Volume 23, Number 4
Volume 23, Number 4, 2018
Miranda E., Planas A.
Equivariant Classification of $b^m$symplectic Surfaces
Abstract
Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of $b^m$Poisson structures which can be also visualized using differential forms with singularities as bmsymplectic structures. In this paper we extend the classification scheme in [24] for $b^m$symplectic surfaces to the equivariant setting. When the compact group is the group of decktransformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct $b^m$symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.

Bai Y., Svinin M., Yamamoto M.
DynamicsBased Motion Planning for a PendulumActuated Spherical Rolling Robot
Abstract
This paper deals with the dynamics and motion planning for a spherical rolling robot with a pendulum actuated by two motors. First, kinematic and dynamic models for the rolling robot are introduced. In general, not all feasible kinematic trajectories of the rolling carrier are dynamically realizable. A notable exception is when the contact trajectories on the sphere and on the plane are geodesic lines. Based on this consideration, a motion planning strategy for complete reconfiguration of the rolling robot is proposed. The strategy consists of two trivial movements and a nontrivial maneuver that is based on tracing multiple spherical triangles. To compute the sizes and the number of triangles, a reachability diagram is constructed. To define the control torques realizing the resttorest motion along the geodesic lines, a geometric phasebased approach has been employed and tested under simulation. Compared with the minimum effort optimal control, the proposed technique is less computationally expensive while providing similar system performance, and thus it is more suitable for realtime applications.

Meyer K. R., Palacián J. F., Yanguas P.
Normalization Through Invariants in $n$dimensional Kepler Problems
Abstract
We present a procedure for the normalization of perturbed Keplerian problems in $n$ dimensions based onMoser regularization of the Kepler problem and the invariants associated to the reduction process. The approach allows us not only to circumvent the problems introduced by certain classical variables used in the normalization of this kind of problems, but also to do both the normalization and reduction in one step. The technique is introduced for any dimensions and is illustrated for $n = 2, 3$ by relating Moser coordinates with Delaunaylike variables. The theory is applied to the spatial circular restricted threebody problem for the study of the existence of periodic and quasiperiodic solutions of rectilinear type.

Jovanović B., Jovanovic V.
Heisenberg Model in PseudoEuclidean Spaces II
Abstract
In the review we describe a relation between the Heisenberg spin chain model on pseudospheres and lightlike cones in pseudoEuclidean spaces and virtual billiards. A geometrical interpretation of the integrals associated to a family of confocal quadrics is given, analogous to Moser’s geometrical interpretation of the integrals of the Neumann system on the sphere.

Yagasaki K.
Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics
Abstract
We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikov’s method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.

Kuznetsov S. P., Sedova Y. V.
Hyperbolic Chaos in Systems Based on FitzHugh–Nagumo Model Neurons
Abstract
In the present paper we consider and study numerically two systems based on model FitzHugh–Nagumo neurons, where in the presence of periodic modulation of parameters it is possible to implement chaotic dynamics on the attractor in the form of a Smale–Williams solenoid in the stroboscopic Poincaré map. In particular, hyperbolic chaos characterized by structural stability occurs in a single neuron supplemented by a timedelay feedback loop with a quadratic nonlinear element.

Kudryashov N. A.
Exact Solutions and Integrability of the Duffing–Van der Pol Equation
Abstract
The forcefree Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some wellknown methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear secondorder ordinary differential equations.

Borisov A. V., Mamaev I. S., Vetchanin E. V.
Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation
Abstract
This paper addresses the problem of selfpropulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasiperiodic regimes (attracting tori) and strange attractors. Oneparameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and perioddoubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermilike acceleration) is possible.
