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Periodic solutions of polynomial first order differential equations
Nonlinear Analysis. Theory, Mrrhods & Applicanons, Vol 0 Pergamon Press Ltd 1981 Prmted in Great Britain
5. No
2. pp.
0362.546X~81.0201.0157
151-165
SO2.0010
PERIODIC SOLUTIONS OF POLYNOMIAL FIRST ORDER DIFFERENTIAL EQUATIONS
S. SHAHSHAHANI Department
of Mathematics Department
and Computer of Mathematics,
(Received
Key words andphrases: bifurcation, hyperbolic
Science, Sharif University University
15 November
of California,
of Technology,
Berkeley, California
P.O. Box 3406, Tehran,
Iran
94720, U.S.A.
1979; revised 20 February 1980)
Polynomial differential equation, periodic solution, equation of variation, periodic solution, multiplicity of a periodic solution, averaging method.
0. INTRODUCTION
been suggested by C. Pugh that as an easier version of Hilbert’s 16th problem [2] on the determination of the number of limit cycles of a polynomial differential equation in the plane, one might investigate the number of periodic solutions of the first-order differential equation IT HAS
1 = xn + U,_i(t)Xn-l + . . . + a,(t)
(0)
in which the coefficients a,(t) are real polynomials in t, or more generally real-analytic functions defined on [0, 11. By a periodic solution of (0), one means a solution x(t) defined on [0, l] for which x(0) = x(1). It is a simple consequence of the equation of variation for (0) that if n < 2, then n itself is an upper bound for the number of periodic solutions. For n = 3, Smale has obtained the same result in an unpublished work. His method consists of extending the time-one map of the differential equation to a map CP’ -+ CP’, studying the behavior of the extension at co, and applying the Lefschetz Index Formula. Note that the fued points of time-one map correspond to periodic solutions. For n = 4, independent examples by Nirenberg, Yorke, and Lins show that n is no longer an upper bound for the number of periodic solutions. If the degree in t of all q(t) is bounded by p, it remains an open problem to obtain an upper bound in terms of p and n, even for n = 4. The present paper contains some new results in this direction, and provides a self-contained introduction to the subject. In Section 1 we state some preliminary ideas. For convenience, a slightly broader class of equations than (0) is considered which, however enjoys the same features. Section 2 is devoted to the local study of periodic solutions. In particular, we investigate the number of periodic solutions that can bifurcate off a given one. In (2.1) and (2.5) we observe that the proliferation of periodic solutions for n z 4 is already present at the local level. In Section 3 we discuss some global results. Theorem (3.1) is analogous to results on the perturbations of Hamiltonian systems and is based on the averaging method. In (3.3) we give a new proof of Smale’s 157
158
S. SHAHSHAHANI
result based on an elementary bifurcation argument that the coefficients a,(t) be C3 functions oft.
and the results of Section
2, requiring
only
1. GENERALITIES
We consider
differential
equations
of the form
i = a&) + a,(t)x + . . . + a,(t)x” in which IZ 3 1, a, are real-valued c’ functions, one may look at the vectorfield X given by
(1)
r 3 1, and u,(t) > 0 for all 0 d t < 1. Equivalently,
t=1 i i = a&) + ul(t)x
+ . . + u,(t)x”
on the strip [0, l] x R. It follows from t = 1 that the trajectories of X point inward on the boundary (0) x R and outward on (1) x R. For n 3 2, a solution may go to infinity in finite time. &t, 5) will denote the solution of (1) with initial condition 4(0,5) = 5. Letting J denote the open interval in R consisting of all 5 for which 4(t, 5) is defined for all 0 d t d 1, we define the growth function g :J + R by g(5) = (b(l, 5) - 5. A solution with initial condition 5 E J is periodic ifs(t) = 0. LEMMA 1.1. If the coefficients finite.
a, of (1) are analytic
then the number
of periodic
solutions
of (1) is
Proof i vanishes at some point of every periodic solution. Since u,(t) is non-zero on [0, 11, it follows from Ii1 = lu,(t)x”l(l + 0(1/x)) that the union of the periodic trajectories of X lies within some bounded set [0, l] x [-K, K]. By the continuous dependence on initial condition this is a closed set. It follows from analytic dependence on initial condition and u,(t) # 0 that the periodic solutions form a discrete set of solutions. The result then follows. 0 Let 4(t, 5) be a periodic solution of (1). The solution is called: (i) hyperbolic if g’(5) # 0, (ii) stable (resp. unstable) if there is an open interval I around 5 so that for each r] E I, y # <, (rl - 5). s(r) < 0 (rev. > 0). The following is an immediate
and standard
consequence
of the transversality
condition
of
s’(5) z 0: LEMMA 1.2. (a) A hyperbolic periodic solution (resp. > 0), then &t, 5) is stable (resp. unstable). then a neighborhood U of (t, 4(t, t)), 0 d t < E-small perturbation of (2) possesses a unique periodic solution of the perturbed system near (resp. unstable). 0 We shall need a stronger
version
is either stable or unstable. In fact if g’(t) < 0 (b) If the periodic solution +(t, 5) is hyperbolic, 1, in [0, l] and an E > 0 exist so that every C’ (hyperbolic) periodic solution in U. The unique &t, 4) is stable (resp. unstable) if 4(t, 5) is stable
of 1.2(b) as follows.
Periodic
solutions
of polynomial
first order differential
equations
159
LEMMA 1.3. Suppose (1) possesses k periodic solutions, all hyperbolic. Then any equation sufficiently C’ close to (1) also possesses k (hyperbolic) periodic solutions.
