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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 [1],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 proposition 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 alge2

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 [2] 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
[1] Hirsch and Smale, Linear Algebra,Di?erential Equation and Dynamical System, Academic Press, 1974. [2] C.C. Pugh, A. Lins and Demelo, On Lienard Equation, Lecture Note in Mathematics, 597.

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