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# 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 N-beads 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 non-autonomous 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. Citation: Cooley B.,  Newton P., Random number generation from chaotic impact collisions, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 199-212 DOI:10.1070/RD2004v009n03ABEH000276
 Kim P.,  Olver P. J. Geometric integration via multi-space Abstract We outline a general construction of symmetry-preserving 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 multi-space, 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 symmetry-preserving geometric integrators offer significant advantages. Citation: Kim P.,  Olver P. J., Geometric integration via multi-space, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 213-226 DOI:10.1070/RD2004v009n03ABEH000277
 Celletti A.,  Falcolini C.,  Locatelli U. On the break-down threshold of invariant tori in four dimensional maps Abstract We investigate the break-down of invariant tori in a four dimensional standard mapping for different values of the coupling parameter. We select various two-dimensional 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 break-down threshold of the invariant tori. In particular, the analyticity domain is recovered by means of a suitable implementation of Padé approximants. The break-down 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. Citation: Celletti A.,  Falcolini C.,  Locatelli U., On the break-down threshold of invariant tori in four dimensional maps, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 227-253 DOI:10.1070/RD2004v009n03ABEH000278
 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. Citation: Dullin H. R., Poisson integrator for symmetric rigid bodies, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 255-564 DOI:10.1070/RD2004v009n03ABEH000279
 Borisov A. V.,  Mamaev I. S.,  Kilin A. A. Two-body 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 Newtonian-type 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. Citation: Borisov A. V.,  Mamaev I. S.,  Kilin A. A., Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 265-279 DOI:10.1070/RD2004v009n03ABEH000280
 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 slow-fast 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 codimension-one bifurcations of a saddle-node 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 saddle-node periodic orbit with non-central homoclinics is behind the phenomena of bi-stability 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 inter-burst interval remains nearly constant. Citation: Shilnikov A. L.,  Cymbalyuk G., Homoclinic bifurcations of periodic orbits en a route from tonic spiking to bursting in neuron models, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 281-297 DOI:10.1070/RD2004v009n03ABEH000281
 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 near-Liouville-integrable Hamiltonian systems is developed in the convex real-analytic 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 $\varepsilon$. Citation: Rudnev M.,  Ten V. V., Sharp upper bounds for splitting of separatrices near a simple resonance, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 299-336 DOI:10.1070/RD2004v009n03ABEH000282
 Morozov A. D. On degenerate resonances in nearly Hamiltonian systems Abstract The problem of topology of a neighborhood of degenerate resonance zone in time-periodic two-dimensional nearly Hamiltonian systems of differential equations is considered. The investigation is based on averaged systems analysis. The role of polyharmonical Hamiltonian and non-Hamiltonian perturbations is discussed. In the Hamiltonian case, conditions of existence of degenerate resonances in non-degenerate and degenerate resonance zones are established. In the case of non-Hamiltonian perturbations, a problem of synchronization on degenerate resonances is considered. Citation: Morozov A. D., On degenerate resonances in nearly Hamiltonian systems, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 337-350 DOI:10.1070/RD2004v009n03ABEH000283
 Cordani B. From order to chaos in a perturbed Kepler problem Abstract The aim of this paper is twofold. First, we want to find angle-action 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 three-dimensional 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 two-dimensional 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. Citation: Cordani B., From order to chaos in a perturbed Kepler problem, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 351-372 DOI:10.1070/RD2004v009n03ABEH000284
 Benet L.,  Merlo O. Phase-space 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 non-interacting particles where open motion (escape) takes place, and includes the quasi-periodic perturbations of the shepherd's Kepler motion with small and zero eccentricity. We concentrate in the phase-space 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. Citation: Benet L.,  Merlo O., Phase-space structure for narrow planetary rings, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 373-383 DOI:10.1070/RD2004v009n03ABEH000285
 Efthymiopoulos C.,  Bountis A.,  Manos T. Explicit construction of first integrals with quasi-monomial 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 "quasi-polynomial" functions, from the information provided solely by the Painlevé–Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasi-monomial terms appearing in a quasi-polynomial 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 quasi-polynomial 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 non-existence of a quasi-polynomial first integral. Examples from specific dynamical systems are given. Citation: Efthymiopoulos C.,  Bountis A.,  Manos T., Explicit construction of first integrals with quasi-monomial terms from the Painlevé series, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 385-398 DOI:10.1070/RD2004v009n03ABEH000286

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