Proof: Let yl, . . . , yk be the periodic solutions of (1) having initial conditions tl, . . . , & respectively, where 5, < . . . < &. For each yi, choose ( Ui, Em)as required by 1.2(b). We may assume that U,‘s are disjoint and that Ui intersects (0) x R in an interval ]qZi_ l, rzi[. Since a,(t) # 0, there is /I > 0 so that if 5 d II, or 5 3 ylZk,then Ig(<)\ > 1~.Since there are no periodic solutions starting in (0) x [qZi, uZi+ J, i = 1,. . . ‘k - 1, it follows that pi > 0 exists so that for all 5 E CV2i,)Izi+ 11, IS(S)\ > Pi.NowCo dependence on parameter implies that any system sufficiently Co close to (1) possesses no periodic solutions with initial condition 5 E Iv,, q2[u . . . u]qlk_ 1, qzk[. This together with 1.2(b) finishes the proof. 0 Remark 1.4.
Denoting the right-hand side of (1) by f(t, x), it follows from the equation of variation
PI : (3)
and the definition of growth function g that 1 div X(t, +(t, x)) dt
g’(5) = - 1 + exp (S 0
>
(4)
so that a periodic solution is hyperbolic if and only if the average of div X along that solution is non-zero.
2. LOCAL
STUDY
OF
PERIODIC
SOLUTIONS
We now investigate the number of periodic solutions that can bifurcate off a given periodic solution under c’ small variations of the coefficients. For this, the knowledge of the higher derivatives of the growth function will be required. Unless stated otherwise, we assume that the coeffrcients of (1) are C” functions oft. Consider the following reduction of (1). Let cc(t),0 < t d 1 be a periodic solution, The map 0: [0, l] x R --f [0, l] x-R given by
e(t, X)
=
(t,
x
-
a(t))
is clearly a diffeomorphism of [0, l] x R onto itself. This diffeomorphism differential equation of the form gi-= a,(t)x + . . . + a,(t)x”
transforms (1) into a (5)
where a,,(t) is the same as in (l), hence a,,(t) > 0, but other coefficients may differ. Note, in particular, that the growth function for (5) differs from that of (1) by a translation in the argument of constant ~(0) = a(1). Hence the periodic solutions of (1) and (5) are in one-one correspondence and the derivatives of the growth functions at corresponding periodic solutions coincide. We shall then limit ourselves to the local study of the periodic solution 4(t, x) = 0 of (5).
160
S.SHAHSELWAN~
is
If the periodic solution +(t, x) 3 0 of (5) is non-hyperbolic,
it follows from (1.4) that :
a,(t)dr = 0.
(6)
0
Again denoting the right-hand side of (5) by f(t, x), (3) implies that:
2 (t, x) =
exp (f
(7)
; g (r, 9(r, x)) dr)
Upon successive differentiation with respect to x, we obtain: 2
g$(t, x) = g
(t, x) = 2 (t, x) * j-1 {a
; g3 SI
(4$%x)) - g (t, x)” +
(t, x) *
tsi
t
x)’
o
ax2 ’
ay
dX2
I
I
(9)
+ 3~(t,~(r,x))~(I,x)~(~,x) (t, x) .
dt + g(t,x)
+ ~(t,cW,.))~~(t,x)
dt
(4 rb@, 4) -g (t, 4 dt
o @(r,&r1x))*$t,x)3
+ gz (t, &t, x,) - b$ (t, x) dt + 2 2
(8)
(t, xl] dt
g2(t, +(t, x))f$ (t, x)
sia-
+e?!(,
g$(t, x) = 2 (t, x) -
k 4% 4) - $
1 g 1-I .
f$
ft, &t, x))fg
(4 46thd) .g
ft, x)2
(t, x)
dt.
(W
We further introduce w(t) = exp(lLa,(s)ds).
P~opostno~ 2.1. Let j < 4, and suppose that the derivatives of g at the zero solution of (5) vanish through g ‘j - ’ ‘(0). Then : g(j)(O) = (j !) ’ w’-‘(t)aj(t) dt s0
is
provided 2 G j d 3,
(11)
1
g(4)(0) = (4 !)
w3(t)a,(t)dt
(12)
+ j,i, w2(t)a3(t) [j-b o(s)a&) ds] dr}.
0
Proof: These follow from (7)-(11), the definition of g, and straightforward Remark 2.2. In order to facilitate the exploitation
computation.
•l
of (11) and (12), we discuss a further simplification of (5). Suppose that g’(0) = 0 so that (6) holds. We consider $: [0, 1) x Iw+ [O, 1] x fw given by $(t, x) = (t, co- ‘(t)x).
Periodic solutions of polynomial first order differential equations
$ is a diffeomorphism
and transforms
161
(5) into the form
1 = &)x2
+ . . . + a,(t)x”
(13)
where the new x replaces o- '(t)x and q(t) stands for old q(t)& l(t). Note that since o(t) > 0, the leading coefficient is still positive. Also observe that by virtue of o(O) = o(l) = 1, II/ keeps the endpoints of solutions of (5) fixed so that the growth function g remains unchanged. Thus under the hypothesis of the existence of a non-hyperbolic periodic solution, the differential equation (1) can be transformed to the form (13) without affecting the number and the nature of the periodic solutions. For (13), (11) and (12) become: 1 g’“(0)
=
aj(t) dt
fj !)
providedj
d 3,
(14)
s gf4’(0) = ,!){[;,Isdt
+ l:a&)[j;
u,(s)ds]dr).
