The main object of this paper is stability and multiplicity of periodic solutions to first order quasi-autonomous scalar differential equations. It has been noticed for ...
Stability and multiplicity of periodic or almost periodic solutions to scalar first order ODE Alain Haraux
R´ esum´ e. Utilisant une propri´et´e de monotonie sp´eciale au premier ordre, on ´etablit une propri´et´e g´en´erale concernant le type de stabilit´e des solutions p´eriodiques successives de l’´equation diff´erentielle scalaire du premier ordre u0 = f (t, u). D’autre part on ´enonce une condition optimale de petitesse sur la source f de l’´equation u0 + g(u) = f (t) o` u g ∈ C 1 (R) et f : R → R est presque p´eriodique pour que cette
´equation ait exactement N solutions presque p´eriodiques sous l’hypoth`ese qu’il existe N points d’´equilibre ci et g0 (ci ) 6= 0 pour tout i.
Abstract. Using a monotonicity property specific to this case we give a general property of the stability pattern of successive periodic solutions of the first order scalar differential equation u0 = f(t, u). We also give an optimal smallness condition in order for the quasi-autonomous equation u0 + g(u) = f(t) where g ∈ C 1 (R) and f : R → R is almost periodic to have exactly N almost periodic solutions on the line assuming that we have exactly N equilibria ci and g 0 (ci ) 6= 0 for all i. Keywords: differential equations, periodic solutions, stability
AMS classification numbers: 34C25, 34C27, 34D05, 34D020, 34D030
1
Introduction The main object of this paper is stability and multiplicity of periodic solutions to first order quasi-autonomous scalar differential equations. It has been noticed for a long time that stability of periodic solutions obey some kind of alternating principle which is the exact analog of the fact that two successive zeroes of a function f cannot give the same sign to the derivative f’. We prove this fact in a completely general framework relying only on some order preserving property of the flow generated by the equation. After some basic preliminaries in Section 1, Sections 2 and 3 are devoted to precise statement and proof of this result. In section 4, we are interested in the case of a small (periodic or almost periodic) quasi-autonomous perturbation of an autonomous equation having only nondegenerate equilibria. Section 5 is devoted to some typical examples and showing the optimality of the results in Section 4.
1- The periodic case: basic properties In this section we consider the first order differential equation u0 = f(t, u)
(1.1)
where f is measurable in t and locally Lipschitz continuous with respect to u, uniformly for t bounded. First we recall a classical result (cf eg Pliss [7], Haraux [5]) Proposition 1.1. Let u, v be two solutions of (1.1) on some interval J. Then u−v
has a constant sign on J
Proof. Indeed the difference z := v − u is a solution of z 0 = f(t, u + z) − f(t, u) = h(t)z(t) Rt with z ∈ L∞ loc (J) and then the result follows at once since z(t) = z(s)exp s h(σ)d(σ) for all s, t in the interior of J. Corollary 1.2.
