Resonance and nonlinearity: a survey
Наведено огляд останніх результатів щодо нерезонансних і резонансних періодично збуджуваних нелінійних осциляторів: існування періодичних необмежених або обмежених розв'язків для обмежених нелінійних збурень лінійних та кусково-лінійних осциляторів, а також деяких класів плоских гамільтонових с...
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irk-123456789-55252020-11-05T20:18:12Z Resonance and nonlinearity: a survey Mawhin, J. Статті Наведено огляд останніх результатів щодо нерезонансних і резонансних періодично збуджуваних нелінійних осциляторів: існування періодичних необмежених або обмежених розв'язків для обмежених нелінійних збурень лінійних та кусково-лінійних осциляторів, а також деяких класів плоских гамільтонових систем. This paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of linear and of piecewise linear oscillators, as well as of some classes of planar Hamiltonian systems. 2007 Article Resonance and nonlinearity: (A survey) / J. Mawhin // Укр. мат. журн. — 2007. — Т. 59, № 2. — С. 190-205. — Бібліогр.: 61 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/5525 517.9 en Інститут математики НАН України |
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Статті Статті Mawhin, J. Resonance and nonlinearity: a survey |
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Наведено огляд останніх результатів щодо нерезонансних і резонансних періодично збуджуваних нелінійних осциляторів: існування періодичних необмежених або обмежених розв'язків для обмежених нелінійних збурень лінійних та кусково-лінійних осциляторів, а також деяких класів плоских гамільтонових систем. |
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Mawhin, J. |
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Mawhin, J. |
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Mawhin, J. |
| title |
Resonance and nonlinearity: a survey |
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Resonance and nonlinearity: a survey |
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Resonance and nonlinearity: a survey |
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Resonance and nonlinearity: a survey |
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Resonance and nonlinearity: a survey |
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resonance and nonlinearity: a survey |
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Інститут математики НАН України |
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2007 |
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Resonance and nonlinearity: (A survey) / J. Mawhin // Укр. мат. журн. — 2007. — Т. 59, № 2. — С. 190-205. — Бібліогр.: 61 назв. — англ. |
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2025-07-02T08:37:04Z |
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UDC 517.9
J. Mawhin (Univ. Catholique de Louvain, Belgium)
RESONANCE AND NONLINEARITY: A SURVEY
REZONANS I NELINIJNIST\: OHLQD
This paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This
includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of
linear and of piecewise linear oscillators, as well as of some classes of planar Hamiltonian systems.
Navedeno ohlqd ostannix rezul\tativ wodo nerezonansnyx i rezonansnyx periodyçno zbudΩuvanyx ne-
linijnyx oscylqtoriv: isnuvannq periodyçnyx neobmeΩenyx abo obmeΩenyx rozv’qzkivdlq obmeΩenyx
nelinijnyx zburen\ linijnyx ta kuskovo-linijnyx oscylqtoriv, a takoΩ deqkyx klasiv ploskyx hamil\-
tonovyx system.
1. Introduction. The concept of resonance in mechanics and physics is defined as fol-
lows: a vibrating system excited by a periodic force, whose frequency is equal or close
to one natural frequency of vibration of the system, exhibits oscillations of increasing
amplitude. According to Richard Feynmann,
If we look in the Physical Reviews, every issue has a resonance curve.
Well known examples are a child pushing another one on a swing, a cristal glass bro-
ken by the tune of a singing diva, Jericho’s wall falling under the sound of trumpets, a
radio or TV antenna and electromagnetic waves, detecting (weak) nuclear magnetism
through nuclear magnetic resonance, a bridge in Tours collapsing when crossed by a
troup of soldiers which did not break the step, Tacoma bridge exhibiting large oscillations
and destruction under the action of the wind, and even the tides: according to Charles
Ed. Guillaume,
the ocean is a swing that Moon pushes in cadence.
We first recall the mathematical theory of resonance for a 2π-periodically forced linear
oscillator
ẍ + λx = e(t).
In this case, nonresonance can be characterized by the existence of a (unique) 2π-periodic,
or by the boundedness of all solutions, for all forcings e(t), and resonance can be charac-
terized by the unboundedness of all solutions.
The problem is to see which part of this picture is preserved or destroyed in the case
of a boundedly perturbed forced linear oscillator
ẍ + λx = f(x) + e(t),
of a forced asymmetric oscillator
ẍ + µx+ − νx− = e(t),
of a boundedly perturbed forced asymmetric oscillator
ẍ + µx+ − νx− = f(x) + e(t),
of a forced positive homogeneous planar Hamiltonian system
Ju̇ = ∇H(u) + p(t),
c© J. MAWHIN, 2007
190 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
RESONANCE AND NONLINEARITY: A SURVEY 191
and of a boundedly perturbed forced positive homogeneous planar Hamiltonian system
Ju̇ = ∇H(u) + g(u) + p(t).