(15)
COROLLARY 2.3. Suppose that 1 d j d 3, the coefficients of (1) are Cj, and (1) possesses a periodic solution 4(t, 5) for which g’“(c) is the first non-vanishing derivative of g at 5. Then a neighborhood U of +(t, 0, 0 d t < 1, and an E > 0 exist so that any system Cj s-close to (1) possesses at most j periodic solutions in U. Moreover, a Cj perturbation in the form (1) with exactly j periodic solutions in a neighborhood of 4(t, 5) always exists.
Proof. The first assertion is of a general nature and requires no proof. For j = 1 both assertions are contained in (1.2). Letting j 3 2, we may assume that the equation is in the form (13) with b(t, 0) = 0 the periodic solution in question. We prove the second assertion for j = 3, the case j = 2 being simpler. If g”‘(0) is the first non-vanishing derivative of g at 0, then +(t, 0) E 0 is either stable or unstable. For definiteness assume g”‘(0) > 0 so that 4(t, 0) 3 0 is unstable. We may then choose a region M bounded by [0, l] x {0}, (0) x [w,{l} x [w,and a C’ curve C, that lies entirely in the lower half-strip, in such a way that the vectorfield points outward on the boundary segment C, of M. Replacing a,(t) by a&t) + Ed, E~ > 0, makes g”(0) > 0 by virtue of (14). We can choose .sl so small that the perturbed vectorfield still points outward on C,. But since g’(0) = 0 and g”(0) > 0, g(5) > 0 for [[I # 0 sufliciently small. It follows that the new equation possesses (at least) one periodic solution lying between C, and 0. We now take a C’ curve C, lying between the new periodic solution and 0 on which the vectorlield points upward. Introducing the coefficient al(t) = .s2 > 0 so small that the new vectorfield still points upward on C,, we note that g’(0) > 0. Thus #t, 0) = 0 is converted into an unstable periodic solution, again forcing the appearance of a second periodic solution between C, and itself. Since &t, 0) s 0 remains a periodic solution throughout, at least 3 periodic solutions exist. By taking s1 and s2 sufficiently small, the first assertion ensures that no more than 3 periodic solutions may exist in a neighborhood of&t, 0) = 0. 0 We remark that the phenomenon considered in (2.3) may be regarded as an instance of Hopf bifurcation for maps. The direct approach considered is needed to ascertain that the perturbations are of the desired type. For a periodic solution of (1) with initial condition 5, if j is the smallest positive integer so that g’“(5) # 0, we call j the multiplicity of the periodic solution.
162
S. SHAHSHAHANI
COROLLARY2.4. If n d 3 in (l), then a periodic Proof
The assumption
a,(t) # 0 ensures
solution
of (1) has multiplicity
via (14) that g’“)(t) # 0.
< n.
q
The appearance of the second term in (15) (and of more complicated expressions for derivatives of order >4) brings about an ‘explosion’ in the number of periodic solutions of (1) when n 3 4, as the following example will show. Example 2.5. Consider 1 = (3t2 - 2t)x’ + 60(2t Straightforward
computation
1)x3 + x4.
shows that for all j d 4,
g(j)(O) = 0 g’f”(0) < 0, i.e. the zero solution form
has multiplicity
5. In the manner
1 = .sqx + (3t2 - 2t + Qx2
of the proof of (2.3), perturbations
+ [60(1 + Q(2t - 1) + aJx3
of the
+ x4
will show the existence of live periodic solutions in a neighborhood of zero, provided E~‘Sare sufficiently small. Incidentally, small perturbations of this form possess a sixth periodic solution not bifurcating off the zero solution. This is seen as follows. Note that i > 0 for x >> 0 and that the zero solution is stable for the original equation since g”‘(O) < 0. Therefore the original equation possesses l(mod 2) periodic solutions lying the half-strip x > 0. This situation is preserved under small perturbations. Further tedious computations lead the author to : Conjecture. For a2(t) and a,(t) polynomials in t of degree p, the best upper bound plicitly of a periodic solution of an equation of the form
for the multi-
x4
1 = a2(t)x2 + a3(t)x3 + is max(4, p + 3). 3. SOME Our first result of this section represents each coefficient q(t) of(l), we define
GLOBAL
an application
Ai=
of the idea of ‘averaging’
[4,6].
For
’ ai(t) dt. s
Then the averaged equation corresponding
RESULTS
0
to (1) is the time-independent
i = A, + A,x + . . . +A,x”.
differential
equation (16)
The periodic solutions of (16) are clearly of the form $(t, a) 3 c1where CYis a real root of A(x) =A, + A,x + . . . + A,x” = 0. Moreover, 4(t, cc)E CIis hyperbolic if and only if CIis a simple root.
Periodic solutions of polynomial first order differential equations
163
THEOREM 3.1. Suppose that A(x) = 0 has k real roots, all simple. Then for p > 0 sufficiently small,
the differential equation with C’ coefficients 1 = &)(t)
+ a,(t)x + . . . + un(t)x”],
a,(t) > 0
(17)
possesses exactly k periodic solutions, all hyperbolic. These solutions tend to the periodic solutions of the averaged equation (16) as p + 0. Proofi We regard (17) as a perturbation of the differential equation i = 0. Thus every solution of (17) defined for 0 Q t < 1 can be written as x(t) = x0 + O(p), where x,, is x(O). 0(/c) depends, in general, on x0. For a periodic solution x(t) of (17): 1
dt
i(t)
o=
s =
0
p
1
f.
4)
4,
+
O(P)]
dt
si
=
p
i
A,xb
( i=O
+
O($).
