The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics that could be expected to be dominant in some ranges of the perturbation parameter actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies.

In 1994, Jürgen Moser generalized Hénon’s area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser’s six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none.
The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser’s map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled Hénon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2D planes through the phase space and by 3D slices through the tori.

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.

We give a new proof of the existence of compact surfaces embedded in $\mathbb{R}^3$ with Anosov geodesic flows. This proof starts with a noncompact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of
explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.

Solutions of the planar Kepler problem with fixed energy $h$ determine geodesics of the corresponding Jacobi–Maupertuis metric. This is a Riemannian metric on $\mathbb{R}^2$ if $h\geqslant 0$ or on a disk $\mathcal{D}\subset \mathbb{R}^2$ if $h<0$. The metric is singular at the origin (the collision singularity) and also on the boundary of the disk when $h<0$. The Kepler problem and the corresponding metric are invariant under rotations of the plane and it is natural to wonder whether the metric can be realized as a surface of revolution in $\mathbb{R}^3$ or some other simple space. In this note, we use elementary methods to study the geometry of the Kepler metric and the embedding problem. Embeddings of the metrics with $h\geqslant0$ as surfaces of revolution in $\mathbb{R}^3$ are constructed explicitly but no such embedding exists for $h<0$ due to a problem near the boundary of the disk. We prove a theorem showing that the same problem occurs for every analytic central force potential. Returning to the Kepler metric, we rule out embeddings in the three-sphere or hyperbolic space, but succeed in constructing an embedding in four-dimensional Minkowski spacetime. Indeed, there are many such embeddings.

We give a review of the Fermi–Pasta–Ulam (FPU) problem from the perspective of its possible impact on the foundations of physics, concerning the relations between classical and quantum mechanics. In the first part we point out that the problem should be looked upon in a wide sense (whether the equilibrium Gibbs measure is attained) rather than in the original restricted sense (whether energy equipartition is attained). The second part is devoted to some very recent results of ours for an FPU-like model of an ionic crystal, which has such a realistic character as to reproduce in an impressively good way the experimental infrared spectra. Since the existence of sharp spectral lines is usually considered to be a characteristic quantum phenomenon, even unconceivable in a classical frame, this fact seems to support a thesis suggested by the original FPU result. Namely, that the relations between classical and quantum mechanics are much subtler than usually believed, and should perhaps be reconsidered under some new light.

We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.

We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasi-periodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this “linearization” (or conjugacy) from knowledge of a single quasi-periodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian problem: determining the characteristics of a quasi-periodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method “the weighted Birkhoff average”.

The goal of this paper is to apply Lagrangian Descriptors (LDs), a technique based on Dynamical Systems Theory (DST) to reveal the phase space structures present in the wellknown Arnold’s cat map. This discrete dynamical system, which represents a classical example of an Anosov diffeomorphism that is strongly mixing, will provide us with a benchmark model to test the performance of LDs and their capability to detect fixed points, periodic orbits and their stable and unstable manifolds present in chaotic maps. In this work we show, both from a theoretical and a numerical perspective, how LDs reveal the invariant manifolds of the periodic orbits of the cat map. The application of this methodology in this setting clearly illustrates the chaotic behavior of the cat map and highlights some technical numerical difficulties that arise in the identification of its phase space structures.

In this paper we prove that generic one-parameter families of vector fields on $\mathbb{S}^2$ in the neighborhood of the fields of classes AH, SN,HC, SC (Andronov–Hopf, saddle-node, homoclinic curve, saddle connection) are structurally stable. We provide a classification of bifurcations in these families.

The application of intersection theory to construction of $n$-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few new discretizations of motion of the Euler top sharing the integrals of motion with the continuous time system and the Poisson bracket up to the integer scaling factor.

A method is presented to establish regions of phase space for 3D vector fields through which pass no co-oriented invariant 2D submanifolds transverse to a given oriented 1D foliation. Refinements are given for the cases of volume-preserving or Cartan–Arnol’d Hamiltonian flows and for boundaryless submanifolds.