Giving answers to those questions require the use of various deep methods of nonlinear
analysis and dynamical systems.
2. Forced linear oscillator. 2.1. Free linear oscillator. It is well known that the free
linear oscillator
ẍ + λx = 0, λ > 0, (1)
has the solutions
x(t) = A sin
√
λ(t + θ), A ≥ 0, θ ∈ R,
which are all periodic with the same period 2π/
√
λ.
It follows from the energy integral that, in the phase place (x, ẋ), the corresponding
orbits are given by the family of ellipses centered at (0, 0)
ẋ2
2
+ λ
x2
2
= E, E ≥ 0,
so that the origin is an isochronous center with period 2π/
√
λ.
Given a fixed period, say 2π, the eigenvalues of (1) are the λ > 0 such that equation
(2) has nontrivial 2π-periodic solutions. They are given by n2, n ∈ N0.
2.2. Forced linear oscillator: nonresonance. Consider the forced linear oscillator
ẍ + λx = e(t) λ > 0, (2)
where e(t) is continuous and 2π-periodic. Its spectrum is the set of λ ∈ R such that
equation (2) has a unique 2π-periodic solution for each 2π-periodic continuous e(t). It
coincides with the set of eigenvalues of equation (1), because of the following classical
elementary result:
Proposition 1. If λ �∈ {n2 : n ∈ N0}, then, for any continuous 2π-periodic e(t),
equation (2):
1) has a (unique) 2π-periodic solution;
2) has all its solutions bounded over R (indeed quasiperiodic).
2.3. Forced linear oscillator: resonance. Any continuous and 2π-periodic function
e(t) has the Fourier expansion
e(t) ∼ e0
2
+
∞∑
n=1
(en cosnt + fn sinnt) =
+∞∑
n=−∞
êne
int,
where
en :=
1
π
2π∫
0
e(t) cosnt dt, fn :=
1
π
2π∫
0
e(t) sinnt dt,
ên :=
1
2π
2π∫
0
e(t)e−int dt,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
192 J. MAWHIN
so that
ên = ê−n, |ên| = |ê−n| =
1
2
(
e2
n + f2
n
)1/2
.
The following result is also well known:
Proposition 2. All solutions of equation
ẍ + n2x = e(t), n ∈ N0, (3)
are
1) unbounded if and only if |en| + |fn| �= 0;
2) 2π-periodic if and only if |en| + |fn| = 0.
Let us formulate this result in another way which makes more transparent the com-
parison with further results for nonlinear oscillators. Define the resonance function Σn
e
for (3) by
Σn
e (θ) :=
2π∫
0
e(t) sinn(t + θ) dt = π[fn cosnθ + en sinnθ], θ ∈ R. (4)
Σn
e is (2π/n)-periodic, identically zero if and only if |en| + |fn| = 0, and changes sign
(with two simple zeros) in [0, 2π/n[ if and only if |en| + |fn| �= 0. Hence Proposition 2
trivially implies the following proposition.
Proposition 3. All solutions of equation (3)
1) are unbounded if and only if Σn
e changes sign;
2) 2π-periodic if and only if Σn
e ≡ 0.
Notice the dual mathematical formulation of the resonance phenomenon for the peri-
odically forced linear oscillator as an alternative between the 2π-periodic or unbounded
character of all responses, depending upon the nature of the forcing term.
Let us finally observe that, for the forced linear operator, either all solutions are
bounded over R, or all solutions are unbounded over R.
3. Perturbed forced linear oscillator. 3.1. Nonresonance. We now consider the
perturbed forced linear oscillator
ẍ + λx = f(x) + e(t), (5)
where λ > 0, f : R → R is continuous, and the unperturbed oscillator is nonresonant.
Theorem 1. If λ �∈ {n2 : n ∈ N0} and f is bounded, equation (5) has a 2π-periodic
solution for any continuous and 2π-periodic forcing e(t).
The proof is a simple consequence of Schauder’s fixed point theorem after reduction
of the problem to an integral equation using Green’s function of the linear part.
So the existence of a 2π-periodic response survives despite of the nonlinear term, but
uniqueness may be lost because of the nonlinear term. For example, the equation
ẍ + (1/2)x = arctanx
has three 2π-periodic (constant) solutions.