Assuming in addition that f is T − periodic in t, for any
solution u of (1.1) global and bounded on R+ , the sequence of functions un (t) := u(nT + t) 2
is monotonic with respect to n. Proof. Introducing v(t) = u(t + T ) , by Proposition 1.1 z = v − u has constant
sign on R+ . This implies immediately the result. Corollary 1.3. (cf e.g.[5, 7])
When f is T-periodic in t, for any solution u of
(1.1) global and bounded on R+ is such that lim |u(t) − ω(t)| = 0
t→+∞
where ω is some T-periodic solution ω Proof. When f is T-periodic in t, for any solution u of (1.1) global and bounded on R+ , the sequence of functions u(nT + t) is monotonically convergent to some solution ω of (1.1), uniformly on compact intervals of R. It is obvious that ω is T-periodic. Then the result follows immediately by writing t = nT + r,
r ∈ [0, T ]
2- Local study near a periodic solution. In this section we examine the stability properties of a periodic solution under a small perturbation of the initial value. We say that a solution u of (1.1) is right-stable (R-stable) if ∀ε > 0,
∃δ > 0 :
0 ≤ v(0) − u(0) ≤ δ =⇒ sup |v(t) − u(t)| ≤ ε
(2.1)
t≥0
where v is a solution of (1.1). Otherwise we say that u is R-unstable. Analogously we define left-stable (L-stable) and L-unstable solutions. Finally we say that a solution u of (1.1) is right-asymptotically stable (R-asymptotically stable) if it is R-stable and in addition ∃η > 0 :
0 ≤ v(0) − u(0) ≤ η =⇒ lim (v(t) − u(t)) = 0 t→∞
(2.2)
The corresponding notion of left-asymptotically stable (L-asymptotically stable) solution is defined similarly. Theorem 2.1. Let ω be a T-periodic solution of (1.1). Then one of the following mutually exclusive properties holds true i) There is no T-periodic solution of (1.1) above ω. In this case either no solution of (1.1) above ω is bounded, or for some ε0 > 0, all solutions u of (1.1) with ω(0) ≤ u(0) ≤ ω(0) + ε0 3
are asymptotic to ω as t → ∞ and ω is R- asymptotically stable.
ii) There exists a smallest T-periodic solution ω1 > ω. In that case, either ω is
R- asymptotically stable and ω1 is L - unstable, or ω1 is L - asymptotically stable and ω is R - unstable. iii) There exists an infinite decreasing sequence ωn of distinct T-periodic solutions of (1.1) above ω such that lim ωn (0) = ω(0)
n→∞
In that case ω is R-stable but not R- asymptotically stable. Proof. The set P of T - periodic solutions of (1.1) is totally ordered by Proposition 1.1. Moreover if B ⊂ P is any bounded subset, a simple application of Gronwall’s lemma shows that
∀(ω1 , ω2 ) ∈ B × B,
kω1 − ω2 k∞ ≤ C(B)|ω1 (0) − ω2 (0)|
(2.3)
Now let S = {z ∈ P,
z > ω}
Either S = ∅ (case i) or S 6= ∅ but ω 6∈ S (case ii), or ω ∈ S. In this last case it is immediate to build by induction a decreasing sequence of T-periodic solutions of (1.1) above ω such that limn→∞ ωn (0) = ω(0), hence we are in case iii). We now examine successively the 3 cases i) In this case, any bounded solution u > ω is asymptotic to a T - periodic solution ω ˜ > ω . By hypothesis ω ˜ = ω and all solutions v of (1.1) with ω(0) ≤ v(0) ≤ u(0) are uniformly asymptotic to ω as t → +∞.
ii) If u(0) ∈ (ω(0), ω1 (0)), the sequence u(nT ) is either always increasing, or
always decreasing since a change in the monotonicity type would imply, by continuity, the existence of a T-periodic solution strictly between ω and ω1 . If u(nT ) is increasing, then u is asymptotic to ω1 at infinity, and due to ordering of the solutions in this case ω1 is L-asymptotically stable. In addition since u(nT ) − ω(nT ) tends to a positive limit no matter how close u(0) is from ω(0), it is clear that ω is R - unstable. If u(nT )
is decreasing,a similar argument shows that ω is R - asymptotically stable and ω1 is L - unstable. iii) It is clear that ω is R - stable since if ω(0) ≤ u(0) ≤ ωn (0) we have by (2.3) ku − ωk∞ ≤ kωn − ωk∞ ≤ C|ωn (0) − ω(0)| → 0 as n → ∞ 4
However if ωn+1 (0) ≤ u(0) ≤ ωn (0) we have u − ω ≥ ωn+1 − ω ≥ δn > 0 It follows easily that the only solution u ≥ ω asymptotic to ω as t → ∞ is ω itself.
