Volume 9, Number 3
Volume 9, Number 3, 2004
Effective computations in modern dynamics
Cooley B., Newton P.
Random number generation from chaotic impact collisions
Abstract
The collision sequence produced by Nbeads sliding on a frictionless hoop gives rise to a dynamical system that can be formulated as a string of matrix products [4, 5]. The matrices that form the product are written in the order in which the collisions unfold and their corresponding eigenvalues on the unit circle are treated as a nonautonomous rotation map on $S^1 $. The problem of three beads of masses $m$, $m  \epsilon $, $m + \epsilon $ is studied computationally for splitting parameter values $ \epsilon \geqslant 0$. When $ \epsilon = 0$ (three equal masses), the spectrum is discrete on the unit circle underlying the fact that the dynamics are regular [5]. For $ \epsilon > 0$, the eigenvalue spectrum produced by a deterministically chaotic trajectory is compared to spectra produced by two different stochastic problems. The first is the spectrum associated with the sequence of matrix products in which a random number algorithm is used to produce the matrix orderings. The second is the spectrum generated from a random walk process on the unit circle. We describe how to use the chaotic collision sequences as the basis for a random number generating algorithm and we conclude, by an examination of both the runs and reverse arrangement tests, that the degree of randomness produced by these sequences is equivalent to Matlab's rand() routine for generating random numbers.

Kim P., Olver P. J.
Geometric integration via multispace
Abstract
We outline a general construction of symmetrypreserving numerical schemes for ordinary differential equations. The method of invariantization is based on the equivariant moving frame theory applied to prolonged symmetry group actions on multispace, which has been proposed as the proper geometric setting for numerical analysis. We explain how to invariantize standard numerical integrators such as the Euler and Runge–Kutta schemes; in favorable situations, the resulting symmetrypreserving geometric integrators offer significant advantages.

Celletti A., Falcolini C., Locatelli U.
On the breakdown threshold of invariant tori in four dimensional maps
Abstract
We investigate the breakdown of invariant tori in a four dimensional standard mapping for different values of the coupling parameter. We select various twodimensional frequency vectors, having eventually one or both components close to a rational value. The dynamics of this model is very reach and depends on two parameters, the perturbing and coupling parameters. Several techniques are introduced to determine the analyticity domain (in the complex perturbing parameter plane) and to compute the breakdown threshold of the invariant tori. In particular, the analyticity domain is recovered by means of a suitable implementation of Padé approximants. The breakdown threshold is computed through a suitable extension of Greene's method to four dimensional systems. Frequency analysis is implemented and compared with the previous techniques.

Dullin H. R.
Poisson integrator for symmetric rigid bodies
Abstract
We derive an explicit second order reversible Poisson integrator for symmetric rigid bodies in space (i.e. without a fixed point). The integrator is obtained by applying a splitting method to the Hamiltonian after reduction by the $S^1$ body symmetry. In the particular case of a magnetic top in an axisymmetric magnetic field (i.e. the Levitron) this integrator preserves the two momentum integrals. The method is used to calculate the complicated boundary of stability near a linearly stable relative equilibrium of the Levitron with indefinite Hamiltonian.

Borisov A. V., Mamaev I. S., Kilin A. A.
Twobody problem on a sphere. Reduction, stochasticity, periodic orbits
Abstract
We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtoniantype potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.

Shilnikov A. L., Cymbalyuk G.
Homoclinic bifurcations of periodic orbits en a route from tonic spiking to bursting in neuron models
Abstract
The methods of qualitative theory of slowfast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. We demonstrate how two different codimensionone bifurcations of a saddlenode periodic orbit with homoclinic orbits can explain transitions between tonic spiking and bursting activities in neuron models following Hodgkin–Huxley formalism. In the first case, we argue that the Lukyanov–Shilnikov bifurcation of a saddlenode periodic orbit with noncentral homoclinics is behind the phenomena of bistability observed in a model of a leech heart interneuron under defined pharmacological conditions. This model can exhibit two coexisting types of oscillations: tonic spiking and bursting. Moreover, the neuron model can also generate weakly chaotic trains of bursting when a control parameter is close to the bifurcation value. In the second case, the transition is continuous and reversible due to the blue sky catastrophe bifurcation. This bifurcation provides a plausible mechanism for the regulation of the burst duration which may increases with no bound as $1/\sqrt{\alpha\alpha_0}$, where $\alpha_0$ is the transitional value, while the interburst interval remains nearly constant.

