Volume 16, Number 12
Volume 16, Number 12, 2011
In honor of Henk Broer for his 60th birthday
Hanssmann H., Homburg A. J., van Strien S.
Foreword
Abstract

Cushman R., Bates L.
Applications of the odd symplectic group in Hamiltonian systems
Abstract
In this paper we give two applications of the odd symplectic group to the study of the linear Poincaré maps of a periodic orbits of a Hamiltonian vector field, which cannot be obtained using the standard symplectic theory. First we look at the geodesic flow. We show that the period of the geodesic is a noneigenvalue modulus of the conjugacy class in the odd symplectic group of the linear Poincaré map. Second, we study an example of a family of periodic orbits, which forms a folded Robinson cylinder. The stability of this family uses the fact that the unipotent odd symplectic Poincaré map at the fold has a noneigenvalue modulus.

Poschel J.
KAM à la R
Abstract
Recently Rüssmann proposed a new variant of kamtheory based on a slowly converging iteration scheme. It is the purpose of this note to make this scheme accessible in an even simpler setting, namely for analytic perturbations of constant vector fields on a torus. As a side effect the result may be the shortest complete kam proof for perturbations of integrable vector fields available so far.

Sevryuk M. B.
The reversible context 2 in KAM theory: the first steps
Abstract
The reversible context 2 in KAM theory refers to the situation where dim Fix $G < \frac{1}{2}$ codim $\mathcal{T}$, here Fix $G$ is the fixed point manifold of the reversing involution $G$ and $\mathcal{T}$ is the invariant torus one deals with. Up to now, this context has been entirely unexplored. We obtain a first result on the persistence of invariant tori in the reversible context 2 (for the particular case where dim Fix $G = 0$) using J. Moser’s modifying terms theorem of 1967.

Broer H. W., Holtman S. J., Vegter G., Vitolo R.
Dynamics and geometry near resonant bifurcations
Abstract
This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf–Neimarck–Sacker bifurcation as developed in [1–4]. The focus is on the local resonance set, i.e., regions in parameter space for which periodic dynamics occurs. A classification of the corresponding geometry is obtained by applying Poincaré–Takens reduction, Lyapunov–Schmidt reduction and contactequivalence singularity theory, equivariant under an appropriate cyclic group. It is a classical result that the local geometry of these sets in the nondegenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our approach also provides a skeleton for the local resonant Hopf–Neimarck–Sacker dynamics in the form of planar Poincaré–Takens vector fields. To illustrate our methods a leading example is used: A periodically forced generalized Duffing–Van der Pol oscillator.

Hanssmann H.
Quasiperiodic bifurcations in reversible systems
Abstract
Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parameterized by phase space variables. This allows for quasiperiodic bifurcations within a single given system, induced by changes of the normal behavior of the tori. It turns out that in a nondegenerate reversible system all semilocal bifurcations of codimension 1 persist, under small nonintegrable perturbations, on large Cantor sets.

Puig J., Simo C.
Resonance tongues in the quasiperiodic Hill–Schrödinger equation with three frequencies
Abstract
In this paper we investigate numerically the following Hill’s equation $x'' + (a + bq(t))x = 0$ where $q(t) = cost + \cos\sqrt{2}t + \cos\sqrt{3}t$ is a quasiperiodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasiperiodic potential.
Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasiperiodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of $b$ the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of $a$, for large $b$, where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case. 
Schmidt K., Verbitskiy E.
New directions in algebraic dynamical systems
Abstract
The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.

Field M. J.
Exponential mixing for smooth hyperbolic suspension flows
Abstract
We present some simple examples of exponentially mixing hyperbolic suspension flows. We include some speculations indicating possible applications to suspension flows of algebraic Anosov systems. We conclude with some remarks about generalizations of our methods.

Borisov A. V., Kilin A. A., Mamaev I. S.
Hamiltonicity and integrability of the Suslov problem
Abstract
The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed.

Chow S., Huang W., Li Y., Zhou H.
A free energy based mathematical study for molecular motors
Abstract
We present a Parrondo’s paradox for free energy in a classical flashing ratchet model and use it as an alternative way to interpret the working mechanism of molecular motors. We also study the efficiency of molecular motors measured by their free energies. Our example demonstrates that a molecular motor can gain up to 20% in its free energy during the process. In addition, we report a noise induced free energy increasing phenomenon, which is similar to the stochastic resonance, in flashing ratchet models.

Ostrovski G., van Strien S.
Piecewise linear Hamiltonian flows associated to zerosum games: transition combinatorics and questions on ergodicity
Abstract
In this paper we consider a class of piecewise affine Hamiltonian vector fields whose orbits are piecewise straight lines. We give a first classification result of such systems and show that the orbitstructure of the flow of such a differential equation is surprisingly rich.

Vitolo R., Broer H. W., Simo C.
Quasiperiodic bifurcations of invariant circles in lowdimensional dissipative dynamical systems
Abstract
This paper first summarizes the theory of quasiperiodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddlenode and perioddoubling type.
