Oscillator Synchronisation under Arbitrary Quasi-periodic Forcing

Corsi, Livia; Gentile, Guido

doi:10.1007/s00220-012-1548-2

Cited by 27 publications

(39 citation statements)

References 29 publications

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“…Such assumption can be seen as a time-reversibility condition on the Hamiltonian system. For non-convex Hamiltonians (1.3) with r = 1, the same result of persistence of at least one invariant torus has been proved in [11] without assuming Hypothesis 2 on the perturbation; the result has been extended to more general one-dimensional systems in [12]. Strictly speaking, Cheng's result does not imply the result in [11,12], because the unperturbed Hamiltonian is not convex; however Cheng's method can be adapted to such a case; see [13] for an explicit implementation.…”

Section: Introductionmentioning

confidence: 56%

“…and set χ n (x) = χ(x/ρ n ) for n ≥ 0 and χ −1 (x) = 1. Set also Ψ n (x) = χ n−1 (x) − χ n (x) for n ≥ −1; see Figure 1 in [11]. Next, we introduce the sequences {m n , p n } n≥0 , with m 0 = 0 and, for all n ≥ 0, m n+1 = m n + p n + 1, where p n := max{q ∈ Z + : α mn (ω) < 2α mn+q (ω)}, with α m (ω) defined in (1.6).…”

Section: Renormalised Trees: Node Factors and Propagatorsmentioning

confidence: 99%

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Resonant tori of arbitrary codimension for quasi-periodically forced systems

Corsi

Gentile

2016

Nonlinear Differ. Equ. Appl.

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We consider a system of rotators subject to a small quasi-periodic forcing. We require the forcing to be analytic and satisfy a time-reversibility 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 quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases.

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Section: Introductionmentioning

confidence: 56%

Section: Renormalised Trees: Node Factors and Propagatorsmentioning

confidence: 99%

Resonant tori of arbitrary codimension for quasi-periodically forced systems

Corsi

Gentile

2016

Nonlinear Differ. Equ. Appl.

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“…It follows that we obtain a well defined approximate solution to all orders under the condition (4.9); for instance one can adapt the argument given in Appendix H of [CG12].…”

Section: Existence Of Solutions Of the Zeroth Order Termmentioning

confidence: 99%

Response Solutions for Quasi-Periodically Forced, Dissipative Wave Equations

Calleja¹,

Celletti²,

Corsi³

et al. 2017

SIAM J. Math. Anal.

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Abstract. We consider several models of nonlinear wave equations subject to very strong damping and quasi-periodic 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., quasi-periodic solutions with the same frequency as the forcing).Under very general non-resonance 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 π/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.

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“…Chaotic motion has been found in some nonintegrable systems [30], and periodic doubling or quasi-periodic doubling sequences have been seen as the different routes to chaos [31,32]. Substituting ψ = ϕ(ξ )e iϑ , n = φ(ξ ) into Equation (4), where ξ = r 11 x + r 12 y + r 13 s − r 14 z and ϑ = r 21 x + r 22 y + r 23 s − r 24 z, we have …”

Section: Chaotic and Periodic Motions Of Equation (4)mentioning

confidence: 99%

Dynamic behavior of the incoherent optical spatial solitons in a nonlocal nonlinear medium

Zhen

Tian

Xie

et al. 2015

Journal of Modern Optics

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Dynamic behavior of the propagation of incoherent optical spatial solitons in a nonlocal nonlinear medium is investigated. By means of the Hirota method, we obtain the soliton solutions, based on which we find that |ψ| is positively related to n 0 , but inversely related to ω and κ, whereas n is independent of them, with ψ as the slowly varying amplitude of the beam, n as the refractive index change, n 0 , ω, and κ as the unperturbed refractive index, frequency of the propagating beam, and beam intensity distribution, respectively. Head-on and bound-state soliton interactions are both given, and one interaction period decreases with ω and κ increasing in the bound state one. With the external perturbations taken into consideration, the associated chaotic motions of the perturbed model are studied, and the corresponding power spectra and phase projections are obtained. Both the weak and developed chaotic states are investigated, and the difference between them roots in the relative magnitude of nonlinearities and perturbations. Such chaotic motions can be weakened via increasing ω and κ or decreasing n 0 . Periodic motion is obtained with the nonlinearities and perturbations balanced.

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Oscillator Synchronisation under Arbitrary Quasi-periodic Forcing

Cited by 27 publications

References 29 publications

Resonant tori of arbitrary codimension for quasi-periodically forced systems

Resonant tori of arbitrary codimension for quasi-periodically forced systems

Response Solutions for Quasi-Periodically Forced, Dissipative Wave Equations

Dynamic behavior of the incoherent optical spatial solitons in a nonlocal nonlinear medium

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