)
Thus the initial condition x0 of a periodic solution of(17) corresponds to a root ofA + O(u) = 0. We claim that a compact region [0, l] x [-K, K] exists so that periodic solutions of (17) stay entirely in that region for all 0 < 1~d 1. Choose K > 0 so large that for 1x13 K:
so(t)
---+ X"
... +
a -l(t) n
Then in each of the regions [0, l] x [K, m[ and [0, l] x ] - co, -K] 1 = px”(a,(t) + a,_l(t)/x
the sign of
+ . . . + a,(t)/x”)
does not change and the claim follows. We conclude then that as 1~+ 0, the periodic solutions of (17) approach the periodic solutions of i = A(x). Since every real root CIof A(x) = 0 is simple, the periodic solutions off = A(x) are hyperbolic and the assertions follow. 0 Our next goal is to determine the maximum possible number of periodic solutions of (1) for n = 3. A key ingredient in applying the local results of the preceding section is the following. PROPOSITION 3.2. Suppose that in(l), it = 3, and the coefficients are C3 functions oft. If (1) possesses a non-hyperbolic periodic solution, then the total number of periodic solutions of (1) counted with multiplicity is three.
Proof: By (2.2), we may assume without the loss of generality that the equation has the form J? = a,(t)x2 + a3(t)x3
(18)
with cj(t, 0) = 0 the non-hyperbolic periodic solution hypothesized. If the total number of periodic solutions counted with multiplicity exceeds three, it follows from (2.4) that a periodic solution x(t) exists for which x(t) # 0, 0 d t d 1. Integrating the differential form y = [a,(t)
164
+ a,(t)x]
S. SHAHSHAHANI
dt along x(t), we see from (18) that y= s
d”=O s
X2
since x(0) = x( 1). Thus 1
a,(t)x(t)dt
s
=
-A,
(19)
0
where A, is the integral of a,(t) over [0, 11.If the multiplicity of c&t, 0) = 0 is 3, we conclude from (14) that the above vanishes, contradicting a,(t) > 0 and x(t) # 0. Thus in this case the zero solution is the only periodic solution and the Proposition is proved. Now suppose the multiplicity of the zero solution is two, i.e. A, # 0. Since two solutions (t, 90, 0) and (t, cb(t, v)) of (2) cannot have any points in common, and a,(t) > 0, it follows that at most one solution x(t) of (18) may satisfy (19). Such a solution in fact exists, as can be seen by noting that for 1x1sufficiently large, x . i > 0. It remains to show that this unique non-zero periodic solution has multiplicity 1. Consider the case A, > 0, the case A, < 0 being similar. If A, > 0, then x(t) < 0 for all 0 < t < 1. Since g”(0) > 0, g(r) > 0 for every { satisfying x(0) < 5 < 0. But g(5) < 0 for 5 < x(0) by noting the sign of i for x e 0. It follows that x(t) is an unstable periodic solution. Therefore the multiplicity j of x(t) must be odd. j > 3 is excluded by (2.4), and we have seen that j = 3 leads to the uniqueness of the periodic solution. Therefore j = 1, and the proof is complete. 0 THEOREM 3.3. Suppose that in(l), n = 3 and the coefficients at most three periodic solutions.
are C3 functions
oft. Then (1) possesses
Proof. Denoting the space of differential equations of type (1) with n = 3 and C3 coefficients with C, we first assume that a C3 curve A: [0, l] -+ C exists so that A(l) is the given differential equation and A(O) has at most three periodic solutions, all hyperbolic. Suppose A(l) possesses more than 3 periodic solutions. Let S consist of those s E [0, l] with the property that for any 0, 0 d (T < s, A(o) possesses no more than 3 periodic solutions. By the assumption on A(O), S is a subinterval of [0, l] with non-empty interior, and 0 E S. Let T = sup S. Clearly r < 1 by the assumption on A(1) and (1.3). If all the periodic solutions of A(S) are hyperbolic, then the definition of z and (1.3) imply that z + E E S for all 0 ,< F < 6 if6 is sufficiently small, contradicting T = sup S. Therefore A(z) must have at least one non-hyperbolic periodic solution. It follows from (3.2) that the total number of periodic solutions of A(r) counted with multiplicity is 3. From (2.3) we deduce that for E > 0 sufficiently small, A(7 + I-:)has no more than 3 periodic solutions, again contradicting the definition of T. It remains to show that a C3 curve il as described above always exists. Consider the corresponding averaged equation (16). If A(x) = 0 has no multiple roots, the existence of A follows from (3.1). Otherwise, we first define a C3 curve A1: [$ l] -+ C for which A,(l) is the given equation, and the averaged equation of A,(+) has no multiple roots. I, can be clearly constructed by altering the coefficients. Next we construct A,: [0, +] + C so that i2(0) has exactly three (hyperbolic) periodic solutions, and A,($) = A,($. By joining A, and A1, and smoothing at A,($ the desired curve is obtained. 0 Acknowledgements-The
author thanks C. Pugh and S. Smale for general conversations on the topic.
Periodic
solutions
of polynomial
first order differential
equations
165
REFERENCES 1. ANDRONOV A. et al., Theory of Birfurcations of Dynamic Systems in the Plane. John Wiley, New York (1973). 2. BROWDW F. (ed.), Mathematical Developments Arising from Hilbert Problems. American Mathematical Society, Providence, R.I., (1974). 3. C~DDINGTON E. & LEVINSON N. Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). 4. HALE J., Ordinary Differential Equations. Wiley-Interscience, New Jersey (1969). 5. MARSDEN J. & MCCRACKEN M., Hopf Bifurcation and Its Applications. Springer-Verlag, New York (1976). 6. MOSER J., Regularization of Kepler’s problem and the averaging method on a manifold, Communs. pure appl. Math. 23,609433 (1970).
