On the reducibility of linear differential equations with quasiperiodic coefficients

Jorba, Àngel; Simó, Carles

doi:10.1016/0022-0396(92)90107-x

Cited by 161 publications

(130 citation statements)

References 3 publications

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“…However, it seems that these results can be carried over to the case of finite a and small b. Indeed, the numer-ical evidence of the present paper for finite values of (a, b), together with KAM-arguments regarding averaging, see e.g., [1,17,22,10], seem to be in favor of the following facts: to test the present examples of p for this phenomenon, but also to widen the class, e.g., by introducing extra parameters. Compare with [15,12].…”

Section: Discussionsupporting

confidence: 51%

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Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

Broer

Simó

1998

Bol. Soc. Bras. Mat

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Abstract. A simple example is considered of Hill's equation Jc + (a 2 + bp@))x = O,where the forcing term p, instead of periodic, is quasi periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small Ib], can be explained by averaging. Next a numerical exploration is given for the global case of arbitrary b, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

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Section: Discussionsupporting

confidence: 51%

“…Several results concerning nonreducibiltiy in similar or wider contexts can be found in [22,10,11], [15], [17,18], [19] and [3,4,8,9].…”

Section: Further Problems a Numerical Investigationmentioning

confidence: 78%

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

Broer

Simó

1998

Bol. Soc. Bras. Mat

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“…From (30) and l e m m a 1 3 w e h ave t hat S is well de ned (for every ' 2 E ), according to (21). From (31) and l e m m a 1 0 w e c a n b o u nd t he expression of (H S ) t hat a p pear in the transformed Hamiltonian, k (H S )k E (1) R (1) The t echniques that w e u s e t o c o n trol t he r e d uction in the di erent d o m ains when we u s e Cauchy e s t imates, are analogous to t he o n es used in all the previous bounds.…”

Section: Iterative Lemmamentioning

confidence: 99%

On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations

Jorba

Villanueva

1997

Journal of Nonlinear Science

105

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In this work w e c o n s i d er time d ependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the e ect that t his kind o f p e r t urbations has on lower dimensional invariant t ori. Our results s h ow t hat, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the p e r t urbation to t he o n es they already have.The paper also contains estimates on the amount of surviving t ori. The w orst situation happens when the initial tori are normally elliptic. In this case, a torus (identi ed by t he v ector of intrinsic frequencies) can be continued with respect to a p e r t urbative p a r a m eter " 2 0 " 0 ], except for a set of " of measure exponentially small with " 0 . I n c a s e t hat " is xed (and su ciently small), we prove t he existence of invariant t ori for every vector of frequencies close to t he o n e o f t he initial torus, except for a set of frequencies of measure exponentially small with t he d i s t ance to the u nperturbed torus. As a particular case, if the p e r t urbation is autonomous, these results also give t he s a m e k i n d o f e s t imates on the m easure of destroyed tori.Finally, t hese results are applied to s o m e problems of celestial mechanics, in order to h elp in the d escripti o n o f t he p h ase space of some concrete m o d els.jorba@ma1.upc.es y jordi@tere.upc.es 2

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“…After the previous step, we have a new approximation to the solution, x 1 , and we want to find a new transformation C 1 (θ) and a new reduced matrix B 1 such that (7) is satisfied with Q 1 ∞ ≈ ε 2 (this situation has already been considered in several places; see [2,34,33]). Let us introduce the matrices R and B 1 as (14) R…”

Section: Correcting the Approximation Of The Floquet Transformationmentioning

confidence: 99%

On the Computation of Reducible Invariant Tori on a Parallel Computer

Jorba¹,

Olmedo²

2009

SIAM J. Appl. Dyn. Syst.

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Abstract. We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections.

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On the reducibility of linear differential equations with quasiperiodic coefficients

Cited by 161 publications

References 3 publications

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations

On the Computation of Reducible Invariant Tori on a Parallel Computer

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