From the beginning of KAM theory, it was realized that its applicability to realistic problems depended on developing quantitative estimates on the sizes of the perturbations allowed. In this paper we present results on the existence of quasi-periodic solutions for conformally symplectic systems in non-perturbative regimes. We recall that, for conformally symplectic systems, finding the solution requires also to find a "drift parameter". We present a proof on the existence of solutions for values of the parameters which agree with more than three figures with the numerically conjectured optimal values.
The verification of the estimates presented here is not completely rigorous since we do not control the round-off error. Nevertheless, running with different precision shows very little difference in the results. Given the high precision of the calculation and the simplicity of the estimates, this does not seem to affect the results. A full verification should be done implementing interval arithmetic.
We make available the approximate solutions, the highly efficient algorithms to generate them (incorporating high precision based on the MPFR library) and the routines used to verify the applicability of the theorem.
Dissipative systems play a very important role in several physical models, most notably in Celestial Mechanics, where the dissipation drives the motion of natural and artificial satellites, leading them to migration of orbits, resonant states, etc. Hence the need to develop theories that ensure the existence of structures such as invariant tori or periodic orbits and device efficient computational methods. In this work we concentrate on the existence of invariant tori for the specific case of dissipative systems known as "conformally symplectic" systems, which have the property that they trans…
Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential eq…
We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and "drift". Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solution with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high order Taylor's method wi…
We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative t…
In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an {\em a posteriori} approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions …
We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $c\in H^1(\mathbb T,\mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(σ)\in H^1(\mathbb T,\mathbb R)$ with $c(σ)$ continuous and monotonic and \[ \partial_tu_c+H(x,\partial_x u_c+c,t)=α(c),\quad \text{a.e.\ } (x,t)\in\mathbb T^2, \] such that sequence of weak KAM solutions $\{u_c\}_{c\in H^1(\mathbb T,\mathbb R)}$ is $1/2-$Hölder continuity of parameter $σ\in \mathbb{R}$. Moreover, for eac…
We investigate the existence of whiskered tori in some dissipative systems, called \sl conformally symplectic \rm systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_μ$ of conformally symplectic maps which depend on a drift parameter $μ$. We fix a Diophantine frequency of the torus and we assume to have a drift $μ_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{μ_0} \circ K_0 - K_0 \circ T_ω$ (where $T_ω$ denotes the shift by $ω$). We also assume to have a splitting of the tangen…
We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can ensure that the torus persists under perturbation. Such models are common in celestial mechanics. In field theory, the adjustable parameter is called the counterterm and in celestial mechanics, the drift. It is known that there are formal expansions in powers of the perturbation both for the quasi-periodic solution and the counterterm. We prove that the asym…
We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an "effective hyperbolicity" condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.
We correct some tables and figures in [A.P. Bustamante and R.C. Calleja, Physica D: Nonlinear Phenomena, 395 (2019), pp. 15-23, arXiv:1712.05476]. We also report on the new computations that verify the accuracy of the data and extend the results. The new computations have led us to find new patterns in the data that were not noticed before. We formulate some more precise conjectures.
We prove the existence of hedgehogs for germs of complex analytic diffeomorphisms of $(\mathbb{C}^{2},0)$ with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of Pérez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of $(\mathbb{C},0)$ with a neutral fixed point.
Deterministic dynamical systems such as the baker maps are useful to shed light on some of the conditions verified by deterministic models in non-equilibrium statistical physics. We investigate a 2D dynamical system, enjoying a weak form of reversibility, with peculiar basins of attraction and steady states. In particular, we test the conditions required for the validity of the Transient Fluctuation Relation. Our analysis illustrates by means of concrete examples why ergodicity of the equilibrium dynamics seems to be a necessary condition for the Transient Fluctuation Relation to hold. This in…
We investigate the bifurcations and basins of attraction in the Bogdanov map, a planar quadratic map which is conjugate to the Hénon area-preserving map in its conservative limit. It undergoes a Hopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homoclinic tangency of the manifolds of a remote saddle point. The Bogdanov map is the Euler map of a two-dimensional system of ordinary differential equations first considered by Bogdanov and Arnold in their study of the versal u…
The behavior of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greene's residue criterion. The results are compared to the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.
We consider the integrable (zero perturbation) two--dimensional standard map, in light of current developments on ergodic sums of irrational rotations, and recent numerical evidence that it might possess non-trivial q-Gaussian statistics. Using both classical and recent results, we show that the phase average of the sum of centered positions of an orbit, for long times and after normalization, obeys the Cauchy distribution (a q-Gaussian with q=2), while for almost all individual orbits such a sum does not obey any distribution at all. We discuss the question of existence of distributions for K…
