# Effects of a quadrupolar magnetic term in a Generalized Störmer problem

A Leghmouche and N Mebarki  Laboratoire de Physique Mathématique et Physique
Subatomique,
Frères Mentouri University of Constantine 1, Algeria<EMAIL_ADDRESS>
###### Abstract

A new generalized Störmer problem is proposed. The charged particles motion
around a rotating axisymmetric magnetic planet is studied using various
conditions mainly in planetary magnetospheres. It is shown that the existence
of specific trajectories is due to the quadrupolar and dipolar magnetic terms.
Moreover, the equilibrium and stability behaviors and the corresponding
orbital frequencies for various kinds of charged dust grains are treated
giving rise to a new model describing the dynamical system and leading to more
interesting results.

##  1 Introduction

Investigation of charged particles dynamics subject to the magnetic dipole has
an important interest in astrophysics. In fact, their tiny fraction of mass in
orbit around a planet [1] was first introduced in the pioneering work of
Störmer(1955) [2] mainly the theoretical study in planetary magnetospheres. It
was also considered by Mendis and Axford(1974)[3] in order to explain the
origin of the accelerated mysterious dark spokes of Saturn’s B ring in the
space region observed by Voyager 1 and 2 [4].This treatment has been of a
great interest to unveil the structure of force fields acting on the particles
and their behaviors[5].It is worth to mention that the starting point to
describe the motion of charged dust particles in a purely magnetic field
orbiting a planet with a magnetosphere is the classical Störmer problem
[6].However, the consideration of the charged dust grain is constrained by the
much small ratios between charge-to-mass which complicate the dynamics for the
heaviest particles case. In the latter case, the planetary gravity and the co-
rotational electric field have also to be taken into account and therefore the
dynamics becomes more rich but complicated in the generalized Störmer problem.
In fact, this treatment is characterized by simplified assumptions of
Keplerian gravity, specific magnetic field (pure dipole), constant charge
without radiation pressure etc.. It is very important to remind that the four
distinct Störmer problems are: CSP(classical Störmer problem) in which a
charged particle moves in a pure dipole magnetic, RSP (rotational Störmer
problem) which is characterized by the electric field,GSP(gravitational
Störmer problem) in which only the Keplerian gravity is included and finally
the most full interesting system RGSP(rotational gravitational Störmer
problem) in which both the fields action is taken into account [7] [8].So far,
many models giving exact analytical expressions have been developed.In fact,
much work has been done on dynamics of charged dust grains near non spherical
planets (prolate and oblate) [9] and also near a black holes and compact stars
under the effect of a strong magnetic field [5].Furthermore, the study of the
stability of charged dust grains orbiting a planet characterized by all charge
to mass ratios (from ions to rocks) [10] and sub-microns circumplanetary dust
grains in multipolar fields case with the inclusion of non-axisymmetric
magnetic field terms were investigated [11].In addition, the behavior of
grains near solar corona with the influence of the solar radiation
pressure[12] and the injection of charged particles into a magnetic dipole
field (MDF) is also studied.[13]

In the present work, we introduce the effect of a quadrupolar magnetic term
(QMT) [14] [15]which is not included in the so called generalized Störmer
problem.The main goal is to perform a detailed analytical treatment for
axisymmetric magnetized planet by considering various parameters present in
the effective potential [16].In section 2 we presente the model, in section 3
we study the relative equilibrium with the effect of QMT.In section 4 the
stability problem is discussed.Finally, in section 5 we draw our conclusion.
Within our present model, we analyze the dynamics of charged dust particles
verifying the magnetic dipole approximation [7] and study their relative
equilibrium and stability in order to map the trajectories of the new
dynamical system by giving specific Hamiltonian and effective potential.

##  2 The model description

Our starting point is based on the approach of ref [7].We assume a particle of
mass $m$ and charge $q$ orbiting around a rotating magnetic planet(aligned
centered planet)[16] with mass $M$ and radius $R$,while the existence of both
gravitational and electromagnetic forces is crucial[17].
The general Hamiltonian of this particle is given by:

$H=\frac{1}{2}m(\vec{P}-\frac{q}{c}\vec{A})^{2}+U(r)$ (1)

Where $c$ is the speed of light in the vacuum,$\vec{r}$ the position vector of
the particle, $\vec{P}$ the conjugate momenta of $\vec{r}$ and $\vec{A}$ the
vector potential with the following expression:

