A NORMAL FORM FOR DEGENERATE ...

Proc. of the 9th Seminar on Differential Equations and Dynamical Systems, Tabriz, Iran, July 11-13, 2012

A NORMAL FORM FOR DEGENERATE MONODROMIC CRITICAL POINTS OF POLYNOMIAL PLANAR SYSTEMS Razieh Shafeii Lashkarian

Daryoush Behmardi

Faculty of Mathematics Alzahra University, Tehran, Iran [email protected]

Faculty of Mathematics Alzahra University , Tehran, Iran [email protected]

ABSTRACT In this paper we will consider the polynomial planar systems with a degenerate critical point at the origin, and by introducing a normal form for these systems we solve the monodromy problem. 1. INTRODUCTION Consider the planar system

x˙ y˙

= =

P(x; y) Q(x; y)

;

(1)

where P and Q are analytic functions in some planar region Ω. We say that (x0 ; y0 ) is a critical point or singular point of this system, if P(x0 ; y0 ) = Q(x0 ; y0 ) = 0. A critical point is called monodromic if there is no solution of the system tending to the critical point with a concrete slope and all the orbits turned around it. In analytic systems a monodromic singular point is either a focus or a center. There are two important problems in qualitative theory of dynamical systems, which are related to the monodromic critical points. One of them is the monodromy problem (distinguishing if the critical point is monodromic or not), and the other is the Poincare-center-focus ´ problem (distinguishing between the centers and the foci of the nonlinear systems). When the determinant of the jacobian matrix at the critical point differs from zero, the critical point is called nondegenerate. In this case the monodromy problem has been solved, in fact when the eigenvalues of the jacobian matrix are nonzero complex numbers, the critical point is monodromic. If the real part of these eigenvalues are different from zero, then the critical point is a focus, while if their real part are zero (i.e. the singular point is hyperbolic), then the critical point may be a focus or a center. When the determinant of the jacobian matrix at the critical point, equals zero but the jacobian matrix is not identically zero, the critical point is called a nilpotent one (in this case at least one of the eigenvalues equals zero), and when the planar system has no linear part the critical point is called strongly degenerate or linearly zero. For the hyperbolic critical point the center- focus problem has been solved theoretically by Poincare´ [5] and Lyapunov [2] by means of the so called Lyapunov constants. For the nilpotent one the monodromy problem was solved by Andrneev [1] and the center focus problem was solved by Moussu [4]. But since their pure solutions is far from to applied in practice, even for simple polynomial systems, these problems has been considered by many authors in these cases. For the degenerate case almost nothing has been solved about these problems and it has been rarely studied in this case. For this type of critical point even the monodromy problem is very difficult, see for instance [3]. In this paper we shall consider this type of critical points of the planar polynomial systems. Without loose of generality and with a linear transform if we need, we can suppose that the critical point is moved to the origin. In section 2, we introduce some normal form for the systems with a monodromic critical point at the origin. 2. THE NORMAL FORM In this section we introduce some normal form for the planar polynomial systems with a degenerate critical point at the origin. Our proofs are based on a classical result which says that in a planar analytic system (1), any trajectory which approaches to the origin as t ! ∞ either spirals toward the origin or it tend to the origin with a definite direction θ that satisfies the equation cos θ Q(cos θ ; sin θ )

sin θ P(cos θ ; sin θ ) = 0:

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(2)


If xQ(x; y) yP(x; y) is not identically zero then the equation (2) has a solution θ0 if and only if there exist some trajectories which approach to the origin with direction θ = θ0 . Theorem 1 (the normal form) If the origin is a monodromic critical point of the system

P¯ (x; y) Q¯ (x; y)

=

x˙ y˙

=

(3)

;

where P¯ ; Q¯ are polynomials which P¯ (0; 0) = 0; Q¯ (0; 0) = 0, then there exist analytic functions P; Q and natural numbers p; q where the system can be written as: x˙ = y2p 1 + P(x; y) : (4) 2q y˙ = x 1 + yQ(x; y) Lemma 1 If the system (3) has a monodromic critical point at the origin then there are polynomials f (y); g(x) which are not identically zero and analytic functions Pˆ ; Qˆ where the system can be written as:

x˙ y˙

f (y) + Pˆ (x; y) g(x) + Qˆ (x; y)

= =

(5)

:

P¯ (x;y) y

proof: Suppose that there are no functions f ; g with the desired property, let P˜ (x; y) = equation (2) can be written as: cos θ sin θ Q˜ (cos θ ; sin θ )

