section.Now We extend g to exterior of γ1 s follows: We

De?ne g(x) = g(x?) +

∞ 0

f

(?t(x))

?

f

(?t(x?)).

x? is the unique point on

γ1 which has the same fate as x:that is their trajectory are asymptotic with

the rate of exp(?t), see ,Chapter 13

The Integrals converge since the corresponding functions approach to zero

with ”exp” rate, For x on γ1 ,x? = x and g was an integral for f restricted

on γ1

This Shows that g described above is an integral for f in a neighborhood of

limit cycle γ1 with the same values on γ1.

g can be de?ne on the whole of the plane since the orbits of exterior

points of γn accumulate to it.

Note that Since Vector ?eld L is analytic,x? Is analytic too,thus the propo-

sition is valid if we de?ne L on smooth or analytic function space

Remark 1.Let F be an even polynomial,then the corresponding Lienard equation L, has a center and both kernel and co-kernel of operator L(f ) = L.f are in?nite dimensional space: We Show that L possesses a global analytic ?rst integral thus kernel of above operator is in?nite dimensional space,furthermore for each set of n closed orbits we present n independent elements in the quotion space of Image of operator L(f ) = L.f : let f be a smooth (analytic or algebraic) maps separates closed the orbits then the elements 1, f, f 2, f 3.....f n?1 are independent in quotion space of image because for each map g ,L.g should vanish in at least one point on each closed orbit It remains to prove the integrability of the Lienard equation with center:Let F (x) = K(x2) for a polynomial K,the square of intersection of orbits with the graph of parabola y = ?x2 de?nes a global ?rst integral,this parabola is not transverse to lienard vector ?eld at the origin so apparently the above ?rst integral is not analytic at the origin.Using Change of coordinate x := x2 y := y we ?nd that this ?rst integral corresponds to intersection of solutions of
x˙ = y ? k(x)
y˙ = ?1
with transversal section y = x,which is analytic

Remark 2.The operator L described above can be restricted to alge-

2

braic functions.Since for each set of n closed orbits we can give n independent element in co-kernel,The index of operator is an upper bound for the number of the limit cycles.In line of conjecture in  on the number of limit cycles of lienard equation,we suggest to compute the index of operator restricted to polynomials maps.Are there uniform upper bounds for this index in terms of degree of F,where the degree of F is odd
Remark 3.The F redholm Index ,mentioned above, is not necessarily ?nite if an arbitrary algebraic Vector Field Possess A limit cycle,for example there is a cubic system with a center and a limit cycle simultaneously ,so in this case the co-kernel’s dimension is in?nite. But not only such ”co-existence” of limit cycle and center cannot occur for quadratic Systems,but also,all quadratic systems with center have been classi?ed with a FINITE number of algebraic condition.Furthermore,since f redholm index is ?xed on the connected component of the space of f redholm operators ,it strongly seems that H(2) is ?nite provided that we prove following problem :
Problem.For a quadratic system without center the corresponding operator X(f ) = X.f is f redholm where X is Considered as an analytic Vector ?eld on 2 dimensional sphere(X is considered as poincare compcti?cation of corresponding quadratic system )
References
 Hirsch and Smale, Linear Algebra,Di?erential Equation and Dynamical System, Academic Press, 1974.
 C.C. Pugh, A. Lins and Demelo, On Lienard Equation, Lecture Note in Mathematics, 597.
3

...An Integral Representation of the Image Field_图....pdf

interpretation in terms of a modified HuygensE"...we shall have to make a number of approximations...denote the position vector of a typical field ...

An Index Interpretation For the Number Of Limit Cycles Of A ....pdf

An Index Interpretation For the Number Of Limit Cycles Of A Vector Field_专业资料。We show that the number of limit cycles of Lienard vector field L ...

...gradient methods for the interpretation of gravi....pdf

Ma et al. (2012) presented tensor local wavenumber method interpret the ...Interpretation of aeromagnetic data using eigenvector analysis of pseudo ...

快速自动机器人人平台外文翻译.doc

The magnitude interpretation as a an importance is useful when vector fields...2.2.5.2 Limit cycle based vector fields Limit cycles are part of ...

vectorCalculus.pdf

o The geometrical interpretation of the grad ...o The curl of a vector field is the vector ...number of identities relating different vector ...

...to-Identifying-Out-of-Control Variables for Multivariate ....pdf

is the interpretation of out-of-control signals.... x is the p×1 mean vector for quality ...The applications of neural networks in the field ...

...wavefield imaging with scalar and vector potenti....pdf

2007_SEG_Abstracts_Elastic wavefield imaging with scalar and vector ...elds contain a mix of P and S wave modes which hampers interpretation of...

section 8 the relations_图文.pdf

finite number of piecewise smooth simple closed ...Proof Physical interpretation of Theorem 8.2 ? v... 8.4 Suppose that we have avector field ? ...

...Comparison of 2D Vector Visualization Methods A Pilot ....pdf

” which was good since this could confuse the interpretation of the data....J. 2001. Quantitative Comparative Evaluation of 2D Vector Field Visualization ...

数学专业词汇(G).doc

number field 高斯数域 gaussian plane 复数平面 ...interpretation 几何解释 geometric mean 比例中项 ...vector 几何向量 geometrization 几何化 geometry 几何...

The Interior Field of a Magnetized Einstein-Maxwell Object_....pdf

(4) are convenient because we can give a direct physical interpretation of...Observe that the norm of the time-like 1 Killing vector X vanishes if f...

Does only the amplitude of the state vector carry the ....pdf

A number of conclusions can be done from Eqs. ...interpretation of QM, the entire state vector is... considering Ψ as an objectively real field [7...

Transport by vector fields with Kolmogorov spectrum.pdf

Transport by vector fields with Kolmogorov spectrum...the number of points in the support of the ...Let us discuss the physical interpretation of ...

Modeling and Temporal Evolution of a Family of Curves.pdf

We show that this vector field framework is well...of a vector eld has many advantages: a vector ...interpretation: in an image sequence, we will ...

definination of Divergence.pdf

In light of the physical interpretation, a vector field with constant zero ...divergence of a vector field can be defined in any number of dimensions. ...

Introduction to the special section on graph algorithms in ....pdf

cycle algorithm or Lawler and Meggido's minimum ...Citing deficiencies in feature vector-based ...interpretation, a classical computer vision problem ...

A Computationally Universal Field Computer That is Purely ....pdf

field computer is a (spatial) continuum-limit ... italic letters indicate components of a vector. ...This con dence-level interpretation doesn't ...

Interpretation of spatio-temporal relations in realtime and ....pdf

vector describes the objects displacement from the ...The number of classes to be distinguished depends...interpretation (a) Cycle 216 (b) Cycle 221 (c...

Failure of microcausality in quantum field theory on ....pdf

Failure of microcausality in quantum field theory ...? interpretation of the symbol x. We will not ...Because of this, no spinor, vector, etc. ?...

...volume of Manati field the challenge of taking_图文.pdf

of Manati field the challenge of taking_专业资料...A number of alternative realizations of the ...a base interpretation and uncertainty vectors1,2....