$\vec{A}(r)=\frac{\mu}{r^{3}}\left(\begin{array}[]{c}-y\\\ x\\\ 0\\\
\end{array}\right)$ (2)

Here $U(r)$ is the scalar potential which generates the electric and
gravitational interactions.The magnetic field $\vec{B}$ of the planet is
supposed to be a perfect magnetic dipole of strength $\mu$ aligned to the
north-south poles of the planet.Otherwise,the magnetosphere surrounding the
planet is taken as rigid conducting plasma that rotates with the same angular
velocity $\vec{\Omega}$ as the planet. The charge ${q}$ is subject to a
corotational electric field which has the general form:

$\vec{E}=-\frac{1}{c}(\vec{\Omega}\times\vec{r})\times\vec{B}$ (3)

In order to characterize the influence of the charge to the mass ratio of the
charged particle, we assume that the gravitational and corotational electric
effects depend on two parameters $\sigma_{g}$ and $\sigma_{r}$ taking their
values from 0 to 1[7].In our treatment and in order to get more interesting
results of the dynamic of charged particles near magnetic planet, we consider
a general expression for the magnetic field taking into account the
geomagnetic field [18]. The expression reads (in the spherical coordinates
system)

$\overrightarrow{B}(r,\theta,\varphi,t)=-\overrightarrow{\nabla}V(r,\theta,\varphi,t)$
(4)

where

$V(r,\theta,\varphi,t)=a\sum_{l=1}^{n}\sum_{m=0}^{l}(\frac{a}{r})^{l+1}(g^{\l}_{m}(t)\cos(m\varphi)+h^{\l}_{m}(t)\sin(m\varphi))P^{\l}_{m}(\cos\theta)$
(5)

and

$\overrightarrow{E}=-\frac{q}{c}\Omega\overrightarrow{\nabla}(\frac{a^{3}}{r}(-g^{0}_{1}\sin^{2}\theta)+(\frac{a^{4}}{2r^{2}}(3g^{0}_{2}\sin^{2}\theta\cos\theta))$
(6)

Here, $r$, $\theta$ and $\varphi$ are the radius, co-latitude and longitude in
spherical coordinates, $P^{\l}_{m}(\cos\theta)$ are the associated Legendre
polynomials , $g^{\l}_{m}$ and $h^{\l}_{m}$ represents the Gaussian
coefficients describing the magnetic field and $a$ is the planet radius, $l$
and $m$ are spherical harmonic degree and order respectively [19]. It is very
important to mention that the treatment used in ref [7] is a special case of
this expression($a^{3}g^{0}_{1}=1$ and $a^{4}g^{0}_{2}=0$ ),where in our case
the magnetic dipole and quadrupole terms are considered.In what follows ,we
try to understand the effect of this QMT on the generalized Störmer problem.
In this paper one can show,the dynamics of charged dust grains is described by
the following Hamiltonian taking into account the effect of a QMT:

$H=\frac{1}{2}(P^{2}_{r}+\frac{P^{2}_{\theta}}{r})+\frac{1}{2}{\omega}^{2}r^{2}\sin^{2}\theta-\frac{\sigma_{g}}{r}+\sigma_{r}\delta\frac{a^{3}g^{0}_{1}\sin^{2}\theta}{r}+\sigma_{r}\delta\frac{3a^{4}g^{0}_{2}\sin^{2}\theta\cos\theta}{2r^{2}}+U_{eff}$
(7)

where the effective potential $U_{eff}$ has the expression

$U_{eff}=\frac{1}{2}{\omega}^{2}r^{2}\sin^{2}\theta+\sigma_{r}\delta\frac{a^{3}g^{0}_{1}\sin^{2}\theta}{r}+\sigma_{r}\delta\frac{3a^{4}g^{0}_{2}\sin^{2}\theta\cos\theta}{2r^{2}}-\frac{\sigma_{g}}{r}$
(8)

Here $\delta$ is the charge of the dust grain and $\omega$ corresponds to the
orbital frequency.The notation $P_{r}$ and $P_{\theta}$ stand for the
derivatives with respect to $r$ and $\theta$ respectively. To make our study
clear and comprehensive and in order to compare our results concerning the
trajectories of charged dust grains with those of ref [7], we take
$a^{3}g_{1}^{0}$=$a^{4}g_{2}^{0}=1$.

