We study in this paper the remnants of the contact partial order on the orbits of the adjoint action of contactomorphism groups on their Lie algebras. Our main interest is a class of noncompact contact manifolds, called convex at infinity.

The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract (via an informal discussion and a simple example) the experts’ attention to the peculiarities of the so-called excitation of elliptic normal modes in the reversible context 2.

We establish Écalle’s mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré–Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.

We present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequence $f_n$ of analytic mappings of ${\mathbb C}^d$ has a common fixed point $f_n(0) = 0$, and the maps $f_n$ converge to a linear mapping $A_\infty$ so fast that $$ \sum_n \|f_m - A_\infty\|_{\mathbf{L}^\infty(B)} < \infty $$ $$ A_\infty = \mathop{\rm diag}( e^{2 \pi i \omega_1}, \ldots, e^{2 \pi i \omega_d}) \qquad \omega = (\omega_1, \ldots, \omega_q) \in {\mathbb R}^d, $$ then $f_n$ is nonautonomously conjugate to the linearization. That is, there exists a sequence $h_n$ of analytic mappings fixing the origin satisfying \[ h_{n+1} \circ f_n = A_\infty h_{n}. \] The key point of the result is that the functions $h_n$ are defined in a large domain and they are bounded. We show that $\sum_n \|h_n - \mathop{\rm Id} \|_{\mathbf{L}^\infty(B)} < \infty$.
We also provide results when $f_n$ converges to a nonlinearizable mapping $f_\infty$ or to a nonelliptic linear mapping.
In the case that the mappings $f_n$ preserve a geometric structure (e.g., symplectic, volume, contact, Poisson, etc.), we show that the $h_n$ can be chosen so that they preserve the same geometric structure as the $f_n$.
We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.

Balanced configurations of $N$ point masses are the configurations which, in a Euclidean space of high enough dimension, i.e., up to $2(N - 1)$, admit a relative equilibrium motion under the Newtonian (or similar) attraction. Central configurations are balanced and it has been proved by Alain Albouy that central configurations of four equal masses necessarily possess a symmetry axis, from which followed a proof that the number of such configurations up to similarity is finite and explicitly describable. It is known that balanced configurations of three equal masses are exactly the isosceles triangles, but it is not known whether balanced configurations of four equal masses must have some symmetry. As balanced configurations come in families, it makes sense to look for possible branches of nonsymmetric balanced configurations bifurcating from the subset of symmetric ones. In the simpler case of a logarithmic potential, the subset of symmetric balanced configurations of four equal masses is easy to describe as well as the bifurcation locus, but there is a grain of salt: expressed in terms of the squared mutual distances, this locus lies almost completely outside the set of true configurations (i. e., generalizations of triangular inequalities are not satisfied) and hence could lead most of the time only to the bifurcation of a branch of virtual nonsymmetric balanced configurations. Nevertheless, a tiny piece of the bifurcation locus lies within the subset of real balanced configurations symmetric with respect to a line and hence has a chance to lead to the bifurcation of real nonsymmetric balanced configurations. This raises the question of the title, a question which, thanks to the explicit description given here, should be solvable by computer experts even in the Newtonian case. Another interesting question is about the possibility for a bifurcating branch of virtual nonsymmetric balanced configurations to come back to the domain of true configurations.

We show how to construct the hyperbolic plane with its geodesic flow as the reduction of a three-problem whose potential is proportional to $I/\Delta^2$ where $I$ is the moment of inertia of this triangle whose vertices are the locations of the three bodies and $\Delta$ is its area. The reduction method follows [11]. Reduction by scaling is only possible because the potential is homogeneous of degree $-2$. In trying to extend the assertion of hyperbolicity to the analogous family of planar $N$-body problems with three-body interaction potentials we run into Mnëv’s astounding universality theorem which implies that the extended assertion is doomed to fail.

A nearly-integrable dynamical system has a natural formulation in terms of actions, $y$ (nearly constant), and angles, $x$ (nearly rigidly rotating with frequency $\Omega(y)$). We study angleaction maps that are close to symplectic and have a twist, the derivative of the frequency map, $D\Omega(y)$, that is positive-definite. When the map is symplectic, NekhoroshevЃfs theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volume-preserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rank-$r$ resonances. A comparison with computations for a generalized Froeschlé map in four-dimensions shows that this theory gives accurate results for the rank-one case.

We construct a family of integrable deformations of the Bogoyavlenskij–Itoh systems and construct a Lax operator with spectral parameter for it. Our approach is based on the construction of a family of compatible Poisson structures for the undeformed systems, whose Casimirs are shown to yield a generating function for the integrals in involution of the deformed systems.We show how these deformations are related to the Veselov–Shabat systems.

Some interesting variants of walking droplet based discrete dynamical bifurcations arising from diffeomorphisms are analyzed in detail. A notable feature of these new bifurcations is that, like Smale horseshoes, they can be represented by simple geometric paradigms, which markedly simplify their analysis. The two-dimensional diffeomorphisms that produce these bifurcations are called sigma maps or double sigma maps for reasons that are made manifest in this investigation. Several examples are presented along with their dynamical simulations.

In this paper, we discuss geometric structures related to the Lagrange multipliers rule. The practical goal is to explain how to compute or estimate the Morse index of the second variation. Symplectic geometry allows one to effectively do it even for very degenerate problems with complicated constraints. The main geometric and analytic tool is an appropriately rearranged Maslov index. We try to emphasize the geometric framework and omit analytic routine. Proofs are often replaced with informal explanations, but a well-trained mathematician will easily rewrite them in a conventional way. We believe that Vladimir Arnold would approve of such an attitude.