46 research outputs found
Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system
We give a precise statement for KAM theorem
in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system.
If the frequencies of the elliptic point are nonresonant up to a certain order
, and a nondegeneracy condition is fulfilled, we get an estimate for
the measure of the complement of the KAM tori in a neighbourhood of given
radius. Moreover, if the frequencies satisfy a Diophantine condition,
with exponent ,
we show that in a neighbourhood of radius the measure of the complement
is exponentially small in .
We also give a related result for quasi-Diophantine frequencies,
which is more useful for practical purposes.
The results are obtained
by putting the system in Birkhoff normal form up to an
appropiate order, and the key point relies on giving accurate bounds for
its terms
Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems
We study the existence of transverse homoclinic orbits in a singular or weakly
hyperbolic Hamiltonian, with degrees of freedom, as a model for the
behaviour of a nearly-integrable Hamiltonian near a simple resonance. The
example considered consists of an integrable Hamiltonian possessing a
-dimensional hyperbolic invariant torus with fast frequencies
and coincident whiskers or separatrices, plus a
perturbation of order , giving rise to an exponentially
small splitting of separatrices. We show that asymptotic estimates for the
transversality of the intersections can be obtained if satisfies
certain arithmetic properties. More precisely, we assume that is a
quadratic vector (i.e.~the frequency ratio is a quadratic irrational number),
and generalize the good arithmetic properties of the golden vector. We provide
a sufficient condition on the quadratic vector ensuring that
the Poincar\'e--Melnikov method (used for the golden vector in a previous
work) can be applied to establish the existence of transverse homoclinic orbits
and, in a more restrictive case, their continuation for all values of
Homoclinic orbits to invariant tori in Hamiltonian systems
We consider a perturbation of an integrable Hamiltonian system which
possesses invariant tori with coincident whiskers (like some rotators and a pendulum).
Our goal is to measure the splitting distance between the perturbed whiskers, putting
emphasis on the detection of their intersections, which give rise to homoclinic orbits
to the perturbed tori. A geometric method is presented which takes into account the
Lagrangian properties of the whiskers. In this way, the splitting distance is the gradient of a splitting potential. In the regular case (also known as a priori-unstable: the
Lyapunov exponents of the whiskered tori remain fixed), the splitting potential is well-
approximated by a Melnikov potential. This method is designed as a first step in the
study of the singular case (also known as a priori-stable: the Lyapunov exponents of the
whiskered tori approach to zero when the perturbation tends to zero)
Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio
The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector , with where the frequency ratio is a quadratic irrational number. Applying the Poincaré--Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov potential. This allows us to provide an asymptotic estimate for the maximal splitting distance and show the existence of transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for the transversality of the splitting. Both estimates are exponentially small in , with the functions in the exponents being periodic with respect to , and can be explicitly constructed from the continued fraction of . In this way, we emphasize the strong dependence of our results on the arithmetic properties of . In particular, for quadratic ratios with a 1-periodic or 2-periodic continued fraction (called metallic and metallic-colored ratios, respectively), we provide accurate upper and lower bounds for the splitting. The estimate for the maximal splitting distance is valid for all sufficiently small values of , and the transversality can be established for a majority of values of , excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur. Read More: http://epubs.siam.org/doi/10.1137/15M1032776Peer ReviewedPostprint (published version
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tort with quadratic and cubic frequencies
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Q is the so-called cubic golden number (the real root of x(3) x - 1= 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.Postprint (published version
Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequenciesPreprin
Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector ¿/ev, with ¿=(1,O,O˜) where O is a cubic irrational number whose two conjugates are complex, and the components of ¿ generate the field Q(O). A paradigmatic case is the cubic golden vector, given by the (real) number O satisfying O3=1-O, and O˜=O2. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors ¿k,¿¿, k¿Z3. Applying the Poincaré–Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on e) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in e, and valid for all sufficiently small values of e. This estimate behaves like exp{-h1(e)/e1/6} and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h1(e) in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector ¿, and proving that it is a quasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.Peer ReviewedPreprin
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
We study the splitting of invariant manifolds of whiskered t
ori with two or three frequencies in nearly-integrable
Hamiltonian systems. We consider 2-dimensional tori with a
frequency vector
ω
= (1
,
Ω) where Ω is a quadratic
irrational number, or 3-dimensional tori with a frequency v
ector
ω
= (1
,
Ω
,
Ω
2
) where Ω is a cubic irrational number.
Applying the Poincar ́e–Melnikov method, we find exponentia
lly small asymptotic estimates for the maximal splitting
distance between the stable and unstable manifolds associa
ted to the invariant torus, showing that such estimates
depend strongly on the arithmetic properties of the frequen
cies. In the quadratic case, we use the continued fractions
theory to establish a certain arithmetic property, fulfille
d in 24 cases, which allows us to provide asymptotic estimate
s
in a simple way. In the cubic case, we focus our attention to th
e case in which Ω is the so-called cubic golden number
(the real root of
x
3
+
x
−
1 = 0), obtaining also asymptotic estimates. We point out the
similitudes and differences
between the results obtained for both the quadratic and cubi
c cases.Preprin
Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type
We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyintegrable
Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We
consider a 2-dimensional torus with a fast frequency vector , with where is an irrational number of constant type, i.e. a number whose continued fraction has bounded
entries. Applying the Poincar´e–Melnikov method, we find exponentially small lower bounds for
the maximal splitting distance between the stable and unstable invariant manifolds associated to
the invariant torus, and we show that these bounds depend strongly on the arithmetic properties
of the frequencies.Preprin