##  3 Relative equilibrium

In order to study charged particles equilibrium,taking into account the QMT
effect and find the critical points,we have to solve the differential
equations system :

$\frac{\partial U_{eff}}{\partial\ r}=0$ (9)

and

$\frac{\partial U_{eff}}{\partial\ \theta}=0$ (10)

The sign of the particle angular velocity $\omega$ determines the nature of
the charged dust grains motion( prograde or retrograde with a positive or a
negative charge respectively) with the direction of planet motion . The
critical points of the $U_{eff}$ are found as the solutions of the system
equations

One can show that the equivalent non linear system Eqs(9),(10) can be
simplified as :

$-{\omega}^{2}r\sin^{2}\theta+\frac{1}{r}(\delta(\omega-\sigma_{r})\sin^{2}\theta+\sigma_{g})+\frac{1}{r^{3}}(3\delta(\omega-\sigma_{r})\sin^{2}\theta\cos\theta=0$
(11) $\frac{-\sin\theta\cos\theta}{r^{2}}({\omega}^{2}{\
r}^{4}+2\delta(\omega-\sigma_{r})r+3\delta(\omega-\sigma_{r})(\cos\theta-\frac{\sin\theta\tan\theta}{2})=0$
(12)

It is worth to mention that the angular velocity $\omega$ is a function of $\
r$ and $\theta$. To obtain the trajectories of charged dust particles within
this approach we follow the method used in ref [7] using the generalized
Störmer problem.Furthermore, if $\theta=\frac{\pi}{2}$
(resp.$\theta\neq\frac{\pi}{2}$) we get equatorial (resp.non equatorial or
Halos) orbits,where their existence is strictly related to the values of
$\omega$ and $\delta$.

###  3.1 Equatorial orbits

Assuming that the motion is in the equatorial plane, such kind of orbits
appear when $\theta=\frac{\pi}{2}$,and Eqs(12) is satisfied leading to the
parametric equation:

${\omega}^{2}{\
r}^{4}+2\delta(\omega-\sigma_{r})r+3\delta(\omega-\sigma_{r})(\cos\theta-\frac{sin\theta\tan\theta}{2})=0$
(13)

Figure 1: Equatorial orbits existence region for RGSP $a^{4}g^{0}_{2}=1$

Figure 2: Equatorial orbits existence region for GSP $a^{4}g^{0}_{2}=1$

The motion of particles is well described in the $(\delta,\omega)$ plane for a
generalized Störmer problem cases and the presence of trajectories depends on
the distance from the center of the planet $r$ with retrograde and prograde
orbits.This situation is similar to the case $a^{3}g^{0}_{1}=a^{4}g^{0}_{2}=1$
where the equatorial orbits exist without the quadrupolar perturbations(the
existence regions for all Störmer problem generalizations are taken into
account).

Figure 3: Equatorial orbits existence region for CSP $a^{4}g^{0}_{2}=1$

Figure 4: Equatorial orbits existence region for RSP $a^{4}g^{0}_{2}=1$

###  3.2 Halos orbits

Halos orbits ( do not cross the equatorial plane) and appear when
$\theta\neq\frac{\pi}{2}$.There description is connected to the solution of
the parametric equations depending on the partial radial $r$ and angular
$\theta$ parameters respectively. They are obtained from the solution of Eqs
(11),(12) system

$\displaystyle Q(r,\omega)$ $\displaystyle=$ $\displaystyle
324{\delta}^{4}{(\sigma_{r}-\omega)}^{4}+4{r}^{4}(-\sigma_{g}+\delta(\sigma_{r}-\omega)+{r}^{3}{\omega}^{2})$
(14) $\displaystyle+$
$\displaystyle(5{r}^{3}{\omega}^{2}+\delta(\omega-\sigma_{r})({{r}^{3}{\omega}^{2}}+{2\delta\omega-\sigma_{r}}^{2})-(27{\delta}^{3}{r}^{2}(\sigma_{r}-\omega)^{2}$
$\displaystyle+$
$\displaystyle((3\sigma_{g}-2\delta\sigma_{r})^{2}+4\delta(3\sigma_{g}-2\delta\sigma_{r})\omega+4(\delta^{2}-3\sigma_{g}{r}^{3}){\omega}^{2}+8{r}^{6}{\omega}^{4}))=0$