5. No
2. pp.
0362.546X~81.0201.0157
151-165
SO2.0010
PERIODIC SOLUTIONS OF POLYNOMIAL FIRST ORDER DIFFERENTIAL EQUATIONS
S. SHAHSHAHANI Department
of Mathematics Department
and Computer of Mathematics,
(Received
Key words andphrases: bifurcation, hyperbolic
Science, Sharif University University
15 November
of California,
of Technology,
Berkeley, California
P.O. Box 3406, Tehran,
Iran
94720, U.S.A.
1979; revised 20 February 1980)
Polynomial differential equation, periodic solution, equation of variation, periodic solution, multiplicity of a periodic solution, averaging method.
0. INTRODUCTION
been suggested by C. Pugh that as an easier version of Hilbert’s 16th problem [2] on the determination of the number of limit cycles of a polynomial differential equation in the plane, one might investigate the number of periodic solutions of the first-order differential equation IT HAS
1 = xn + U,_i(t)Xn-l + . . . + a,(t)
(0)
in which the coefficients a,(t) are real polynomials in t, or more generally real-analytic functions defined on [0, 11. By a periodic solution of (0), one means a solution x(t) defined on [0, l] for which x(0) = x(1). It is a simple consequence of the equation of variation for (0) that if n < 2, then n itself is an upper bound for the number of periodic solutions. For n = 3, Smale has obtained the same result in an unpublished work. His method consists of extending the time-one map of the differential equation to a map CP’ -+ CP’, studying the behavior of the extension at co, and applying the Lefschetz Index Formula. Note that the fued points of time-one map correspond to periodic solutions. For n = 4, independent examples by Nirenberg, Yorke, and Lins show that n is no longer an upper bound for the number of periodic solutions. If the degree in t of all q(t) is bounded by p, it remains an open problem to obtain an upper bound in terms of p and n, even for n = 4. The present paper contains some new results in this direction, and provides a self-contained introduction to the subject. In Section 1 we state some preliminary ideas. For convenience, a slightly broader class of equations than (0) is considered which, however enjoys the same features. Section 2 is devoted to the local study of periodic solutions. In particular, we investigate the number of periodic solutions that can bifurcate off a given one. In (2.1) and (2.5) we observe that the proliferation of periodic solutions for n z 4 is already present at the local level. In Section 3 we discuss some global results. Theorem (3.1) is analogous to results on the perturbations of Hamiltonian systems and is based on the averaging method. In (3.3) we give a new proof of Smale’s 157
158
S. SHAHSHAHANI
result based on an elementary bifurcation argument that the coefficients a,(t) be C3 functions oft.
and the results of Section
2, requiring
only
1. GENERALITIES
We consider
differential
equations
of the form
i = a&) + a,(t)x + . . . + a,(t)x” in which IZ 3 1, a, are real-valued c’ functions, one may look at the vectorfield X given by
(1)
r 3 1, and u,(t) > 0 for all 0 d t < 1. Equivalently,
t=1 i i = a&) + ul(t)x
+ . . + u,(t)x”
on the strip [0, l] x R. It follows from t = 1 that the trajectories of X point inward on the boundary (0) x R and outward on (1) x R. For n 3 2, a solution may go to infinity in finite time. &t, 5) will denote the solution of (1) with initial condition 4(0,5) = 5. Letting J denote the open interval in R consisting of all 5 for which 4(t, 5) is defined for all 0 d t d 1, we define the growth function g :J + R by g(5) = (b(l, 5) - 5. A solution with initial condition 5 E J is periodic ifs(t) = 0. LEMMA 1.1. If the coefficients finite.
a, of (1) are analytic
then the number
of periodic
solutions
of (1) is
Proof i vanishes at some point of every periodic solution. Since u,(t) is non-zero on [0, 11, it follows from Ii1 = lu,(t)x”l(l + 0(1/x)) that the union of the periodic trajectories of X lies within some bounded set [0, l] x [-K, K]. By the continuous dependence on initial condition this is a closed set. It follows from analytic dependence on initial condition and u,(t) # 0 that the periodic solutions form a discrete set of solutions. The result then follows. 0 Let 4(t, 5) be a periodic solution of (1). The solution is called: (i) hyperbolic if g’(5) # 0, (ii) stable (resp. unstable) if there is an open interval I around 5 so that for each r] E I, y # <, (rl - 5). s(r) < 0 (rev. > 0). The following is an immediate
and standard
consequence
of the transversality
condition
of
s’(5) z 0: LEMMA 1.2. (a) A hyperbolic periodic solution (resp. > 0), then &t, 5) is stable (resp. unstable). then a neighborhood U of (t, 4(t, t)), 0 d t < E-small perturbation of (2) possesses a unique periodic solution of the perturbed system near (resp. unstable). 0 We shall need a stronger
version
is either stable or unstable. In fact if g’(t) < 0 (b) If the periodic solution +(t, 5) is hyperbolic, 1, in [0, l] and an E > 0 exist so that every C’ (hyperbolic) periodic solution in U. The unique &t, 4) is stable (resp. unstable) if 4(t, 5) is stable
of 1.2(b) as follows.
Periodic
solutions
of polynomial
first order differential
equations
159
LEMMA 1.3. Suppose (1) possesses k periodic solutions, all hyperbolic. Then any equation sufficiently C’ close to (1) also possesses k (hyperbolic) periodic solutions.
