RESONANT FORCED OSCILLATIONS IN SYSTEMS WITH PERIODIC NONLINEARITIES

We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations L(p)x = f(x) + b(t), p = d/dt with 2π-periodic forcing b and periodic f we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of 2π-periodic solutions.


Introduction. Consider the equation
where L is a polynomial with constant coefficients, deg L = ≥ 2, f is continuous and periodic with a period T , and b is continuous and periodic with the period 2π. We study 2π-periodic solutions of this equation: their existence, the finiteness and boundedness of the set Π of all 2π-periodic solutions, the asymptotic behavior of 2π-periodic solutions with increasing to infinity amplitudes. If L(ki) = 0 for integer k, then this problem is non-resonant, the set Π is non-empty and bounded in any reasonable sense. We study the resonant case L(±i) = 0 and L(ki) = 0 for integer k = ±1.
In [2] (see also the references therein) the authors consider related problems with the use of abstract results based on variational arguments, the periodic problem for the case L(p) = p 2 + 1 is studied in details. To apply the technique from [2] to 2π-periodic problems for (1), the polynomial L must be even, in this case the linear operator L(d/dt) with periodic boundary conditions is self-adjoint and the equation is Hamiltonian. We study (1) for generic resonant L.

ALEXANDER KRASNOSEL'SKII
Let b(t) = β sin(t + ψ) +b(t) where the functionb does not contain the first harmonics: 2π 0b (t) cos t dt = 2π 0b (t) sin t dt = 0. ( If β = 0, then the set Π is bounded (e.g., in C ), moreover it is empty if |β| > 4 sup |f |/π. We give generic conditions, under which Π is infinite and unbounded if β = 0, and it is bounded and contains arbitrary finite number of distinct elements if |β| = 0 is small enough. The main results can be generalized in various directions, some of them are discussed in Section 3.

2.
The main results. Throughout the paper we use the notation ω = 2π/T . According to the Fredholm Alternative, the linear equation L(p)x = b(t) has a unique 2π-periodic solution b 1 (t) ∈ C +1 satisfying Consider the Fourier series of f : without loss of generality we assume µ 0 = 0: the constant is included in the forcing term b. Such representation is unique if µ s ≥ 0. Fourier series (4) and the function b 1 define the functions All these functions are continuous and periodic, the function B ∈ C is π-periodic, it contains even harmonics only starting from the second, the functions q j are T -periodic in ξ and 2π-periodic in ϕ. Let β = 0.
Let us proceed to the case β = 0.
Theorem 2.2. Under the conditions of Theorem 2.1 for any integer N there exists a σ > 0 such that for any |β| ∈ (0, σ) equation (1) has at least N distinct 2π-periodic solutions.
If β → 0, then the diameter (e.g., in L 2 ) of the set Π tends to infinity. With a linear change of variable (y = ωx) the equation with a periodic nonlinearity with the period T may be reduced to the case T = 2π (we get L(p)y = ωb(t)+ωf (ω −1 y), the function f (ω −1 y) is 2π-periodic). We preserve a generic value for the period T in the formulations and proofs to stress the difference between the period 2π of oscillations and the period T of the nonlinearity.

Generalizations and comments.
3.1. Example. Generically all zeros of the function B are robust, there is a finite number of such zeros. The Sturm-Hurwitz theorem 2 implies the existence of at least four such zeros on [0, 2π): constant disappears due to the differentiation, the first harmonics (as well as other odd ones) disappear due to the specific form of the function B. According the same Sturm-Hurwitz theorem, the function q 1 (ϕ * , ξ) + q 2 (ϕ * , ξ) generically has at least two zeros.
If β = 0, then the change of variables x = x + b 1 leads us to the equivalent equation 1 An isolated zero of a scalar function is call robust if the function changes sign in a vicinity of this zero. 2 The number of sign changes on a period for any periodic continuous function is not less that the lowest order of its harmonics [1,9].

3.2.
Equations from control theory. Consider the equation  The proof of this theorem almost coincides with the presented proofs of Theorems 2.1-2.2. The assumption > m + 1 is particularly used in estimates (10) and (11).

Other resonant cases.
It is possible to obtain analogs of Theorems 2.1-2.2 for the case L(±si) = 0 for some integer s > 1 and L(±ki) = 0 for any k = ±s, but the final formulations are much more cumbersome. For the case s = 1 the principal part of the bifurcation system (it appears in the proofs) has the form According to the first equation, the second one may be rewritten as B(ϕ)q 1 (ϕ, ξ) = 0. Since q 1 (ϕ, ξ) = 0 it is equivalent to B(ϕ) = 0, and the answer has more or less explicit form as in Theorem 2.1. For s > 1 the analogous principal part contains equations with 2s terms, it is impossible to separate the variables and to formulate simple enough conditions of solvability for the bifurcation system. The possible condition for the case s > 1 may have the form: 'Let the bifurcation system of two variables (ϕ, ξ) have an isolated zero (ϕ * , ξ * ) of a nonzero index'.

Almost periodic nonlinearities.
Instead of periodic f it is possible to consider the functions f = f 1 + f 2 with a periodic f 1 and sufficiently rapidly decreasing f 2 : if the function f 2 (x)|x| 1+δ is uniformly bounded for some δ > 0. It is also possible to consider equations with almost periodic f = f k and T k -periodic f k . It would be interesting to study more general equations L(p)x = f (t, x) with periodic in both variables nonlinearities: the principal part of the bifurcation system has the form For various partial cases this system can be studied in an explicit form.