A delicate question in this case is to know if, like in the unperturbed case (f ≡
≡ 0), all solutions of (5) are bounded (in C1-norm) over R. This is called Littlewood’s
problem because it was prompted by questions of Littlewood in [32]. The first positive
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
RESONANCE AND NONLINEARITY: A SURVEY 193
answer was given by Morris [40] in 1976 for the superlinear problem ẍ + 2x3 = e(t),
and many generalizations have been given for this superlinear case. The idea consists
in transforming the equation, outside of a large ball of the phase plane (x, x′), into a
perturbation of an integrable Hamiltonian system, and to apply Arnold – Kolmogorov –
Moser’s twist theorem (see, e.g., [41]) to its Poincaré’s map, which is shown to be closed
to a twist map outside this ball. The idea was successfully applied by Bin Liu [33] for
some class of unbounded perturbations f in equation (5) and, in a subsequent paper [34],
to some class of smooth bounded perturbations f and for all smooth 2π-periodic forcings
e(t). See also [30], who assumes f and e smooth and odd, limx→+∞ f(x) �= 0, some
growth conditions at infinity upon the sixth derivative of f and some growth condition
upon the first six derivatives of the indefinite integral of f.
3.2. Resonance: periodic solutions. The study of the perturbed resonant linear
oscillator
ẍ + n2x = f(x) + e(t), (6)
where n ∈ N0 and f : R → R is continuous is more delicate. As shown by the unper-
turbed case, some restriction must be made upon e(t) and f.
The following result was proved in 1969 by Lazer and Leach [28]:
Theorem 2. If f : R → R is bounded and
f(±∞) := lim
u→±∞
f(u) (7)
exist, equation (6) has a 2π-periodic solution for each continuous and 2π-periodic e(t)
such that
|ên| <
1
π
∣∣∣f(+∞) − f(−∞)
∣∣∣. (8)
Lazer – Leach’s original proof was based upon a quite involved argument using Lya-
punov – Schmidt’s decomposition into the subspaces of constant functions and of func-
tions with mean value zero, and Schauder’s fixed point theorem. Simpler subsequent
proofs have used topological degree arguments (see, e.g., [25]).
Condition (8), nowadays called a Landesman – Lazer condition (and not a Lazer –
Leach condition), because of its (more popular) subsequent version for semilinear elliptic
Dirichlet problems, can be written equivalently
(e2
n + f2
n)1/2 <
2
π
∣∣∣f(+∞) − f(−∞)
∣∣∣,
and hence excludes the case where f ≡ 0.
Condition (8) is necessary for the class of f such that
min{f(−∞), f(+∞)} < f(u) < max
{
f(−∞), f(+∞)
}
, u ∈ R.
For example, equation
ẍ + n2x = a arctanx + e(t), a �= 0,
has a 2π-periodic solution if and only if |ên| < |a|. This shows that the presence of the
nonlinearity extends the range of the nonlinear differential operator
d2
dt2
+ n2I + f(·) in
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
194 J. MAWHIN
the space C2π of continuous and 2π-periodic functions, from the co-dimension two vector
subspace
{
e ∈ C2π(R) : ên = 0
}
, to the open strip
{
e ∈ C2π(R) : |ên| < |a|
}
.
Define the resonance function Λn
e,f for (6) by
Λn
f,e(θ) := 2 [f(+∞) − f(−∞)] + Σn
e (θ), θ ∈ R. (9)
Landesman – Lazer’s condition can trivially be written in the following equivalent form:
Theorem 3. Equation (6) has a 2π-periodic solution if its resonance function Λn
f,e
does not vanish.
Notice that, when f(+∞) �= f(−∞), Λn
f,e can have a constant sign for a nonzero Σn
e .
Again, in the case where a 2π-periodic solution exists, the question of the bounded-
ness over R (in C1-norm) of all solutions of (6) is a delicate one. Ortega [42] has shown in
1999, using similar techniques as for the nonresonance case, that all solutions are bounded
in R in the C1-norm under condition (8) for the special case where f(u) = −L if x ≥ 1,
−Lx if −1 ≤ x ≤ 1, L if x ≤ −1, and e is of class C5. Bin Liu [36] has improved the
same year Ortega’s result as follows:
Theorem 4. If f ∈ C6(R), lim|x|→∞ x6f (6)(x) = 0, e ∈ C7(R), the limits (7)
exist, Landesman – Lazer’s condition
|ên| <
[
f(−∞) − f(+∞)
]
and inequality
f(+∞) ≤ f(x) ≤ f(−∞), x ∈ R,
holds, then supt∈R(|x(t)| + |x′(t)|) for all solutions x of (6).
See also [26].
3.3. Resonance: unbounded solutions. The following nonexistence result for 2π-
periodic solutions of (6) given in the same paper [28] of Lazer and Leach has been less
noticed:
Theorem 5. Equation (6) has no 2π-periodic solution if
|ên| ≥
1
π
[
sup
R
f − inf
R
f
]
> 0. (10)
Inequality (10) looks like a negation of Landesman – Lazer condition (8), except that
supR f and infR f replace the limits of f at ±∞.
Let us now recall a classical result of Massera [39]:
Lemma 1. Equation
ẍ = h(t, x, ẋ)
with h 2π-periodic in t, has a 2π-periodic solution if its Cauchy problem is uniquely
globally solvable, and if it has a solution bounded in the future or in the past.
