This paper considers a class of nearly integrable reversible systems whose unperturbed part has a degenerate frequency mapping and a degenerate equilibrium point. Based on some KAM techniques and the topological degree theory, we prove the persistence of multiscale degenerate hyperbolic lower-dimensional invariant tori with prescribed frequencies.
In this note, we discuss the topology of Diophantine numbers, giving simple explicit examples of Diophantine isolated numbers (among those with the same Diophantine constants), showing that Diophantine sets are not always Cantor sets.
General properties of isolated Diophantine numbers are also briefly discussed.
We review Kolmogorov’s 1954 fundamental paper On the persistence of conditionally periodic motions under a small change in the Hamilton function (Dokl. akad. nauk SSSR, 1954, vol. 98, pp. 527–530), both from the historical and the mathematical point of view. In particular, we discuss Theorem 2 (which deals with the measure in phase space of persistent tori), the proof of which is not discussed at all by Kolmogorov, notwithstanding its centrality in his program in classical mechanics.
In Appendix, an interview (May 28, 2021) to Ya. Sinai on Kolmogorov’s legacy in classical mechanics is reported.
In the last years substantial mathematical progress has been made in KAM theory for quasi-linear/fully nonlinear Hamiltonian partial differential equations, notably for water waves and Euler equations. In this survey we focus on recent advances in quasi-periodic vortex patch solutions of the (2d)-Euler equation in (mathbb{R}^{2}) close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux – Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in an infinite-dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself.
We study the existence of infinite-dimensional invariant tori in a mechanical system of infinitely many rotators weakly interacting with each other. We consider explicitly interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy. The proof we provide of the persistence of the invariant tori implements the renormalisation group scheme based on the tree formalism, i. e., the graphical representation of the solutions of the equations of motion in terms of trees, which has been widely used in finite-dimensional problems. The method is very effectual and flexible: it naturally extends, once the functional setting has been fixed, to the infinite-dimensional case with only minor technical-natured adaptations.
We construct (C^{infty}) time-periodic fluctuating surfaces in (mathbb{R}^{3}) such that the corresponding non-autonomous geodesic flow has orbits along which the energy, and thus the speed goes to infinity. We begin with a static surface (M) in (mathbb{R}^{3}) on which the geodesic flow (with respect to the induced metric from (mathbb{R}^{3})) has a hyperbolic periodic orbit with a transverse homoclinic orbit. Taking this hyperbolic periodic orbit in an interval of energy levels gives us a normally hyperbolic invariant manifold (Lambda), the stable and unstable manifolds of which have a transverse homoclinic intersection. The surface (M) is embedded into (mathbb{R}^{3}) via a near-identity time-periodic embedding (G:Mtomathbb{R}^{3}). Then the pullback under (G) of the induced metric on (G(M)) is a time-periodic metric on (M), and the corresponding geodesic flow has a normally hyperbolic invariant manifold close to (Lambda), with stable and unstable manifolds intersecting transversely along a homoclinic channel. Perturbative techniques are used to calculate the scattering map and construct pseudo-orbits that move up along the cylinder. The energy tends to infinity along such pseudo-orbits. Finally, existing shadowing methods are applied to establish the existence of actual orbits of the non-autonomous geodesic flow shadowing these pseudo-orbits. In the same way we prove the existence of oscillatory trajectories, along which the limit inferior of the energy is finite, but the limit superior is infinite.
In this paper we consider the dynamics of a roller bicycle on a horizontal plane. For this bicycle we derive a nonlinear system of equations of motion in a form that allows us to take into account the symmetry of the system in a natural form. We analyze in detail the stability of straight-line motion depending on the parameters of the bicycle. We find numerical evidence that, in addition to stable straight-line motion, the roller bicycle can exhibit other, more complex, trajectories for which the bicycle does not fall.
We prove here the criterion of (C^{1})- (Omega)-stability of self-maps of a 3D-torus, which are skew products of circle maps. The (C^{1})- (Omega)-stability property is studied with respect to homeomorphisms of skew products type. We give here an example of the (Omega)-stable map on a 3D-torus and investigate approximating properties of maps under consideration.
This paper discusses a range of questions concerning the application of solvable Lie algebras of vector fields to exact integration of systems of ordinary differential equations. The set of (n) independent vector fields generating a solvable Lie algebra in (n)-dimensional space is locally reduced to some “canonical” form. This reduction is performed constructively (using quadratures), which, in particular, allows a simultaneous integration of (n) systems of differential equations that are generated by these fields. Generalized completely integrable systems are introduced and their properties are investigated. General ideas are applied to integration of the Hamiltonian systems of differential equations.
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