3- The case of a finite number of periodic solutions. In this section we assume for simplicity that (1.1) has a finite number of Tperiodic solutions ω0 < ω1 ... < ωN This is the case for instance if f(t, u) = P (u) + h(t) where P is a polynomial, cf. e.g. Pliss [7] and Brull-Mawhin [3] To each T- periodic solution ω we attach a ”stability signature” σ(ω) = {ε(ω), η(ω)} ∈ {+, −}2 defined as follows ε(ω) = +
if
ω is L-stable;
ε(ω) = −
if
ω is L-unstable
η(ω) = +
if ω is R-stable;
η(ω) = −
if
ω is R-unstable
The following result is an easy consequence of Theorem 2.1 Proposition 3.1. For any j ∈ {0, 1...N − 1} we have ε(ωj+1 ) = −η(ωj ) Moreover 1) Either ω0 is asymptotically L-stable, or no solution u < ω0 is bounded from below as t goes to infinity. 2) Either ωN is asymptotically R-stable, or no solution u > ωN is bounded from above as t goes to infinity. 5
Remark 3.2. 1) All the situations are possible even in the autonomous case. For instance if f(t, u) = f (u) = C
Y (u − ωj )αj j
then near ωj we have f(u) ∼ cj (u − ωj )αj and depending on the parity of αj and the sign of cj we find all 4 possible values for X σ(ωj ). For u > ωN or u < ω0 the behavior depends on the degree d = αj and j
the sign of C. Proposition 3.1 somehow means that the general periodic case with a
C 1 function f is not really more complicated than the simple case of a polynomial autonomous ODE. 2) A related result, with a quite different technique and under different hypotheses, can be found in Ph. Holmes & D.C. Lewis [6], lemma 3 p. 235.
4- Quasi-autonomous problems with a small forcing. We now consider the equation u0 + g(u) = f (t)
(4.1)
where g ∈ C 1 (R) and f : R → R bounded. We set E = g−1 (0) and we assume ∀x ∈ E,
g0 (x) 6= 0
(4.2)
For all x ∈ E we set τ + (x) = inf{y > x,
g 0 (y)g 0 (x) ≤ 0}
with the convention τ + (x) = +∞ ⇐⇒ ∀y > x,
g0 (y)g 0 (x) > 0
(=⇒ E ∩ (x, +∞) = ∅)
and similarly τ − (x) = sup{y < x, 6
g 0 (y)g0 (x) ≤ 0}
with the convention τ − (x) = −∞ ⇐⇒ ∀y < x,
g 0 (y)g 0 (x) > 0 (=⇒ E ∩ (−∞, x) = ∅)
Finally we set ∀x ∈ E,
m(x) = min{|g(τ + (x))|, |g(τ − (x))|} ∈ [0, ∞]
with the obvious conventions τ + (x) = +∞ =⇒ g(τ + (x)) = lim g(y) ∈ [−∞, +∞] x→+∞
τ − (x) = −∞ =⇒ g(τ − (x)) = lim g(y) ∈ [−∞, +∞] x→−∞
It is clear by (4.2) that ∀x ∈ E, Theorem 4.1.
