Volume 16, Numbers 1-2

Volume 16, Numbers 1-2, 2011
In honor of Henk Broer for his 60th birthday

Hanßmann H.,  Homburg A. J.,  van Strien S.
Citation: Hanßmann H.,  Homburg A. J.,  van Strien S., Foreword, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 1
Cushman R.,  Bates L.
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.
Keywords: Hamiltonian systems, periodic orbits, odd symplectic normal forms
Citation: Cushman R.,  Bates L., Applications of the odd symplectic group in Hamiltonian systems, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 2-16
Poschel J.
Recently Rüssmann proposed a new variant of kam-theory 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.
Keywords: KAM theory, invariant tori, small divisors, weighted norms
Citation: Poschel J., KAM à la R, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 17-23
Sevryuk M. B.
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.
Keywords: KAM theory, Moser’s modifying terms theorem, reversible systems, reversible contexts, fixed point manifold, invariant torus
Citation: Sevryuk M. B., The reversible context 2 in KAM theory: the first steps, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 24-38
Broer H. W.,  Holtman S. J.,  Vegter G.,  Vitolo R.
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 contact-equivalence 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.
Keywords: periodically forced oscillator, resonant Hopf–Neimarck–Sacker bifurcation, geometric structure, Lyapunov–Schmidt reduction, equivariant singularity theory
Citation: Broer H. W.,  Holtman S. J.,  Vegter G.,  Vitolo R., Dynamics and geometry near resonant bifurcations, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 39-50
Hanßmann H.
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 quasi-periodic bifurcations within a single given system, induced by changes of the normal behavior of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.
Keywords: invariant tori, KAM theory, versal unfolding, persistence
Citation: Hanßmann H., Quasi-periodic bifurcations in reversible systems, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 51-60
Puig J.,  Simó C.
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 quasi-periodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic 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 quasi-periodic 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.
Keywords: quasi-periodic Schrödinger operators, quasi-periodic cocycles and skew-products, spectral gaps, resonance tongues, rotation number, Lyapunov exponent, numerical explorations
Citation: Puig J.,  Simó C., Resonance tongues in the quasi-periodic Hill–Schrödinger equation with three frequencies, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 61-78
Schmidt K.,  Verbitskiy E.
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.
Keywords: dimer matchings, domino tilings, Mahler measure, algebraic dynamics, homoclinic points
Citation: Schmidt K.,  Verbitskiy E., New directions in algebraic dynamical systems, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 79-89
Field M. J.
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.
Keywords: exponential mixing, suspension flow, subshift of finite type, algebraic Anosov flow
Citation: Field M. J., Exponential mixing for smooth hyperbolic suspension flows, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 90-103
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
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.
Keywords: Hamiltonian system, Poisson bracket, nonholonomic constraint, invariant measure, integrability
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Hamiltonicity and integrability of the Suslov problem, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 104-116
Chow S.,  Huang W.,  Li Y.,  Zhou H.
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.
Keywords: molecular motors, Parrondo’s paradox
Citation: Chow S.,  Huang W.,  Li Y.,  Zhou H., A free energy based mathematical study for molecular motors, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 117-127
Ostrovski G.,  van Strien S.
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 orbit-structure of the flow of such a differential equation is surprisingly rich.
Keywords: Hamiltonian systems, non-smooth dynamics, Filippov systems, piecewise affine, Arnol’d diffusion, fictitious play, best-response dynamics, learning process
Citation: Ostrovski G.,  van Strien S., Piecewise linear Hamiltonian flows associated to zero-sum games: transition combinatorics and questions on ergodicity, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 128-153
Vitolo R.,  Broer H. W.,  Simó C.
This paper first summarizes the theory of quasi-periodic 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, saddle-node and period-doubling type.
Keywords: bifurcations, invariant tori, resonances, KAM theory
Citation: Vitolo R.,  Broer H. W.,  Simó C., Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 1-2, pp. 154-184

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