Combining Theorem 5 with Lemma 1, we obtain the following result, which can
essentially be traced to Seifert [48]:
Corollary 1. Equation (6) with f locally Lipschitzian has all solutions unbounded
in ]−∞, 0] and in [0,+∞ [ if condition (10) holds.
Condition (10) can be sharpened when f(±∞) exist in a generalized sense introduced
in 1996 by Alonso – Ortega [1]. For f : R → R continuous and bounded, define
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
RESONANCE AND NONLINEARITY: A SURVEY 195
F (s) :=
s∫
0
f(v) dv, F± := lim
s→±∞
F (s)
s
, (11)
when those limits exist. Using l’Hospital’s rule, it is easy to see that F± = f(±∞) if
f(±∞) exist, but, for f(s) = a arctan s + b sin s, F± = ±aπ
2
although f(±∞) do not
exist.
When those generalized limits exist, the resonance function Ωn
f,e for (6) is defined by
Ωn
f,e(θ) := 2
[
F+ − F−]
+ Σn
e (θ), θ ∈ R,
and reduces to Λn
f,e when f(±∞) exist.
Alonso and Ortega [1] have generalized as follows Corollary 1.
Theorem 6. If f is locally Lipschitzian and Ωn
f,e(θ) changes sign, there exists R >
> 0 such that for any solution of (6) with x2(0) + ẋ2(0) > R, one has either
lim
t→+∞
[
x2(t) + ẋ2(t)
]
= +∞
or
lim
t→−∞
[
x2(t) + ẋ2(t)
]
= +∞.
The proof uses properties of Poincaré’s map which associates to each (x0, ẋ0) ∈ R
2
the point (x(2π;x0, ẋ0), ẋ(2π;x0, ẋ0)), where x(t;x0, ẋ0) denotes the solution of (6)
with initial conditions x0 and ẋ0.
The assumption upon Ωn
f,e in Theorem 6 is of course equivalent to
|ên| >
1
π
∣∣F+ − F−∣∣ .
The example
ẍ + x = − sinx + sin(sin t)
which satisfies the conditions of Theorem 6 with n = 1 and has the 2π-periodic solu-
tion x(t) = sin t shows that, in contrast to the linear case, 2π-periodic and unbounded
solutions may coexist in a perturbed resonant oscillator.
4. Forced asymmetric oscillator. 4.1. Free asymmetric oscillator. The free linear
oscillator can be generalized to the free asymmetric oscillator
ẍ + µx+ − νx− = 0, µ > 0, ν > 0, (12)
where x+ := max(x, 0), x− := max(−x, 0). The linear restoring force is replaced by a
piecewise linear one.
It easily follows from the energy integral
ẋ2
2
+ µ
(x+)2
2
+ ν
(x−)2
2
= E, E ≥ 0,
that the orbits of (12) in the phase plane (x, ẋ) are egg-shaped closed curves around the
origin made by half-ellipses of equation
ẋ2
2
+ µ
x2
2
= E
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
196 J. MAWHIN
in the right-hand half plane and
ẋ2
2
+ ν
x2
2
= E
in the left-hand half plane. Hence, the origin is an isochronous center with period π
(
1√
µ
+
+
1√
ν
)
.
A generalized concept of eigenvalue has been introduced by Fučik [23] and by Dancer
[12] for the asymmetric oscillator with various boundary conditions. A Fučik’s eigenvalue
for (12) is any (µ, ν) ∈ R
2 such that equation (12) has a nontrivial 2π-periodic solution.
It is easy to see that the set σF of positive Fučik eigenvalues is given by the following
family of hyperbolic-like curves (Fučik’s curves)
⋃
n∈N0
{
(µ, ν) ∈ R
2
+ :
1√
µ
+
1√
ν
=
2
n
}
:=
⋃
n∈N0
σn
F .
Notice that σn
F intersects the diagonal in the (µ, ν)-plane at the point (n2, n2). If one
defines the corresponding Fučik’s eigenfunctions by
σn(t) :=
1√
µ
sin (
√
µ t) for t ∈
[
0,
π√
µ
]
,
− 1√
ν
sin
(√
ν
(
t− π√
µ
))
for t ∈
[
π√
µ
,
2π
n
]
,
the nontrivial solutions of (12) are given by
x(t) = Aσn(t + θ), A > 0, θ ∈ R,
and are (2π/n)-periodic.
4.2. Nonresonance: periodic solutions. Consider now the forced asymmetric oscil-
lator
ẍ + µx+ − νx− = e(t), µ > 0, ν > 0. (13)
The following result is essentially due to Fučik [23]:
Theorem 7. If (µ, ν) �∈ σF , equation (13) has a 2π-periodic solution for any e(t).
The proof of the first part is based upon Leray – Schauder degree, using a homotopy
which deforms (13) into a linear problem of the form ẍ + λx = 0, with (λ, λ) located
between the same two Fučik curves as (µ, ν). It is also a trivial consequence of Corollary 8
in [8].