Proof: Let yl, . . . , yk be the periodic solutions of (1) having initial conditions tl, . . . , & respectively, where 5, < . . . < &. For each yi, choose ( Ui, Em)as required by 1.2(b). We may assume that U,‘s are disjoint and that Ui intersects (0) x R in an interval ]qZi_ l, rzi[. Since a,(t) # 0, there is /I > 0 so that if 5 d II, or 5 3 ylZk,then Ig(<)\ > 1~.Since there are no periodic solutions starting in (0) x [qZi, uZi+ J, i = 1,. . . ‘k - 1, it follows that pi > 0 exists so that for all 5 E CV2i,)Izi+ 11, IS(S)\ > Pi.NowCo dependence on parameter implies that any system sufficiently Co close to (1) possesses no periodic solutions with initial condition 5 E Iv,, q2[u . . . u]qlk_ 1, qzk[. This together with 1.2(b) finishes the proof. 0 Remark 1.4.
Denoting the right-hand side of (1) by f(t, x), it follows from the equation of variation
PI : (3)
and the definition of growth function g that 1 div X(t, +(t, x)) dt
g’(5) = - 1 + exp (S 0
>
(4)
so that a periodic solution is hyperbolic if and only if the average of div X along that solution is non-zero.
2. LOCAL
STUDY
OF
PERIODIC
SOLUTIONS
We now investigate the number of periodic solutions that can bifurcate off a given periodic solution under c’ small variations of the coefficients. For this, the knowledge of the higher derivatives of the growth function will be required. Unless stated otherwise, we assume that the coeffrcients of (1) are C” functions oft. Consider the following reduction of (1). Let cc(t),0 < t d 1 be a periodic solution, The map 0: [0, l] x R --f [0, l] x-R given by
e(t, X)
=
(t,
x
-
a(t))
is clearly a diffeomorphism of [0, l] x R onto itself. This diffeomorphism differential equation of the form gi-= a,(t)x + . . . + a,(t)x”
transforms (1) into a (5)
where a,,(t) is the same as in (l), hence a,,(t) > 0, but other coefficients may differ. Note, in particular, that the growth function for (5) differs from that of (1) by a translation in the argument of constant ~(0) = a(1). Hence the periodic solutions of (1) and (5) are in one-one correspondence and the derivatives of the growth functions at corresponding periodic solutions coincide. We shall then limit ourselves to the local study of the periodic solution 4(t, x) = 0 of (5).
160
S.SHAHSELWAN~
is
If the periodic solution +(t, x) 3 0 of (5) is non-hyperbolic,
it follows from (1.4) that :
a,(t)dr = 0.
(6)
0
Again denoting the right-hand side of (5) by f(t, x), (3) implies that:
2 (t, x) =
exp (f
(7)
; g (r, 9(r, x)) dr)
Upon successive differentiation with respect to x, we obtain: 2
g$(t, x) = g
(t, x) = 2 (t, x) * j-1 {a
; g3 SI
(4$%x)) - g (t, x)” +
(t, x) *
tsi
t
x)’
o
ax2 ’
ay
dX2
I
I
(9)
+ 3~(t,~(r,x))~(I,x)~(~,x) (t, x) .
dt + g(t,x)
+ ~(t,cW,.))~~(t,x)
dt
(4 rb@, 4) -g (t, 4 dt
o @(r,&r1x))*$t,x)3
+ gz (t, &t, x,) - b$ (t, x) dt + 2 2
(8)
(t, xl] dt
g2(t, +(t, x))f$ (t, x)
sia-
+e?!(,
g$(t, x) = 2 (t, x) -
k 4% 4) - $
1 g 1-I .
f$
ft, &t, x))fg
(4 46thd) .g
ft, x)2
(t, x)
dt.
(W
We further introduce w(t) = exp(lLa,(s)ds).
P~opostno~ 2.1. Let j < 4, and suppose that the derivatives of g at the zero solution of (5) vanish through g ‘j - ’ ‘(0). Then : g(j)(O) = (j !) ’ w’-‘(t)aj(t) dt s0
is
provided 2 G j d 3,
(11)
1
g(4)(0) = (4 !)
w3(t)a,(t)dt
(12)
+ j,i, w2(t)a3(t) [j-b o(s)a&) ds] dr}.
0
Proof: These follow from (7)-(11), the definition of g, and straightforward Remark 2.2. In order to facilitate the exploitation
computation.
•l
of (11) and (12), we discuss a further simplification of (5). Suppose that g’(0) = 0 so that (6) holds. We consider $: [0, 1) x Iw+ [O, 1] x fw given by $(t, x) = (t, co- ‘(t)x).
Periodic solutions of polynomial first order differential equations
$ is a diffeomorphism
and transforms
161
(5) into the form
1 = &)x2
+ . . . + a,(t)x”
(13)
where the new x replaces o- '(t)x and q(t) stands for old q(t)& l(t). Note that since o(t) > 0, the leading coefficient is still positive. Also observe that by virtue of o(O) = o(l) = 1, II/ keeps the endpoints of solutions of (5) fixed so that the growth function g remains unchanged. Thus under the hypothesis of the existence of a non-hyperbolic periodic solution, the differential equation (1) can be transformed to the form (13) without affecting the number and the nature of the periodic solutions. For (13), (11) and (12) become: 1 g’“(0)
=
aj(t) dt
fj !)
providedj
d 3,
(14)
s gf4’(0) = ,!){[;,Isdt
+ l:a&)[j;
u,(s)ds]dr).
