Volume 25, Number 2
Volume 25, Number 2, 2020
CárcamoDíaz D., Palacián J. F., Vidal C., Yanguas P.
On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Threebody Problem
Abstract
The wellknown problem of the nonlinear stability of $L_4$ and $L_5$ in the circular
spatial restricted threebody problem is revisited. Some new results in the light of the concept of
Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates
for three specific values of the mass ratio that remained uncovered. Moreover, in many cases
we improve the estimates found in the literature.

Lim C. C.
General Jacobi Coordinates and Herman Resonance for Some Nonheliocentric Celestial $N$body Problems
Abstract
The general Jacobi symplectic variables generated by a
combinatorial algorithm from the full binary tree $T(N)$ are used
to formulate some nonheliocentric gravitational $N$body problems in
perturbation form. The resulting uncoupled term $H_U$ for $(N1)$
independent Keplerian motions and the perturbation term $H_P$ are
both explicitly dependent on the partial ordering induced by the
tree $T(N)$. This leads to suitable conditions on separations of the
$N$ bodies for the perturbation to be small. We prove the Herman resonance for a
new approximation of the 5body problem.
Full details of the
derivations of the perturbation form and Herman resonance are given
only in the case of five bodies using the caterpillar binary tree
$T_c(5)$.

Biswas A., Kara A. H., Zhou Q., Alzahrani A. K., Belic M. R.
Conservation Laws for Highly Dispersive Optical Solitons in Birefringent Fibers
Abstract
This paper reports conservation laws for highly dispersive optical solitons in
birefringent fibers. Three forms of nonlinearities are studied which are Kerr, polynomial and
nonlocal laws. Power, linear momentum and Hamiltonian are conserved for these types of
nonlinear refractive index.

Grammaticos B., Willox R., Satsuma J.
Revisiting the Human and Nature Dynamics Model
Abstract
We present a simple model for describing the dynamics of the interaction between
a homogeneous population or society, and the natural resources and reserves that the society
needs for its survival. The model is formulated in terms of ordinary differential equations,
which are subsequently discretised, the discrete system providing a natural integrator for the
continuous one. An ultradiscrete, generalised cellular automatonlike, model is also derived.
The dynamics of our simple, threecomponent, model are particularly rich exhibiting either a
route to a steady state or an oscillating, limit cycletype regime or to a collapse. While these
dynamical behaviours depend strongly on the choice of the details of the model, the important
conclusion is that a collapse or near collapse, leading to the disappearance of the population or
to a complete transfiguration of its societal model, is indeed possible.

Simó C.
Simple Flows on Tori with Uncommon Chaos
Abstract
We consider a family of simple flows in tori that display chaotic behavior in a wide
sense. But these flows do not have homoclinic nor heteroclinic orbits. They have only a fixed
point which is of parabolic type. However, the dynamics returns infinitely many times near the
fixed point due to quasiperiodicity. A preliminary example is given for maps introduced in a
paper containing many examples of strange attractors in [6]. Recently, a family of maps similar
to the flows considered here was studied in [9]. In the present paper we consider the case of 2D
tori and the extension to tori of arbitrary finite dimension. Some other facts about exceptional
frequencies and behavior around parabolic fixed points are also included.

Mamaev I. S., Vetchanin E. V.
Dynamics of Rubber Chaplygin Sphere under Periodic Control
Abstract
This paper examines the motion of a balanced spherical robot under the action of
periodically changing moments of inertia and gyrostatic momentum. The system of equations
of motion is constructed using the model of the rolling of a rubber body (without slipping and
twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the
equations of motion admit three invariant submanifolds corresponding to planeparallel motion
of the sphere with rotation about the minor, middle and major axes of inertia. The abovementioned
motions are quasiperiodic, and for the numerical estimate of their stability charts
of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and
amplitude of the moments of inertia. It is shown that rotations about the minor and major axes
of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In
this case, the socalled “Arnol’d tongues” arise in the stability chart. Stabilization of the middle
unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments
of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It
is shown that the nonconservativeness of the system manifests itself in the occurrence of limit
cycles, attracting tori and strange attractors in phase space. Numerical calculations show that
strange attractors may arise through a cascade of perioddoubling bifurcations or after a finite
number of torusdoubling bifurcations.