When (µ, ν) ∈ σn
F for some n ∈ N0, Dancer [12] has given an example of e(t) such
that equation (13) has no 2π-periodic solution.
Because of this result, the set σF is also called the positive Fučik’s spectrum of (12).
The boundedness over R of all solutions of (13) has been also considered. Ortega
[42] has shown in 1993 that (13) has all its solutions bounded (in C1-norm) over R if
e(t) = 1 + p(t) is of class C4 and p is sufficiently small in C4-norm.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
RESONANCE AND NONLINEARITY: A SURVEY 197
4.3. Resonance: periodic solutions. If (µ, ν) ∈ σn
F for some n ∈ N0, Dancer [12]
has introduced in 1977 the resonance function ∆n
e for (13) defined by
∆n
e (θ) := n
2π∫
0
e(t)σn(t + θ) dt, θ ∈ R. (14)
∆n
e is (2π/n)-periodic and reduces to Σn
e for µ = ν = n2. Because of the richer harmonic
structure of σ(t) with respect to sinnt, ∆n
e may have an arbitrary large (even) number of
zeros in [0, 2π/n[.
The following existence result for 2π-periodic solutions of a forced resonant asym-
metric oscillator is due to Dancer [12].
Theorem 8. If (µ, ν) ∈ σn
F for some n ∈ N0, and ∆n
e does not vanish, equa-
tion (13) has a 2π-periodic solution.
The proof is based upon some perturbation technique and homotopy to a linear prob-
lem with Leray – Schauder degree one.
A further existence condition has been obtained in 1998 by Fabry and Fonda [15].
Theorem 9. If (µ, ν) ∈ σn
F for some n ∈ N0, and ∆n
e has more than two zeros in
[0, 2π/n[, all simple, equation (13) has a 2π-periodic solution.
The proof is based upon the fact that Brouwer’s degree of the associated Poincaré’s
map minus identity on large balls is shown to be equal to 1 − z, with 2z the number of
zeros of ∆n
e in [0, 2π/n[.
For further results see [57].
We will see in the next section that unbouded solutions exist when ∆n
e changes sign.
The question of boundedness over R of all solutions when ∆n
e does not vanish has been
considered by in 1999 by Bin Liu [35], who has proved, using twist maps techniques, the
following result:
Theorem 10. If e is of class C6, (µ, ν) ∈ σn
F for some n ∈ N0, and ∆n
e does not
vanish, then all solutions of (13) are bounded over R in C1-norm.
A similar result holds when
1√
µ
+
1√
ν
= 2
m
n
for some relativitely prime positive integers. For related results, see [5].
4.4. Resonance: unbounded solutions. Alonso and Ortega [2] have obtained in
1998 the following sufficient condition for the existence of unbounded solutions.
Theorem 11. Let (µ, ν) ∈ σn
F for some n ∈ N0. If ∆n
e has zeros, all simple, there
exists R > 0 such that for any solution x(t) of (13) with x2(t0) + ẋ2(t0) > R for some
t0 ∈ R, one has either
lim
t→+∞
[x2(t) + ẋ2(t)] = +∞
or
lim
t→−∞
[x2(t) + ẋ2(t)] = +∞.
The proof is based on study of the dynamics of a class of planar mappings associated
to Poincaré’s map.
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198 J. MAWHIN
Alonso and Ortega have also shown that 2π-periodic and unbounded solutions may
coexist in (13), e.g. for infinitely many r in equation
ẍ + µx+ − νx− = cos rt.
For further results see [55, 56].
5. Perturbed forced asymmetric oscillator. 5.1. Nonresonance. Consider the
perturbed forced asymmetric oscillator
ẍ + µx+ − νx− = f(x) + e(t), µ > 0, ν > 0, (15)
where f : R → R is continuous and e(t) is continuous and 2π-periodic.
Theorem 12. If (µ, ν) �∈ σF and f is bounded, equation (15) has a 2π-periodic
solution for any e(t).
The proof of this result is an application of Leray – Schauder degree. For example, it
is a consequence of Corollary 8 in [8].
So Fučik’s existence result for g ≡ 0 survives despite of the nonlinear perturbation g.
5.2. Resonance: periodic solutions. In 2000, Fabry and Mawhin [18] have ex-
tended Landesman – Lazer’s and Fabry – Fonda’s existence conditions to the resonant per-
turbed forced asymmetric oscillator, in terms of the resonance function Ψn
f,e for (15) de-
fined by
Ψn
f,e(θ) := 2n2
[
F+
µ
− F−
ν
]
+ ∆n
e (θ), θ ∈ R, (16)
where F± is defined in (11). Ψn
f,e is 2π/n-periodic and reduces to Ωn
f,e for µ = ν = n2.