m(x) > 0
Assume f ∈ L∞ with kf k∞ < µ = inf {m(x)} x∈E
(4.3)
Then for each x ∈ E there exists one and only one bounded solution ωx with
range contained in [τ − (x), τ + (x)]. In addition if f is almost periodic, then ωx is almost periodic and satisfies the module containment property. In particular if f is T-periodic, is T- periodic. In addition if g0 (x) > 0, ωx is asymptotically stable, and if g 0 (x) < 0, ωx is unstable and backward asymptotically stable Proof. We proceed in three steps. Step 1: Uniqueness. Let x ∈ E and let u, v be two bounded solutions of (4.1)
with range contained in [τ − (x), τ + (x)]. Then u − v has a constant sign and since g is monotone on [τ − (x), τ + (x)] the equation
(u − v)0 + g(u) − g(v) = 0 shows that u−v is monotone and tends to a limit at +∞ and −∞. If one of these limits
is not 0, an easy compactness argument, since g is strictly monotone on [τ − (x), τ + (x)] shows that |g(u) − g(v)| ≥ α > 0 on a halfline. Then |(u − v)0 | is bounded away from 0 on a halfline, and it follows that u − v is unbounded, a contradiction. 7
Step 2: Existence. We use a combination of the method of invariant sets and the classical translation method of Amerio-Biroli and al. Assuming first g 0 (x) > 0 using (4.3) we can select a, b such that τ − (x) < a < b < τ + (x) and g(b) > kfk∞ , Then if u(t0 ) ∈ (a, b)
g(a) < −kfk∞
u0 (t) = f (t) − g(u(t))
is negative as u approaches b and positive as u approaches a and this implies u(t) ∈ (a, b) for all t ≤ t0 : the set [a, b] is positively invariant starting at any t0 . Now select any c ∈ (a, b) and consider the solution un of un (−n) = c,
∀t > −n,
u0n + g(un ) = f(t)
By Ascoli’s theorem there is a sequence nk for which unk converges to a limit u in C 1 (R). The limit u is a solution of (4.1) with range included in [a, b]. The case g 0 (x) < 0 can be handled in the same way by changing t to −t. At this stage the T-
periodic case is obvious by step 1 since both ωx and ωx (T + .) are solutions of (4.1). In the general almost periodic case we can apply Dafermos [4], Theorem 2.5 p. 46 to obtain that ωx is almost periodic and satisfies the module containment property. However in order to have a completely self contained proof we pursue with step 3. Step 3: Continuous dependence. In the construction of Step 2, the points a, b can be chosen independently of f assuming kf k∞ ≤ ρ < µ Since |g 0 | is uniformly bounded away from zero on [a, b], we obtain that the map f −→ ωx (f ) is Lipschitz continuous from the ball {kf k∞ ≤ ρ} to Cb (R). It is then completely
obvious, comparing the solution with its translates, to check that the solution ωx is almost periodic with the module containment property with respect to f . 8
Step 4: stability. Since |g 0 | is uniformly bounded away from zero on [a, b], it
is obvious to check that if g 0 (x) > 0 the linearized equation around ωx displays an
exponentially decaying system, and a classical argument gives that ωx is exponentially stable. For the case g 0 (x) < 0 it is sufficient to change t to−t. Theorem 4.2.
In addition to the hypotheses of Theorem 4.1, we assume kf k∞ < min{µ, inf |g(y)|} = m y∈K
with K = {y ∈ R,
∀x ∈ E,
y 6∈ (τ − (x), τ + (x))}
Then any (positively and negatively) recurrent bounded solution of (4.1) is one of the a.p. solutions given by Theorem 4.1 Proof. any solution u remaining in K satisfies |u0 | = |g(u) − f | ≥ |g(u)| − kf k∞ ≥ ε0 > 0 which is impossible if u is recurrent. Therefore if u is recurrent ∃x ∈ E, ∃t0 ∈ R,
u(t0 ) ∈ (τ − (x), τ + (x)) = J
Now if g0 (x) > 0, we have g(τ + (x)) > 0,
g(τ − (x)) < 0
and u0 (t) = f (t) − g(u(t)) is negative as u approaches τ + (x) and positive as u approaches τ − (x) . Therefore g0 (x) > 0 ⇒
∀t ≥ t0 ,
u(t) ∈ J
since u is positively recurrent this implies u(R) ⊂ [τ − (x), τ + (x)] In the same way if g0 (x) < 0, we obtain ∀t ≤ t0 ,
u(t) ∈ J 9
and since u is negatively recurrent this implies u(R) ⊂ [τ − (x), τ + (x)] The conclusion follows from Theorem 4.1. Corollary 4.3.