Rudnev M., Ten V. V.
Sharp upper bounds for splitting of separatrices near a simple resonance
Abstract
General theory for the splitting of separatrices near simple resonances of nearLiouvilleintegrable Hamiltonian systems is developed in the convex realanalytic setting. A generic estimate
$$\mathfrak{S}_k\,\leqslant\,O(\sqrt{\varepsilon}) \, \times \, \exp\left[ \,{\left k\cdot\left(c_1{\omega\over\sqrt{\varepsilon}}+c_2\right)\right}  k\sigma\right],\;\,k\in\mathbb{Z}^n\setminus\{0\}$$ is proved for the
Fourier coefficients of the splitting distance measure $\mathfrak{S}(\phi),\,\phi\in\mathbb{T}^n,$ describing the intersections of Lagrangian manifolds, asymptotic to invariant $n$tori, $\varepsilon$ being the perturbation parameter.
The constants $\omega\in\mathbb{R}^n,$ $c_1,\sigma>0,\,c_2\in\mathbb{R}^n$ are characteristic of the given problem (the Hamiltonian and the resonance), cannot be improved and can be calculated explicitly, given an example. The theory allows for optimal parameter dependencies in the smallness condition for $\varepsilon$. 
Morozov A. D.
On degenerate resonances in nearly Hamiltonian systems
Abstract
The problem of topology of a neighborhood of degenerate resonance zone in timeperiodic twodimensional nearly Hamiltonian systems of differential equations is considered. The investigation is based on averaged systems analysis. The role of polyharmonical Hamiltonian and nonHamiltonian perturbations is discussed. In the Hamiltonian case, conditions of existence of degenerate resonances in nondegenerate and degenerate resonance zones are established. In the case of nonHamiltonian perturbations, a problem of synchronization on degenerate resonances is considered.

Cordani B.
From order to chaos in a perturbed Kepler problem
Abstract
The aim of this paper is twofold. First, we want to find angleaction variables suitable for the study of a generic perturbed Kepler problem: indeed, the unperturbed problem is degenerate, since its Hamiltonian depends on only one action variable (instead of three), and only a circle (instead of a threedimensional torus) is intrinsically defined. Fortunately, the manifold of the orbits is compact, so the perturbed averaged system has always elliptic equilibrium points: nearby these points the reduced system behaves like a twodimensional harmonic oscillator, which bears naturally the variables we seek. Second, we will apply the method of Numerical Frequencies Analysis in order to detect the transition from order to chaos. Four numerical examples are examined, by means of the free programs KEPLER and NAFF.

Benet L., Merlo O.
Phasespace structure for narrow planetary rings
Abstract
We address the occurrence of narrow planetary rings under the interaction with shepherds. Our approach is based on a Hamiltonian framework of noninteracting particles where open motion (escape) takes place, and includes the quasiperiodic perturbations of the shepherd's Kepler motion with small and zero eccentricity. We concentrate in the phasespace structure and establish connections with properties like the eccentricity, sharp edges and narrowness of the ring. Within our scattering approach, the organizing centers necessary for the occurrence of the rings are stable periodic orbits, or more generally, stable tori. In the case of eccentric motion of the shepherd, the rings are narrower and display a gap which defines different components of the ring.

Efthymiopoulos C., Bountis T., Manos T.
Explicit construction of first integrals with quasimonomial terms from the Painlevé series
Abstract
The Painlevé and weak Painlevé conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlevé test, the calculation of the integrals relies on a variety of methods which are independent from Painlevé analysis. The present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed as "quasipolynomial" functions, from the information provided solely by the Painlevé–Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasimonomial terms appearing in a quasipolynomial integral are obtained by an application of a theorem by Yoshida (1983). The integrals are obtained by a proper balancing of the coefficients in a quasipolynomial function selected as initial ansatz for the integral, so that all dependence on powers of the time $\tau = t – t_0$ is eliminated. Both right and left Painlevé series are useful in the method. Alternatively, the method can be used to show the nonexistence of a quasipolynomial first integral. Examples from specific dynamical systems are given.