Theorem 13. If (µ, ν) ∈ σn
F for some n ∈ N0, f is bounded and locally Lip-
schitzian, F± exist, and Ψn
f,e has no zero or more than two zeros in
[
0,
2π
n
[
, all simple,
then equation (15) has a 2π-periodic solution.
The proof is based upon the fact that if 2z denotes the number of zeros of Ψn
f,e, the
Brouwer degree of Poincaré’s map minus identity is shown to be equal to 1 − z.
A basic ingredient for this proof and the one of the results of next subsection is the use
of the averaging method [3] to compare the solutions of large amplitude of (15) written in
the equivalent form
ρ̇ε = εf
(ρε
ε
σn(t + θ)
)
σ̇n(t + θ) + εe(t)σ̇n(t + θ),
θ̇ = − ε
ρε
f
(ρε
ε
σn(t + θ)
)
σn(t + θ) − ε
ρε
e(t)σn(t + θ),
(17)
through the change of unknowns
x(t) =
ρε(t)
ε
σn(t + θ(t)), ẋ(t) =
ρε(t)
ε
σ̇n(t + θ(t)),
to the averaged equation over [0, 2π] of an asymptotic version of system (17), which
happens to be
˙̃ρ =
ε
2nπ
(
Ψn
f,e
)′ (θ̃), ˙̃
θ = − ε
2nπρ̃
Ψn
f,e(θ̃).
For further related results, see [11, 49, 4, 6].
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RESONANCE AND NONLINEARITY: A SURVEY 199
The existence of infinitely many subharmonic solutions has also been obtained in [18].
For other results based upon twist maps techniques, see [50, 51, 7]. The boundedness of
all solutions of equation (15) has been recently considered by Xiaojing Yang [60].
5.3. Resonance: unbounded solutions. Fabry and Mawhin [18] have also extended
Alonso – Ortega’s result about unbounded solutions to the resonant perturbed forced asym-
metric oscillator.
Theorem 14. If (µ, ν) ∈ σn
F for some n ∈ N0, f is bounded and locally Lip-
schitzian, F± exist, and Ψn
f,e has zeros in
[
0,
2π
n
[
, all simple, any solution x(t) =
= ρ(t)σn(t + θ(t)) of (15) with ρ(0) sufficiently large is unbounded either in the past
or in the future.
The proof is a delicate application of the averaging method indicated above, together
with Riemann – Lebesgue type theorems for oscillatory integrals.
6. Forced planar Hamiltonian systems. 6.1. Free planar Hamiltonian systems.
The free linear and asymmetric oscillators are special cases of free planar Hamiltonian
systems of the form
Ju̇ = ∇H(u), (18)
where J =
(
0 −1
1 0
)
, H : R
2 → R is positive and positively homogeneous of degree
two, and ∇H : R
2 → R
2 is continuous.
Because of the energy integral H(u) = E (E ≥ 0), the orbits of the solutions of
(18) in the phase plane R
2 are closed curves surrounding the origin, and, because of the
positive homogeneity of degree two of H, the corresponding solutions all have the same
minimal period τ. Hence the origin is an isochronous center.
6.2. Nonresonance: periodic solutions. We now consider the forced planar Hamil-
tonian system
Ju̇ = ∇H(u) + p(t), (19)
where J and H are like above and p : R → R
2 is 2π-periodic and continuous.
The following results were proved by Fonda [20] in 2004.
Theorem 15. If H : R
2 → R is positive and positively homogeneous of degree two,
∇H : R
2 → R
2 is continuous, and
2π
τ
�∈ N, system (19) has a 2π-periodic solution for
any p(t).
The proof uses Corollary 8 of [8].
Theorem 16. If H : R
2 → R is positive and positively homogeneous of degree two,
∇H : R
2 → R
2 is locally Lipschitzian and
2π
τ
�∈ N, there exists p(t) for which all
solutions of system (19) are unbounded, namely limt→+∞ ‖u(t)‖ = +∞.
The proof consists in constructing such a function p by a technique generalizing ex-
amples of Dancer [13] and of Ortega [47] for second order equations.
6.3. Resonance: periodic solutions. When
2π
τ
= n ∈ N0, Fonda [20] has intro-
duced in 2004 the resonance function Φn
p : R
2 → R of (19) defined by
Φn
p (θ) =
2π∫
0
〈
p(t)|ϕn(t + θ)
〉
dt, θ ∈ R, (20)
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200 J. MAWHIN
where ϕn is a fixed solution of (18) such that H(ϕn(t)) =
1
2
. Notice that Φn
p is (2π/n)-
periodic.
Fonda [20] has also proved the following result.
Theorem 17. If H : R
2 → R is positive and positively homogeneous of degree two,
∇H : R
2 → R
2 is locally Lipschitzian,
2π
τ
= n ∈ N0, and Φn
p (θ) never changes sign or
has at least four zeros in [0, τ [ , all simple, then system (19) has a 2π-periodic solution.