and

$\displaystyle A(\theta,\omega)$ $\displaystyle=$
$\displaystyle\frac{27}{128}\delta{(\sigma_{r}-\omega)}^{4}(7923\delta^{3}\omega^{2}+1408\delta^{4}(\sigma_{r}-\omega)\cos\theta+13320{\delta}^{3}{\omega}^{2}\cos
2\theta$ (15) $\displaystyle+$ $\displaystyle
576\delta^{4}(\sigma_{r}-\omega)\cos
3\theta+(7900\delta^{3}{\omega}^{2}\cos(4\theta)+(64\delta^{4}(\sigma_{r}-\omega)\cos
5\theta)$ $\displaystyle-$
$\displaystyle(8\delta(\sigma_{g}(\sigma_{r}-\omega)(2\sigma_{g}\cos\theta({\sigma_{g}}^{2}+18(\delta^{2}{(\sigma_{r}-\omega)}^{2}\sin^{4}\theta+\sin\theta(8\delta{\sigma_{g}}^{2}$
$\displaystyle(\omega-\sigma_{r})\sin(2\theta))-({\sigma_{g}}^{3}+2\delta(\sigma_{r}-\omega)\sin^{2}\theta(-5{\sigma_{g}}^{2}$
$\displaystyle+$ $\displaystyle
9\delta(\sigma_{r}-\omega)\sin^{2}\theta(2\sigma_{g}+3\delta(-\sigma_{r}+\omega)\sin^{2}\theta)))\tan\theta=0$

To calculate halos orbits around a magnetic planet, the constraint of the QMT
is also considered, In this paper we use analytical solutions to check the
existence of these trajectories which gives Eqs(16) and (17) explicitly. It is
important to remark that the obtained family of curves for the four known
generalized Störmer cases with the inclusion of the gravitational and co-
rotational fields and the individual effect are different from those of ref
[7].

Figure 5: Curves of constant $r$ and $\theta$ for Halos orbits (CSP ) with
$a^{4}g^{0}_{2}=1$

Figure 6: Curves of constant $r$ and $\theta$ for Halos orbits (GSP ) with
$a^{4}g^{0}_{2}=1$

Figure 7: Curves of constant $r$ and $\theta$ for Halos orbits (RGSP ) with
$a^{4}g^{0}_{2}=1$

Figure 8: Curves of constant $r$ and $\theta$ for Halos orbits (RSP ) with
$a^{4}g^{0}_{2}=1$

Our analysis has shown the possible families of curves defining different
particles trajectories related to both their nature and the planet rotation
direction for the four Störmer problem. For the (CSP) (classical Störmer
problem) in which a charged particle moves in a pure dipole magnetic field,
figure (5) displays in addition to the orbits found in ref [7] they are others
where the QMT is taken into account.In fact,it appears new orbits in regions
which are not allowed in the model of ref [7]. Notice that, even in the
absence of both co-rotational and gravitational fields,orbits exist for
constant radius $r$ and angle $\theta$. For the individual effect of the
gravitational force, the existence region is different from the Störmer
generalizations where the orbits(get different behaviors for fixed radius
values $r$ and $\theta$ (see figure (6)).In the case of the presence of both
the electric and magnetic fields(adding the rotational and magnetic field
(RGSP)), the non-equatorial orbits exists in all regions for various
charges.This is the most interesting case in the DMF because it introduces
simultaneous fields where the original case is modified (and may be not
circular orbits for all cases see figure (7) ). Figure (8) shows trajectories
halos in the presence of the co-rotational electric field and absence of the
gravitational field in this case for fixed $r$ and $\theta$,negative as well
as positives charges retrograde motion and prograde motion for positive
charges respectively. The used method in this work has the advantage to be
clear and helps to describe the profile of the trajectories and/or phase
spaces more easily than the numerical one.Our analysis has shown possible
families curves defining various particles trajectories related to their
nature and the planet rotation direction for the four Störmer problems cases.

Figure 9: 3D plot for RGSP with $a^{4}g^{0}_{2}=0$

Figure 10: 3D plot for GSP with $a^{4}g^{0}_{2}=0$

Figure 11: 3D plot for CSP with $a^{4}g^{0}_{2}=0$

Figure 12: 3D plot for RSP with $a^{4}g^{0}_{2}=0$

###  3.3 Dynamical behavior and 3d plots

Figures [9–16] (resp.[13–16]) show the dynamical behavior with DMF (resp.QMT)
of radius $r$ as a function of the orbital frequency $\omega$ and the charge
$\delta$. Notice that,the individual and simultaneous effect of the
gravitational and corotational electric fields modify the dynamics of charged
dust grains particles located near a rotating magnetic planet.Furthermore,the
introduction of a quadrupolar perturbations gives important results.