In addition to the hypotheses of Theorem 4.2, assume #E < ∞;
lim inf |g(u)| > 0 |u|→∞
Then for each f small enough there exist exacty N = #E almost periodic solutions of (4.1). In addition, each almost periodic solution u has exactly the same stability signature (+, +) or (−, −) as the unique x ∈ E for which u(R) ⊂ [τ − (x), τ + (x)] Remark 4.4. One should not confuse the uniqueness result of Theorem 4.1, valid for any bounded solution confined to the (forward or backward) attraction basin of ωx with the uniqueness result of Theorem 4.2 which is valid without restriction on the range, but only for bounded recurrent trajectories. For instance any function of the form u(t) = √
1 1 + ce−2t
with c > 0 is a (nonrecurrent) bounded solution of the equation u0 + u3 − u = 0 but the point is that u crosses the value τ − (1) = τ + (0) =
√1 3
and consequently does
not fulfill the condition of Theorem 4.1.
5- Examples and counterexamples. Example 5.1. Let us consider the simple case u0 + u3 − λu = f(t) which corresponds to problem (4.1) with g(u) = u3 − λu 10
(5.1)
In this case
√ √ E = {− λ, 0, + λ}
and we have √ τ (0) = τ ( λ) = +
−
and
r
r √ λ − τ (− λ) = τ (0) = − 3
λ ; 3
+
√ τ + ( λ) = +∞;
√ τ − (− λ) = −∞
q ¯ q ¯³q ´3 ¯ λ λ¯ 2 Hence K = ∅ and m = µ = ¯ − λ 3 ¯ = 3 λ λ3 . Therefore, by Theorem 3
4.2, for any f almost periodic such that
kfk∞
2 < λ 3
r
λ 3
equation (5.1) has exactly 3 almost periodic solutions (ω− , ω0 , ω+ ) which are Tperiodic if f is T-periodic. In addition we have r r λ λ ω− ≤ − ≤ ω0 ≤ ≤ ω+ 3 3 which means that all 3 solutions remain in one of the 3 ”stable” regions . Moreover (ω− , ω+ ) are (positively) stable and ω0 is (positively) right and left unstable. Finally the condition on kfk∞ is easily seen to be sharp when f is constant. Example 5.2. Let us consider (5.1) with λ = 1. The function 1 u(t) = √ + ε sin ωt 3 is a solution of 1 1 u0 + u3 − u = ε cos ωt + ( √ + ε sin ωt)3 − √ + ε sin ωt =: f(t) 3 3 with
1 1 2 kf k∞ ≤ εω + | √ ( − 1)| + Cε = (C + ω)ε + √ 3 3 3 3
The solution u is
2π ω
periodic and its set of values is not contained in any of the 3
invariant intervals. More precisely the zones 1 {u > √ } 3
1 and {|u| < √ } 3 11
are both positively and negatively unstable, no matter how close kfk∞ is from the critical number
2 √ . 3 3
Finally the condition on kf k∞ in example 5.1 is sharp not only
as a condition to have exactly 3 periodic solutions, but also for all periodic solutions to take values in one of the attraction basins of the equilibria. Example 5.3. Let us consider the equation u0 + u4 − 2u2 − 3 = f(t)
(5.2)
which corresponds to problem (4.1) with g(u) = u4 − 2u2 − 3 = (u2 − 1)2 − 4 In this case
and we have
and
√ √ E = {− 3, + 3} √ τ − ( 3) = 1;
√ τ + (− 3) = −1
√ τ + ( 3) = +∞;
√ τ − (− 3) = −∞
Hence K = [−1, 1] and it is easily seen that m = 4, by Theorem 4.1, for any f almost periodic such that
µ = |g(0)| = 3. Therefore,
kf k∞ < 4 equation (5.2) has at least 2 almost periodic solutions (ω− , ω+ ) which satisfy ω− ≤ −1;
ω+ ≥ 1
and are T-periodic if f is T-periodic. In addition if kf k∞ < 3 there is no other almost periodic solution. However if f(t) ≡ c with −4 < c < −3 12
the equation for stationary solutions is (u2 − 1)2 = 4 + c which has four solutions. Theorem 4.2 gives again an optimal condition.
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