6.4. Resonance: unbounded solutions. The existence of unbounded solutions has
also been proved by Fonda [20].
Theorem 18. If H : R
2 → R is positive and positively homogeneous of degree two,
∇H : R
2 → R
2 is locally Lipschitzian,
2π
τ
= n ∈ N0, and Φn
p (θ) has zeros in [0, τ [ , all
simple, then all solutions of system (19) with sufficiently large amplitude are unbounded,
either in the past, or in the future.
The proof applies a result of Alonso – Ortega [2] to the equivalent system written in
generalized polar coordinates associated to the solutions of (18).
Comparing Theorems 17 and 18, one sees that, again, periodic and unbounded solu-
tions may coexist.
7. Perturbed forced planar Hamiltonian systems. 7.1. Nonresonance. We now
consider the perturbed forced planar Hamiltonian system
Ju̇ = ∇H(u) + g(u) + p(t), (21)
where g : R
2 → R
2 is continuous and p : R → R
2 is continuous and 2π-periodic.
Theorem 19. If 2π/τ �∈ N0, H : R
2 → R is of class C1, positive and positively
homogeneous of degree two, g : R
2 → R
2 is bounded, then system (21) has a 2π-peridoic
solution for any p(t).
The proof uses Corollary 8 of [8].
7.2. Resonance: periodic solutions. In order to generalize Landesman – Lazer’s
and Fabry – Fonda’s conditions to system (21), Fonda and Mawhin [22] have introduced
in 2006 the following class of bounded nonlinear perturbations g, which extends, in our
setting, systems of Lur’e type in control theory [38], and contains, as we will see, the
special case of second order differential equations with separated nonlinearities.
Assume that there exist directions
0 ≤ ϑ1 < ϑ2 < ... < ϑm < 2π,
and locally Lipschitzian functions gk : R → R
2 such that
g(u) =
m∑
k=1
gk
(
〈u|eiϑk〉
)
. (22)
If τ =
2π
n
for some n ∈ N0, and if
Gk(x) :=
x∫
0
gk(s) ds, G±
k := lim
s→±∞
Gk(s)
s
, 1 ≤ k ≤ m, (23)
exist, Fonda and Mawhin have defined as follows the resonance mapping Γg,p : R
2 → R
2
for (21). Let
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RESONANCE AND NONLINEARITY: A SURVEY 201
A+
k := {t ∈ [0, τ ] : 〈ϕn(t)|eiϑk〉 > 0},
A−
k := {t ∈ [0, τ ] : 〈ϕn(t)|eiϑk〉 < 0},
κg
1 := n
m∑
k=1
(〈
G+
k
∣∣∣∣∣
∫
A+
k
ϕn
〉
+
〈
G−
k
∣∣∣∣∣
∫
A−
k
ϕn
〉)
,
κg
2 := n
m∑
k=1
(〈
G+
k
∣∣∣∣∣
∫
A+
k
ϕ̇n
〉
+
〈
G−
k
∣∣∣∣∣
∫
A−
k
ϕ̇n
〉)
,
where ϕn is, like above, a fixed solution of (18) such that H(ϕn(t)) =
1
2
. Then Γg,p is
defined by
Γg,p
1 (θ) := −κg
1 − Φn
p (θ), Γg,p
2 (θ) := κg
2 + (Φn
p )′(θ). (24)
In the special case of equation
ẍ + µx+ − νx− = f(x) + e(t)
written as a Hamiltonian system, Γf,e reduces to
(Ψn
f,e,−(Ψn
e )′) = (Ψn
f,e,−(∆n
e )′).
The following results of Landesman – Lazer’s and Fabry – Fonda’s types hold.
Theorem 20. If H : R
2 → R is positive and positively homogeneous of degree
two, ∇H : R
2 → R
2 is locally Lipschitzian,
2π
τ
= n ∈ N0, g : R
2 → R
2 is locally
Lipschitzian, bounded, of type (22), the sets{
u ∈ R
2 : ‖u‖ = 1 and
∇H(u)
‖∇H(u)‖ = ±eiϑk
}
, 1 ≤ k ≤ m,
have only isolated points, limits (23) exist, and if
1) either Γg,p
1 has constant sign or
2) Γg,p does not vanish and either:
a) Γg,p
2 has constant sign or
b) Γg,p
2 changes sign more than twice on the zeros of Γg,p
1 ,
then system (21) has a 2π-periodic solution.