Figure 13: 3D plot for RGSP with $a^{4}g^{0}_{2}=1$

Figure 14: 3D plot for GSP with $a^{4}g^{0}_{2}=1$

Figure 15: 3D plot for CSP with $a^{4}g^{0}_{2}=1$

Figure 16: 3D plot for RSP with $a^{4}g^{0}_{2}=1$

Figure 17: Stabilty regions for RGSP with $a^{4}g^{0}_{2}=1$

Figure 18: Stabilty regions for GSP $a^{4}g^{0}_{2}=1$

Figure 19: Stabilty regions for RSP with $a^{4}g^{0}_{2}=1$

##  4 Stability

In our treatment,stability of equatorial and halos orbits is related to
$\delta$ and $\omega$ parameters.Following ref [7],their stability is
connected to the local minimum of the effective potential $U_{eff}$.The
hessian matrix is defined by the second derivatives of $U_{eff}$ potential and
given by:

$\frac{\partial^{2}U_{eff}}{\partial\
r^{2}}=\frac{1}{r^{8}}\bigg{(}-5\sigma_{g}+\bigg{(}r^{2}(\delta^{2}+2\delta\
r^{3}(\sigma_{r}-3\omega)+3\ r^{6}{\omega}^{2})+3\delta
r(2\delta+r^{3}(3\sigma_{r}-7\omega))\cos\theta+9\delta^{2}\cos^{2}\theta\bigg{)}\sin^{2}\theta\bigg{)}$
(16)

$\displaystyle\frac{\partial^{2}U_{eff}}{\partial\theta^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{32r^{6}}\bigg{(}99\delta^{2}+64\delta^{2}r^{2}+128\delta
r^{5}\omega+64r^{8}\omega^{2}+240\delta^{2}r\cos\theta-12\delta
r^{4}\sigma_{r}\cos\theta+252\delta r^{4}\omega\cos\theta$ (17)
$\displaystyle+$ $\displaystyle 108\delta^{2}\cos
2\theta+64\delta^{2}r^{2}\cos 2\theta+64\delta r^{5}\sigma_{r}\cos
2\theta+64\delta r^{5}\omega\cos 2\theta+32r^{8}\omega^{2}\cos 2\theta$
$\displaystyle+$ $\displaystyle 144\delta^{2}r\cos 3\theta+108\delta
r^{4}\sigma_{r}\cos 3\theta+36\delta r^{4}\omega\cos 3\theta+81\delta^{2}\cos
4\theta\bigg{)}$

$\displaystyle\frac{\partial^{2}U_{eff}}{\partial r\partial\theta}$
$\displaystyle=$
$\displaystyle\frac{1}{8r^{8}}\bigg{(}(\delta^{2}(387+112r^{2})+32\delta
r^{5}(\sigma_{r}-3\omega)-48r^{8}\omega^{2})\cos\theta+3\delta(76\delta
r+12r^{4}\sigma_{r}$ (18) $\displaystyle+$ $\displaystyle
16r^{4}\omega+4r(29\delta+r^{3}(9\sigma_{r}-16\omega))\cos
2\theta+87\delta\cos 3\theta)\bigg{)}\sin\theta$

Figure 20: saddle unstable for RGSP

Figure 21: 3d Curve of saddle unstable RGSP

Figure 22: saddle unstable for GSP

Figure 23: 3d Curve of saddle unstable GSP

Figure 24: saddle unstable for RSP

Figure 25: 3d Curve of saddle unstable RSP

Figure 26: saddle unstable for CSP

Figure 27: 3d Curve of saddle unstable CSP

###  4.1 Stability of equatorial orbits

For the equatorial orbits ,we take $\theta=\frac{\pi}{2}$ and the radius $r$
from the generalization of kepler’s third law [7].Due to the obtained
complicated equations where the mixed derivatives are not vanishing,the
Hessian matrix determinant will allow to describe new stability regions in the
equatorial plane (which does not exist in the work of ref[7]) taking into
account the QMT and for each case of a generalized Störmer problem,the curves
obtained from the Hessian matrix determinant are shown in figures
[17–19].Notice that,for the two kinds of charged particles in various ranges
of the orbital frequency $\omega$.Significants modifications appear in the
region of stability treated with each effect of dipolar field.However,for the
classical Störmer case the trivial solutions can not explain the stability in
these conditions.