(15)
COROLLARY 2.3. Suppose that 1 d j d 3, the coefficients of (1) are Cj, and (1) possesses a periodic solution 4(t, 5) for which g’“(c) is the first non-vanishing derivative of g at 5. Then a neighborhood U of +(t, 0, 0 d t < 1, and an E > 0 exist so that any system Cj s-close to (1) possesses at most j periodic solutions in U. Moreover, a Cj perturbation in the form (1) with exactly j periodic solutions in a neighborhood of 4(t, 5) always exists.
Proof. The first assertion is of a general nature and requires no proof. For j = 1 both assertions are contained in (1.2). Letting j 3 2, we may assume that the equation is in the form (13) with b(t, 0) = 0 the periodic solution in question. We prove the second assertion for j = 3, the case j = 2 being simpler. If g”‘(0) is the first non-vanishing derivative of g at 0, then +(t, 0) E 0 is either stable or unstable. For definiteness assume g”‘(0) > 0 so that 4(t, 0) 3 0 is unstable. We may then choose a region M bounded by [0, l] x {0}, (0) x [w,{l} x [w,and a C’ curve C, that lies entirely in the lower half-strip, in such a way that the vectorfield points outward on the boundary segment C, of M. Replacing a,(t) by a&t) + Ed, E~ > 0, makes g”(0) > 0 by virtue of (14). We can choose .sl so small that the perturbed vectorfield still points outward on C,. But since g’(0) = 0 and g”(0) > 0, g(5) > 0 for [[I # 0 sufliciently small. It follows that the new equation possesses (at least) one periodic solution lying between C, and 0. We now take a C’ curve C, lying between the new periodic solution and 0 on which the vectorlield points upward. Introducing the coefficient al(t) = .s2 > 0 so small that the new vectorfield still points upward on C,, we note that g’(0) > 0. Thus #t, 0) = 0 is converted into an unstable periodic solution, again forcing the appearance of a second periodic solution between C, and itself. Since &t, 0) s 0 remains a periodic solution throughout, at least 3 periodic solutions exist. By taking s1 and s2 sufficiently small, the first assertion ensures that no more than 3 periodic solutions may exist in a neighborhood of&t, 0) = 0. 0 We remark that the phenomenon considered in (2.3) may be regarded as an instance of Hopf bifurcation for maps. The direct approach considered is needed to ascertain that the perturbations are of the desired type. For a periodic solution of (1) with initial condition 5, if j is the smallest positive integer so that g’“(5) # 0, we call j the multiplicity of the periodic solution.
162
S. SHAHSHAHANI
COROLLARY2.4. If n d 3 in (l), then a periodic Proof
The assumption
a,(t) # 0 ensures
solution
of (1) has multiplicity
via (14) that g’“)(t) # 0.
< n.
q
The appearance of the second term in (15) (and of more complicated expressions for derivatives of order >4) brings about an ‘explosion’ in the number of periodic solutions of (1) when n 3 4, as the following example will show. Example 2.5. Consider 1 = (3t2 - 2t)x’ + 60(2t Straightforward
computation
1)x3 + x4.
shows that for all j d 4,
g(j)(O) = 0 g’f”(0) < 0, i.e. the zero solution form
has multiplicity
5. In the manner
1 = .sqx + (3t2 - 2t + Qx2
of the proof of (2.3), perturbations
+ [60(1 + Q(2t - 1) + aJx3
of the
+ x4
will show the existence of live periodic solutions in a neighborhood of zero, provided E~‘Sare sufficiently small. Incidentally, small perturbations of this form possess a sixth periodic solution not bifurcating off the zero solution. This is seen as follows. Note that i > 0 for x >> 0 and that the zero solution is stable for the original equation since g”‘(O) < 0. Therefore the original equation possesses l(mod 2) periodic solutions lying the half-strip x > 0. This situation is preserved under small perturbations. Further tedious computations lead the author to : Conjecture. For a2(t) and a,(t) polynomials in t of degree p, the best upper bound plicitly of a periodic solution of an equation of the form
for the multi-
x4
1 = a2(t)x2 + a3(t)x3 + is max(4, p + 3). 3. SOME Our first result of this section represents each coefficient q(t) of(l), we define
GLOBAL
an application
Ai=
of the idea of ‘averaging’
[4,6].
For
’ ai(t) dt. s
Then the averaged equation corresponding
RESULTS
0
to (1) is the time-independent
i = A, + A,x + . . . +A,x”.
differential
equation (16)
The periodic solutions of (16) are clearly of the form $(t, a) 3 c1where CYis a real root of A(x) =A, + A,x + . . . + A,x” = 0. Moreover, 4(t, cc)E CIis hyperbolic if and only if CIis a simple root.
Periodic solutions of polynomial first order differential equations
163
THEOREM 3.1. Suppose that A(x) = 0 has k real roots, all simple. Then for p > 0 sufficiently small,
the differential equation with C’ coefficients 1 = &)(t)
+ a,(t)x + . . . + un(t)x”],
a,(t) > 0
(17)
possesses exactly k periodic solutions, all hyperbolic. These solutions tend to the periodic solutions of the averaged equation (16) as p + 0. Proofi We regard (17) as a perturbation of the differential equation i = 0. Thus every solution of (17) defined for 0 Q t < 1 can be written as x(t) = x0 + O(p), where x,, is x(O). 0(/c) depends, in general, on x0. For a periodic solution x(t) of (17): 1
dt
i(t)
o=
s =
0
p
1
f.
4)
4,
+
O(P)]
dt
si
=
p
i
A,xb
( i=O
+
O($).