Like in paper [18], the joint main ingredient of the proof of this result and of next
one is the following. By the change of variable v = δu, for some δ > 0, equation (21)
becomes
Jv̇ = ∇H(v) + δg
(v
δ
)
+ δp(t) . (25)
If v(t) is a solution of (25) with starting point v(0) �= 0, we can write
v(t) = r(t)ϕn(t + θ(t)),
with r(0) > 0. As long as r(t) > 0, the functions θ(t) and r(t) are of class C1 and satisfy
θ′ =
δ
r
〈
g
(r
δ
ϕn(t + θ)
)
+ p(t)
∣∣∣ϕn(t + θ)
〉
, (26)
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202 J. MAWHIN
r′ = −δ
〈
g
(r
δ
ϕn(t + θ)
)
+ p(t)
∣∣∣ϕ̇n(t + θ)
〉
. (27)
Denote by (θ(t; θ0, r0; δ), r(t; θ0, r0; δ)) the solution of (26) with starting point
θ(0; θ0, r0; δ) = θ0 ∈ [0, τ [ , r(0; θ0, r0; δ) = r0 > 0.
For δ small enough, writing θ(t) for θ(t; θ0, r0; δ), and r(t) for r(t; θ0, r0; δ), and setting
θ1 = θ(2π; θ0, r0; δ), r1 = r(2π; θ0, r0; δ), we have
θ1 = θ0 + δ
T∫
0
1
r(t)
〈
f
(
r(t)
δ
ϕ(t + θ(t))
)
+ p(t)
∣∣∣∣∣ϕ(t + θ(t))
〉
dt ,
r1 = r0 − δ
T∫
0
〈
f
(
r(t)
δ
ϕ(t + θ(t))
)
+ p(t)
∣∣∣∣∣ϕ̇(t + θ(t))
〉
dt .
Through a detailed study of the involved oscillatory integrals, one proves the following
lemma.
Lemma 2. We have
θ1 = θ0 −
δ
r0
[Γ1(θ0) + R1(θ0, r0; δ)] ,
r1 = r0 − δ[Γ2(θ0) + R2(θ0, r0; δ)] ,
where R1 and R2 are such that
lim
δ→0+
R1(θ0, r0; δ) = lim
δ→0+
R2(θ0, r0; δ) = 0 ,
uniformly for θ0 ∈ [0, τ ] and r0 in a compact subset of R
+.
The proof of Theorem 20 is a combination of Lemma 2 and of a formula for the
computation of the Brouwer degree of Poincaré’s map minus identity on large balls in
terms of the properties of Γg,p.
For other results, see [17, 21, 59].
7.3. Resonance: unbounded solutions. The following existence results for unboun-
ded solutions are proved in [22].
Theorem 21. If H : R
2 → R is positive and positively homogeneous of degree
two, ∇H : R
2 → R
2 is locally Lipschitzian,
2π
τ
= n ∈ N0, g : R
2 → R
2 is locally
Lipschitzian, bounded, of type (22), the sets{
u ∈ R
2 : ‖u‖ = 1 and
∇H(u)
‖∇H(u)‖ = ±eiϑk
}
, 1 ≤ k ≤ m,
have only isolated points, limits (23) exist, and Γg,p
1 is of constant sign, then:
1) if κg
2 > 0, all solutions of system (21) are bounded in the future, and those with
sufficiently large amplitude are unbounded in the past;
2) if κg
2 < 0, all solutions of system (21) are bounded in the past, and those with
sufficiently large amplitudes are unbounded in the future.
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RESONANCE AND NONLINEARITY: A SURVEY 203
Theorem 22. If H : R
2 → R is positive and positively homogeneous of degree
two, ∇H : R
2 → R
2 is locally Lipschitzian,
2π
τ
= n ∈ N0, g : R
2 → R
2 is locally
Lipschitzian, bounded, of type (22), the sets{
u ∈ R
2 : ‖u‖ = 1 and
∇H(u)
‖∇H(u)‖ = ±eiϑk
}
, 1 ≤ k ≤ m,
have only isolated points, limits (23) exist, and if
1) Γg,p does not vanish;
2) Γg,p
1 (θ) changes sign, only having simple zeros,
then all solutions of system (21) with sufficiently large amplitude are unbounded, either
in the future, or in the past.
The proofs of those theorems are based upon some results of [17].
For other statements, see [17].
7.4. Applications to second order differential equations. Those theorems can be
applied to second order differential equations with separated nonlinearities of the form
ẍ + b(x)ẋ + µx+ − νx− + a(x) = e(t)
(Liénard equations),
ẍ + c(ẋ) + µx+ − νx− + a(x) = e(t)
(Rayleigh equations) or to second order differential equations with mixed nonlinearities
of the form
ẍ + µx+ − νx− + a(x + cẋ) = e(t).
For Liénard equations, the obtained existence conditions depend upon
lim
x→±∞
A(x)
x
:= A±,
lim
x→±∞
B(x)
x
:= B±,
where
A(x) :=
x∫
0
a(s) ds, B(x) =
x∫
0
. . .
s∫
0
. . . b(σ) dσ
ds.
For related work about second order equations with separated nonlinearities, see [9,
10, 27, 29, 30, 37, 31, 52, 53, 58].
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Received 27.10.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
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