###  4.2 Stability of Halos orbits

It is more complicated to define stability regions for this kind of
trajectories.This is due to expression of the Hessian matrix with non
vanishing partial derivatives.The substitution of the radial and angular
parameters $r$ and $\theta$ leads to a relationship between $\omega$ and
$\delta$ .In this paper,we have treated the fourth cases considered in ref [7]
with the individual and simultaneous effects of gravitational and corotational
electric fields.Furthermore,to give a detailed physical description on the
stability of Halos orbits parallel to equatorial one needs more computational
sophisticated methods.Here,we have investigated the problem for both negative
and positive charged particles stability using boundary conditions of the
Hessian matrix and critical points.It is very important to point out that
contrary to the dipolar approximation which present some stability regions
(see figures [20–27]) the non equatorial orbits are unstable for positive
and/or negative charged particles.

##  5 Conclusion

In this work we have studied new contributions compared to those of ref [7]
due to the quadrupolar component of the magnetic field with axisymmetric
combination of gravitational and co-rotational electric fields for typical
charge to mass ratio.It turns out that,the QMT dynamics perturbation give
interesting results where the dynamics is more sensitive to the space
parameters compared to the dipolar one and the occurrence of new orbits in the
$\delta$\- $\omega$ plane.Regarding the equilibrium state, the existence of
equatorial orbits of charged dust grains is mainly related to DMF conditions
for all Störmer cases.For Halos(non-equatorial) orbits the situation is more
interesting,and their existence is significant with the simultaneous influence
of co-rotational and gravitational fields besides the cases where the
individual effects of gravity and electric field are not neglected giving rise
to extra trajectories which are not allowed in the regions presented in ref
[7].Moreover,their behavior is more complicated leading to new interesting
physical results. Furthermore and contrary to ref [7],small charge to mass
constraint in all cases of fields can give halos orbits at fixed $(r,\theta)$
parameters even if the gravity is not present. Notice that the stability of
trajectories is strongly dependent to the effect of $g_{2}^{0}$ . Finally,our
approach can be useful for the dynamics near spherical rotating magnetic
planets by changing parameters $g_{m}^{l}$ in particular $g_{1}^{0}$ where in
ref [7] are taken to be $a^{3}g_{1}^{0}=1$ without any justification.Moreover
it gives the nature of the motion of charged particles direction (prograde or
retrograde) which can be very interesting from observational point of view
taking into account not only the dipolar (like in ref [7] ) but the new
quadrupolar contribution.We believe that other new results can be obtained for
multipolar contributions (more study is under investigation) .

##  6 Acknowledgments

We are very grateful to the Algerian Ministry of Higher Educations and
Scientific Research and DGRSDT for the financial support.

## References

## References

  * [1] Hamilton D P 1993 _Icarus_ 101 244-2
  * [2] Mitchell C J, Horãnyi M and Howard J E 2003 _J Geophys Res: Space Physics_ 108 1-10
  * [3] Mendis D A, Axford W I 1974 _Ann. Rev. of Earth and Planets. Sci_ 2 419-474
  * [4] Shukla P K and Mamun A A 2002 Introduction to Dusty Plasma Physics  (Bristol: IOP Publ)
  * [5] Kovãr J, Stuchlik Z and Karas V 2012 _Class Quantum Grav_ 25 1-17
  * [6] Inãrrea M, Lanchares V, Palacian J F, Pascual A I, Pablo Salas J and Yanguas P 2012 _Physica._ D 241 1026-1042
  * [7] Dullin H R, Horãnyi M and Howard J E 2002 _Physica._ D 171 178-195
  * [8] Inãrrea M, Lanchares V, Palacian J F, Pascual A I, Pablo Salas J and Yanguas P 2004 _Physica._ D 197 242-268
  * [9] Inãrrea M, Lanchares V, Palacian J F, Pascual A I, Pablo Salas J and Yanguas P 2009 _Chaos Solitons and Fractals_ 42 155-169
  * [10] Jontof-Hutter D and Hamilton D P 2012 _Icarus_ 218 420-432
  * [11] Jontof-Hutter D and Hamilton D P 2012 _Icarus_ 220 487-502
  * [12] Maravilla D 1999 _Geofisica internacional_ 38 1-11
  * [13] Kolesnikov E K 2017 _J Geomagnetism and Aeronomy_ 57 137-146
  * [14] Leghmouche A, Mebarki N and Bouldjedri A , 2015 _Proc. Int, Conf on Conférence ”Lumiére et Astronomie”_(Algeria) ISBN 978-9947-0-4967-9
  * [15] Leghmouche A, Mebarki N, 2017 _Proc. Int, Conf on 4th Algerian Conference on Astronomy and Astrophysics_(Algeria) (to appear in IOP journal of conference series publishing)
  * [16] Howard J E, Dullin H R and Horãnyi M 2000 _Phys. Rev. Lett_ 84 3244-3247
  * [17] Howard J E, Horãnyi M and Stewart G R 1999 _Phys. Rev. Lett_ 83 3993-3996
  * [18] Alfvén H and Falthammar C G 1963 Cosmical Electrodynamics: Fundamental Principles (Oxford:Clarendon Press)
  * [19] Schubert G and Soderlund K.M. 2011 _Physics of the Earth and Planetary Interiors_ 187 92-108