, Q˜ (x; y) =

Q¯ (x;y) x ,

then the

sin θ cos θ P˜ (cos θ ; sin θ ) = 0:

and this equation has at least four solutions in [0; 2π ℄. Lemma 2 Consider the system

x˙ y˙

= =

f (y) + Pˆ (x; y) g(x) + Qˆ (x; y)

(6)

;

where f ; g are polynomials and Pˆ; Qˆ are analytic functions. If the origin is a monodromic critical point of the system then both f ; g are odd functions. proof: Suppose that f (y) has a monomial ay2p and let f¯(y) = f (y)

ay2p, then the equation (2) can be written as:

cos θ g(cos θ ) + cos θ Qˆ (cos θ ; sin θ ) a sin2p+1 (θ ) sin θ f¯(sin θ ) sin θ Pˆ (cos θ ; sin θ ) in the interval

π 3π 2; 2

=

0;

(7)

we can divide the equation by cos2p+1 (θ ) and we have: a tan2p+1(θ ) + H (cos θ ; sin θ ) = 0;

where H is a bounded analytic function in the interval

π 3π 2; 2

. Let

G(θ ) := a tan2p+1 (θ ) + H (cos θ ; sin θ ); then we have lim

θ ! π2 +

G(θ ) = ∞ and lim

θ ! 32π

G(θ ) = ∞,where the sign of ∞ at the first limit is

one is sgn(a). By the mean value theorem G(θ ) has at least one roots in point of the system. Similarly g(x) is an odd function. Lemma 3 any system of the form

x˙ y˙

= =

π 3π 2; 2

f (y) + Pˆ (x; y) g(x) + Qˆ (x; y)

sgn(a) and in the second

and the origin can’t be a monodromic critical

(8)

;

where f ; g are odd polynomials and Pˆ ; Qˆ are analytic functions can be written as:

x˙ y˙

= =

f¯(y) + P¯ (x; y) x2q 1 + yQ¯ (x; y)

where q 2 N and f¯(y) is an odd polynomial and P¯ ; Q¯ are analytic functions.

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;

(9)


proof: Let g(x) = bx2q

1 + O(x2q+1 )

without loose of generality one can suppose that b > 0. Consider the change of variables r

u=

2q

Z

x

g(s)ds;

2q 0

dt dt1

u2q 1 : g(x)

=

It can be shown easily that in this coordinate the desired form for the system obtains. Remark 1 If f (y) = ay2p

1

+ O(y2p+1),

let P(x; y) = P¯(x; y) + O(y2p+1) and the system can be written as

x˙ y˙

= =

ay2p 1 + P(x; y) x2q 1 + yQ(x; y)

:

(10)

Lemma 4 Consider the system (10), if the origin is a monodromic critical point of the system then a < 0. proof: Suppose that a > 0, then the equation (2) can be written as cos2q θ + cos θ sin θ Q(cos θ ; sin θ )

a sin2p θ

sin θ P(cos θ ; sin θ ) = 0:

π 3π let G(θ ) = cos2q θ + cos θ sin θ Q(cos θ ; sin θ ) 2; 2

In the interval a sin2p θ sin θ P(cos θ ; sin θ )= cos2p θ we have G(0) = 1 and since a > 0, limθ !+∞ G(θ ) = ∞ and G(θ ) has at least one root in the interval, therefore the origin can’t be monodromic. proof of theorem 1: By the previous lemmas the proof is obvious. 3. REFERENCES [1] A. F. Andrneev, investigation of the behavior of the integral curves of a system of two differential equations in the neighborhood of a singular point, AMS translation, (1958), no. 8, pp. 183-207. 1 [2] A. M. Lyapunov, stability of motion, Mathematics in Science and Engineering, Vol 30. Academic Press, New YorkLondon (1996). 1 [3] N. B. Medvedeva, A monodromic criterion for a singular point of a vector field on the plane, St. Peterburg Math. J., 2 (2002), no. 13, pp. 253-268. 1 [4] R. Moussu, symmretrie ´ et forme normale des centers et foyers dregr ´ enr ´ err ´ es, ´ Ergodic Theory Dynam. Systems, 2 (1982), pp. 241-251. 1 [5] H. Poincare, ´ Memoir sur les courbes definies ´ par les requations ´ diffrerentielles, ´ J. Math. Pure Appl. 3(1881), no. 7, pp. 375-422. 1

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