Nonlinearities with saturation.
Here we present a result concerning the equation L(p)x = b(t) + F (x) where the function F = f + g is the sum of a T -periodic function f and a function g with saturation: lim |x|→∞ |g(x) − sign(x)| = 0, and b(t) = β sin(t + ψ) +b(t) where β > 0 andb again satisfies (2). If β < 4, then Lazer-Leach condition holds, the set of 2π-periodic solutions is bounded and non-empty, if β > 4, then the set is bounded and may be empty; it is empty if β is large enough. The case β = 4 is twice-degenerate: the linear part is degenerate together with the principal nonlinear terms. Twice-degenerate systems without periodic term f were studied in [6] under special one-side conditions on the term Consider the function Theorem 3.2. Let condition (6) be valid for some ν > 1/4. Let for some δ > 0. Let there exist a robust zero ξ * of the function q * . Then for β = 4 the equation L(p)x = b(t) + F (x) has an infinite sequence x n of 2π-periodic solutions satisfying x n L 2 → ∞. For any integer N there exists a σ > 0 such that for any |β − 4| ∈ (0, σ) the equation For β = 4 solutions x n from Theorem 3.2 have the form where n is large enough, ξ n −nT −ξ * → 0, ϕ n → ψ +π, and h n C → 0. To prove Theorem 3.2 it is possible to use the same approaches as in the proof of Theorems 2.1 and 2.2 and partially the same auxiliary statements.
Under the assumptions of Theorems 2.1 and 2.2 the index at infinity for reasonable equivalent vector fields is undefined for β = 0 and is equal to 0 for β = 0. Under the assumptions of Theorem 3.2 the index Unlike Theorems 2.1-2.2, Theorem 3.2 may be easily reformulated for the case L(±si) = 0 with integer s > 1 (see Subsection 3.3).
3.6. Inverse theorem. As it follows from the computation part of the proof, Theorem 2.1 is 'almost invertible' in the following sense. Theorem 3.3. Let β = 0 and let the set Π be unbounded in L 2 . Let (6) be valid for some ν > 1/4. Then there exist solutions ϕ * , ξ * of (9) and the sequence x n ∈ Π of the form 4. Proof of Theorem 2.1.

New variables and linear operators.
We use the spaces C, C 1 , Denote by E ⊂ L 2 the linear span of the functions sin t and cos t, denote by E ⊥ ⊂ L 2 the orthogonal complement of the plane E. Then and Q = I − P are orthogonal projectors onto the subspaces E and E ⊥ of L 2 . Both projectors act in C and in C 1 . We search for 2π-periodic solutions in the form x(t) = ξ sin(t + ϕ) + h(t) where ξ > 0 and h = Qx does not contain the first harmonics. Below the real variables ξ, ϕ and the function h are considered as unknowns.
Denote by A the linear operator that maps any function u ∈ E ⊥ to a unique solution x = Au ∈ E ⊥ of the linear equation L(p)x = u. The existence of the solution x = Au follows from u ∈ E ⊥ , the uniqueness follows from x ∈ E ⊥ . The projectors P and Q commute with differentiation and with the operator A in any reasonable spaces.
The operator A : E ⊥ → E ⊥ is completely continuous. The operator AQ is well-defined in L 2 , it is completely continuous in L 2 , in C, and as an operator from L 2 to C −1 .
The operator A Q : u(t) → d dt AQu(t) is completely continuous in L 2 and in C, it is continuous as an operator from C to C −1 .
Consider a function u ∈ L 2 . If its Fourier coefficients ν k satisfy the estimate |ν k | ≤ ζ k , then the Fourier coefficientsν k andν k of the functions AQu and A Qu satisfy These estimates follow from the equalities Constant r 1 may be defined as Moreover, from the equalities If the polynomial L is even, the constant r 2 in the last estimate equals zero. Otherwise, value is finite: if = 2, then L is even and r 2 = 0, if ≥ 3, then the degree of the polynomial p 3 L(p) is not greater than 2 (may be equal for = 3).

Topological lemma.
For the sequel, we need the following auxiliary statement on the solvability of a system of two scalar equations and an equation in a Banach space H. This lemma contains the sufficient for our goals part of more general statements from [5].

Now rewrite this system in the final equivalent form
4.5. Auxiliary statements. The following 3 auxiliary statements are proved in Section 6.
Then for some ρ > 0 Lemma 4.4. There exists K 1 such that for any ρ > 0 and ξ ≥ 1 The constant K 1 is independent from θ and ρ. The most cumbersome parts (Lemma 6.1 and Lemma 6.3) of the proofs of Lemmas 4.3 and 4.4 are related to the Kelvin method of stationary phase ( [8], § §11-14). We repeat some constructions of the method to obtain necessary uniform estimates.
Equivalent system (16) by construction satisfies all the assumptions of Lemma 4.1 on the rectangles R n for any sufficiently large n and H = L 2 . Therefore there exist 2π-periodic solution x n (t) = ξ n sin(t + ϕ n ) + b 1 (t) + h n (t), (ϕ n , ξ n ) ∈ R n , by construction x n L 2 ≥ √ πξ n → ∞, Theorem 2.1 is proved.
Let us follow the proof of Theorem 2.1 for the equation L(p)x = b + f (x). Construct the orthogonal projectors P and Q, the linear operators A, AQ, and A Q. Create the equivalent (for the equation L(p)x =b + f (x)) system (16) on the set R n × L 2 .