Purdue University, West Lafayette, IN, USA. He is also working as a Technical
Specialist in the Connected and Intelligent Systems group, Cummins Research
and Technology, Columbus, IN, USA. His research interests are in the areas of
Federated learning, distributed optimization, systems, and optimal controls.
---|---
|  Liangqi Yuan (S’22) received the B.E. degree from the Beijing Information
Science and Technology University, Beijing, China, in 2020, and the M.S.
degree from the Oakland University, Rochester, MI, USA, in 2022. He is
currently pursuing the Ph.D. degree with the School of Electrical and Computer
Engineering, Purdue University, West Lafayette, IN, USA. His research
interests are in the areas of sensors, the Internet of Things, signal
processing, and machine learning.
---|---
|  Christopher G. Brinton (S’08, M’16, SM’20) received the M.S. and Ph.D.
(with Hons.) degrees in electrical engineering from Princeton University,
Princeton, NJ, USA, in 2013 and 2016, respectively. He is the Elmore Assistant
Professor with the School of Electrical and Computer Engineering, Purdue
University, West Lafayette, IN, USA. Prior to joining Purdue University, he
was the Associate Director of the EDGE Lab and a Lecturer of Electrical
Engineering with Princeton University. His research interest is at the
intersection of networked systems and machine learning, specifically in
distributed machine learning, fog/edge network intelligence, and data-driven
network optimization. Dr. Brinton is a recipient of the NSF CAREER Award, the
ONR Young Investigator Program Award, the DARPA Young Faculty Award, and the
Intel Rising Star Faculty Award. He currently serves as an Associate Editor
for IEEE Transactions on Wireless Communications, in the ML and AI for
wireless area.
---|---
|  Stanislaw H. Żak (LM’21) received the Ph.D. degree in electrical
engineering from Warsaw University of Technology, Warsaw, Poland, in 1977.
From 1977 to 1980, he was an Assistant Professor with the Institute of Control
and Industrial Electronics, Warsaw University of Technology. From 1980 to
1983, he was a Visiting Assistant Professor with the Department of Electrical
Engineering, University of Minnesota, Minneapolis, MN, USA. In 1983, he joined
the School of Electrical and Computer Engineering, Purdue University, West
Lafayette, IN, USA, where he is currently a Professor. He has been involved in
various areas of control, optimization, fuzzy systems, and neural networks. He
has coauthored Topics in the Analysis of Linear Dynamical Systems (Polish
Scientific Publishers, 1984) and An Introduction to Optimization: With
Applications to Machine Learning, 5th edition (Wiley, 2024), and has authored
Systems and Control (Oxford University Press, 2003). He was an Associate
Editor of Dynamics and Control and the IEEE Transactions on Neural Networks.
---|---
|  Ziran Wang (S’16-M’19) received the Ph.D. degree in Mechanical Engineering
from the University of California, Riverside in 2019. He is a tenure-track
assistant professor in the College of Engineering at Purdue University, where
he directs the Purdue Digital Twin Lab. Prior to this, Dr. Wang was a
principal researcher at Toyota Motor North America R&D in Mountain View,
California. Dr. Wang serves as the founding chair of the IEEE Technical
Committee on Internet of Things in Intelligent Transportation Systems, and an
associate editor of four journals. His research focuses on automated driving,
human-autonomy teaming, and digital twins.
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