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Convergent series for quasiperiodically forced strongly dissipative systems, Preprint, 2012, mp arc 12
"... Abstract We study the ordinary differential equation εẍ +ẋ + ε g(x) = εf (ωt), with f and g analytic and f quasiperiodic in t with frequency vector ω ∈ R d . We show that if there exists c 0 ∈ R such that g(c 0 ) equals the average of f and the first nonzero derivative of g at c 0 is of odd order ..."
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Abstract We study the ordinary differential equation εẍ +ẋ + ε g(x) = εf (ωt), with f and g analytic and f quasiperiodic in t with frequency vector ω ∈ R d . We show that if there exists c 0 ∈ R such that g(c 0 ) equals the average of f and the first nonzero derivative of g at c 0 is of odd order n, then, for ε small enough and under very mild Diophantine conditions on ω, there exists a quasiperiodic solution close to c 0 , with the same frequency vector as f . In particular if f is a trigonometric polynomial the Diophantine condition on ω can be completely removed. This extends results previously available in the literature for n = 1. We also point out that, if n = 1 and the first derivative of g at c 0 is positive, then the quasiperiodic solution is locally unique and attractive.
Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers
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Resonant tori of arbitrary codimension for quasiperiodically forced systems
"... Abstract We consider a system of rotators subject to a small quasiperiodic forcing. We require the forcing to be analytic and satisfy a timereversibility property and we assume its frequency vector to be Bryuno. Then we prove that, without imposing any nondegeneracy condition on the forcing, ther ..."
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Abstract We consider a system of rotators subject to a small quasiperiodic forcing. We require the forcing to be analytic and satisfy a timereversibility property and we assume its frequency vector to be Bryuno. Then we prove that, without imposing any nondegeneracy condition on the forcing, there exists at least one quasiperiodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lowerdimensional tori of arbitrary codimension in degenerate cases.
LOCAL BEHAVIOR NEAR QUASI–PERIODIC SOLUTIONS OF CONFORMALLY SYMPLECTIC SYSTEMS
"... Abstract. We study the behavior of conformally symplectic systems near rotational Lagrangian tori. We recall that conformally symplectic systems appear for example in mechanical models including a friction proportional to the velocity. We show that in a neighborhood of these quasi–periodic solutions ..."
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Abstract. We study the behavior of conformally symplectic systems near rotational Lagrangian tori. We recall that conformally symplectic systems appear for example in mechanical models including a friction proportional to the velocity. We show that in a neighborhood of these quasi–periodic solutions (either transitive tori of maximal dimension or periodic solutions), one can always find a smooth symplectic change of variables in which the time evolution becomes just a rotation in some direction and a linear contraction in others. In particular quasi–periodic solutions of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, C r or C ∞. We emphasize that the results presented here are non–perturbative and apply to systems that are far from integrable; moreover, we do not require any assumption on the frequency and in particular we do not assume any non–resonance condition. We also show that the system of coordinates can be computed rather explicitly and we provide iterative algorithms, which allow to generalize the notion of “isochrones”. We conclude by showing that the above results apply to quasi–periodic conformally symplectic flows.
CONSTRUCTION OF RESPONSE FUNCTIONS IN FORCED STRONGLY DISSIPATIVE SYSTEMS
"... Abstract. We study the existence of quasi–periodic solutions x of the equation ε¨x + ˙x + εg(x) = εf(ωt), where x: R → R is the unknown and we are given g: R → R, f: T d → R, ω ∈ R d. We assume that there is a c0 ∈ R such that g(c0) = ˆ f0 (where ˆ f0 denotes the average of f) and g ′ (c0) = 0. ..."
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Abstract. We study the existence of quasi–periodic solutions x of the equation ε¨x + ˙x + εg(x) = εf(ωt), where x: R → R is the unknown and we are given g: R → R, f: T d → R, ω ∈ R d. We assume that there is a c0 ∈ R such that g(c0) = ˆ f0 (where ˆ f0 denotes the average of f) and g ′ (c0) = 0. Special cases of this equation, for example when g(x) = x 2, are called the “varactor problem ” in the literature. We show that if f, g are analytic, and ω satisfies some very mild irrationality conditions, there are families of quasi–periodic solutions with frequency ω. These families depend analytically on ε, when ε ranges over a complex domain that includes cones or parabolic domains based at the origin. The irrationality conditions required in this paper are very weak. They allow that the small denominators ω · k  grow exponentially with k. In the case that f is a trigonometric polynomial, we only need that ω · k is not zero for k  ≤ K0, where K0 is a multiple of the degree of the polynomial. This answers a delicate question raised in [8]. We also consider the periodic case, when ω is just a number (d = 1). We obtain that there are solutions that depend analytically in a domain which is a disk removing countably many disjoint disks. This shows that in this case there is no Stokes phenomenon (different resummations on different sectors) for the asymptotic series. The approach we use is to reduce the problem to a fixed point theorem. This approach also yields results in the case that g is a finitely differentiable function; it provides also very effective numerical algorithms and we discuss how they can be implemented. 2000 Mathematics Subject Classification. 70K43, 70K20, 34D35. Key words and phrases. Strongly dissipative systems, quasi–periodic solutions, fixed point theorem. R. L. was partially supported by NSF grants DMS0901389, DMS1162544.
Stabilità dell’integrabilità Hamiltoniana:
"... Hamiltoniani la cui funzione di Hamilton è “vicina ” a quella di un sistema analiticamente integrabile. Un sistema a ℓ gradi di libertà e con Hamiltoniana H(p,q) analitica sullo spazio delle fasi è analiticamente integrabile su una regione W dello spazio delle fasi se (1) i punti di W sono rappresen ..."
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Hamiltoniani la cui funzione di Hamilton è “vicina ” a quella di un sistema analiticamente integrabile. Un sistema a ℓ gradi di libertà e con Hamiltoniana H(p,q) analitica sullo spazio delle fasi è analiticamente integrabile su una regione W dello spazio delle fasi se (1) i punti di W sono rappresentabili via ℓ coordinate A = (A1,...,Aℓ) ∈ R ℓ, detteazioni,eℓ angoliα = (α1,...,αℓ) ∈ [0,2π] ℓ nelsensocheipunti(p,q) ∈ W possono essere rappresentati, a mezzo di una trasformazione analitica 1 di coordinate p = P ( A,α),q = Q ( A,α) con ( A,α) ∈ U × T ℓ ove U ⊂ R ℓ
RESPONSE SOLUTIONS FOR QUASIPERIODICALLY FORCED, DISSIPATIVE WAVE EQUATIONS
"... We consider several models of nonlinear wave equations subject to very strong damping and quasiperiodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasiperiodic solutions with the same fr ..."
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We consider several models of nonlinear wave equations subject to very strong damping and quasiperiodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasiperiodic solutions with the same frequency as the forcing). Under very general nonresonance conditions on the frequency, we show the existence of asymptotic expansions of the response solution; moreover, we prove that the response solution indeed exists and depends analytically on ε (where ε is the inverse of the coefficient multiplying the damping) for ε in a complex domain, which in some cases includes disks tangent to the imaginary axis at the origin. In other models, we prove analyticity in cones of aperture pi/2 and we conjecture it is optimal. These results have consequences for the asymptotic expansions of the response solutions considered in the literature. The proof of our results relies on reformulating the problem as a fixed point problem, constructing an approximate solution and studying the properties of iterations that converge to the solutions of the fixed point problem.