)
Thus the initial condition x0 of a periodic solution of(17) corresponds to a root ofA + O(u) = 0. We claim that a compact region [0, l] x [-K, K] exists so that periodic solutions of (17) stay entirely in that region for all 0 < 1~d 1. Choose K > 0 so large that for 1x13 K:
so(t)
---+ X"
... +
a -l(t) n
Then in each of the regions [0, l] x [K, m[ and [0, l] x ] - co, -K] 1 = px”(a,(t) + a,_l(t)/x
the sign of
+ . . . + a,(t)/x”)
does not change and the claim follows. We conclude then that as 1~+ 0, the periodic solutions of (17) approach the periodic solutions of i = A(x). Since every real root CIof A(x) = 0 is simple, the periodic solutions off = A(x) are hyperbolic and the assertions follow. 0 Our next goal is to determine the maximum possible number of periodic solutions of (1) for n = 3. A key ingredient in applying the local results of the preceding section is the following. PROPOSITION 3.2. Suppose that in(l), it = 3, and the coefficients are C3 functions oft. If (1) possesses a non-hyperbolic periodic solution, then the total number of periodic solutions of (1) counted with multiplicity is three.
Proof: By (2.2), we may assume without the loss of generality that the equation has the form J? = a,(t)x2 + a3(t)x3
(18)
with cj(t, 0) = 0 the non-hyperbolic periodic solution hypothesized. If the total number of periodic solutions counted with multiplicity exceeds three, it follows from (2.4) that a periodic solution x(t) exists for which x(t) # 0, 0 d t d 1. Integrating the differential form y = [a,(t)
164
+ a,(t)x]
S. SHAHSHAHANI
dt along x(t), we see from (18) that y= s
d”=O s
X2
since x(0) = x( 1). Thus 1
a,(t)x(t)dt
s
=
-A,
(19)
0
where A, is the integral of a,(t) over [0, 11.If the multiplicity of c&t, 0) = 0 is 3, we conclude from (14) that the above vanishes, contradicting a,(t) > 0 and x(t) # 0. Thus in this case the zero solution is the only periodic solution and the Proposition is proved. Now suppose the multiplicity of the zero solution is two, i.e. A, # 0. Since two solutions (t, 90, 0) and (t, cb(t, v)) of (2) cannot have any points in common, and a,(t) > 0, it follows that at most one solution x(t) of (18) may satisfy (19). Such a solution in fact exists, as can be seen by noting that for 1x1sufficiently large, x . i > 0. It remains to show that this unique non-zero periodic solution has multiplicity 1. Consider the case A, > 0, the case A, < 0 being similar. If A, > 0, then x(t) < 0 for all 0 < t < 1. Since g”(0) > 0, g(r) > 0 for every { satisfying x(0) < 5 < 0. But g(5) < 0 for 5 < x(0) by noting the sign of i for x e 0. It follows that x(t) is an unstable periodic solution. Therefore the multiplicity j of x(t) must be odd. j > 3 is excluded by (2.4), and we have seen that j = 3 leads to the uniqueness of the periodic solution. Therefore j = 1, and the proof is complete. 0 THEOREM 3.3. Suppose that in(l), n = 3 and the coefficients at most three periodic solutions.
are C3 functions
oft. Then (1) possesses
Proof. Denoting the space of differential equations of type (1) with n = 3 and C3 coefficients with C, we first assume that a C3 curve A: [0, l] -+ C exists so that A(l) is the given differential equation and A(O) has at most three periodic solutions, all hyperbolic. Suppose A(l) possesses more than 3 periodic solutions. Let S consist of those s E [0, l] with the property that for any 0, 0 d (T < s, A(o) possesses no more than 3 periodic solutions. By the assumption on A(O), S is a subinterval of [0, l] with non-empty interior, and 0 E S. Let T = sup S. Clearly r < 1 by the assumption on A(1) and (1.3). If all the periodic solutions of A(S) are hyperbolic, then the definition of z and (1.3) imply that z + E E S for all 0 ,< F < 6 if6 is sufficiently small, contradicting T = sup S. Therefore A(z) must have at least one non-hyperbolic periodic solution. It follows from (3.2) that the total number of periodic solutions of A(r) counted with multiplicity is 3. From (2.3) we deduce that for E > 0 sufficiently small, A(7 + I-:)has no more than 3 periodic solutions, again contradicting the definition of T. It remains to show that a C3 curve il as described above always exists. Consider the corresponding averaged equation (16). If A(x) = 0 has no multiple roots, the existence of A follows from (3.1). Otherwise, we first define a C3 curve A1: [$ l] -+ C for which A,(l) is the given equation, and the averaged equation of A,(+) has no multiple roots. I, can be clearly constructed by altering the coefficients. Next we construct A,: [0, +] + C so that i2(0) has exactly three (hyperbolic) periodic solutions, and A,($) = A,($. By joining A, and A1, and smoothing at A,($ the desired curve is obtained. 0 Acknowledgements-The
author thanks C. Pugh and S. Smale for general conversations on the topic.
Periodic
solutions
of polynomial
first order differential
equations
165
REFERENCES 1. ANDRONOV A. et al., Theory of Birfurcations of Dynamic Systems in the Plane. John Wiley, New York (1973). 2. BROWDW F. (ed.), Mathematical Developments Arising from Hilbert Problems. American Mathematical Society, Providence, R.I., (1974). 3. C~DDINGTON E. & LEVINSON N. Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). 4. HALE J., Ordinary Differential Equations. Wiley-Interscience, New Jersey (1969). 5. MARSDEN J. & MCCRACKEN M., Hopf Bifurcation and Its Applications. Springer-Verlag, New York (1976). 6. MOSER J., Regularization of Kepler’s problem and the averaging method on a manifold, Communs. pure appl. Math. 23,609433 (1970).