The uncertainty dimension and fractal boundaries for chargedparticle dynamics in the magnetotail

D. L. Holland,1 M. E. Presley,1 R. F. Martin,1 and H. Matsuoka1

Received 27 September 2010; revised 17 March 2011; accepted 3 April 2011; published 3 August 2011.

[1] In this paper we examine the fractal nature of the basin boundary between forwardscattered and backscattered particles as measured in the asymptotic region of the modifiedHarris model of the Earth magnetotail. It is shown that, in order to enter the chaotic regionof phase space, an incoming ion (launched from above the midplane) must have anasymptotic pitch angle below a certain maximum value. This maximum pitch angledepends on the underlying structure of the phase space and takes on minimum (maximum)values at off‐resonant (resonant) energies. Examples of the fractal basins are shown forboth a resonant and off‐resonant energy. Furthermore, we calculate the uncertaintyexponent and the associated fractal dimension of the basin boundary as a function of theion energy. We find that the uncertainty exponent takes on maximum (minimum) valuesat off‐resonant (resonant) energies indicating that the box counting dimension of thebasin boundary is farthest from integer values at the off‐resonant energies. Finally, weshow that in the integrable limit of vanishing normal component of the magnetic field,the uncertainty exponent approaches zero.

Citation: Holland, D. L., M. E. Presley, R. F. Martin, and H. Matsuoka (2011), The uncertainty dimension and fractalboundaries for charged particle dynamics in the magnetotail, J. Geophys. Res., 116, A08207, doi:10.1029/2010JA016146.

1. Introduction

[2] The motion of charged particles in the Earth’s magne-totail has received a lot of attention over the years. Earlyresearch in the area [Speiser, 1965;West et al., 1978;Wagneret al., 1979] focused on single particle trajectories in the con-text of adiabatic theory and deviations therefrom [Sonnerup,1971; Birmingham, 1984]. While sensitive dependence oninitial conditions was noted in these early works, it was onlyin the late 1980’s that the particle motion was shown to bechaotic [Chen and Palmadesso, 1986; Büchner and Zeleny,1986; Martin, 1986].[3] A simple model for single particle dynamics in the

magnetotail consists of Newton’s 2nd Law with the LorentzForce

mdvdt

¼ q

cv� B ð1Þ

coupled with the modified Harris model of the magneticfield,

B ¼ B0 tanh z=�ð Þex þ bzezð Þ ð2Þ

where we are using standard GSE coordinates. B0 is theasymptotic value of the x component of the magnetic field farfrom the current sheet, d is the scale length of the fieldreversal, Bz is the magnitude of the constant magnetic field

normal to the current sheet, and bz = Bz /B0. Alternatively, wemay write the Lagrangian for the motion as

L ¼ 1

2m v2x þ v2y þ v2z

� �þ q

cvyAy x; zð Þ ð3Þ

where the vector potential is given by

Ay x; zð Þ ¼ �B0 � ln cosh z=�ð Þ½ � � bzxð Þ ð4ÞMany of the important dynamical features of this system aredescribed in detail in the review byChen [1992]. In this paperwe will briefly review those aspects that are relevant to ourproblem. Since the Lagrangian is independent of time and yis a cyclic coordinate, we can immediately observe thatboth the total energyH = 1

2mv2 and the y canonical momentum

Py = mvy +qc Ay(x, z) are constants of the motion. In addition,

since @Ay

@x = Bz is a constant, a third conserved quantity is givenby Cx = m(vx − Wzy) where Wz = qBz /mc is the cyclotronfrequency in the weak normal component of the magneticfield. It should be noted, that since Cx and Py are not ininvolution, i.e. {Cx, Py} = −mWz ≠ 0, the constants are notindependent of each other and thus the system is in generalnonintegrable.[4] It is often convenient to write equations (1) and (2) in

dimensionless form as

d2X

d�2¼ dY

d�ð5Þ

d2Y

d�2¼ d

d�b�2z ln cosh bzZð Þ½ � � X

� � ð6Þ

d2Z

d�2¼ �b�1

z tanh bzZð Þ dYd�

ð7Þ

1Department of Physics, Illinois State University, Normal, Illinois,USA.

Copyright 2011 by the American Geophysical Union.0148‐0227/11/2010JA016146

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where we have used the traditional normalizations X ≡ (x −Py /mWz)/bzd, Y ≡ (y + Cx /mWz)/bzd, Z ≡ z/bzd, t ≡ Wz t andwe have explicitly assumed that bz ≠ 0. Using these nor-malizations, we find that the normalized energy of the par-ticle is given by

H � H= mb2zW2z �

2� � ð8Þ

[5] Two important symmetry properties of equations (5)–(7)are that (1) the equations are invariant under the replacementZ → −Z and (2) the equations are invariant under the simul-taneous replacements of Y → −Y and t → −t. The lattersymmetry will be important because it indicates that Y reversaland t reversal are equivalent. Additionally, at the midplane,Z = 0, to within a constant _X = Y and _Y = −X.[6] Numerical solutions to these equations of motion

show that there are three basic types of orbits, commonlyreferred to as (1) transient (Speiser) orbits, (2) chaotic orbits,and (3) integrable (regular, trapped) orbits. Figures 1a, 1b,and 1c show examples of each of these types of orbits. Afourth class of orbit (Figure 1d) is actually a stronglytrapped chaotic orbit. Since these orbits are caught forlong periods of time in the current sheet, we refer to themas sticky orbits. While the particle is caught, these orbitsare almost indistinguishable from regular orbits.

[7] Due to the wide variety of possible orbits, detailedanalysis of individual orbits is of somewhat limited practicaluse. Instead, we follow the method of Chen and Palmadesso[1986] and study the dynamical system using Poincarésurfaces of section (SoS). As is usual, the surface of sectionis constructed at the midplane, Z = 0 for a fixed energy Hand a fixed value of bz. Every time the particle passesthrough midplane we mark its location in the (X, _X ) plane.(Note that due to the properties of the equations discussedabove, we could also consider this to be the (X, Y) plane orthe (− _Y , _X ) plane, however, it is more common to useconjugate variables.) On first crossing the midplane, allparticles will enter the surface of section through the regionC1, and due to the symmetries discussed above, their lastcrossing will be in the mirror image of C1 across the line_X = 0. After entering the SoS, the particles will proceedthrough the regions Ci in order of increasing i. If a portion ofa region Ci falls within the exit region, then the particles inthat region will escape after a single transit of the midplane.For those portions of the last region Ci that do not fall withinthe exit region, the particles in those regions will enter intothe chaotic region of the SoS and become trapped for anindeterminate amount of time. These chaotic particlescan end up making from one additional transit of the mid-

Figure 1. (a–d) Examples of the different types of orbits. Each of the orbits has energy H = 500. Figures 1a,1b, and 1d only vary in initial pitch and phase angles. Figure 1a is a transient orbit which has a single interactionwith the current sheet. Figure 1b is a chaotic orbit which is mirrored back into the current sheet multiple times.Figure 1c is a regular or integrable orbit, which is trapped in the neighborhood of the midplane for all time.Figure 1d is a sticky orbit whose trajectory brings it close to the integrable region of phase space. Sticky orbitsin reality are just chaotic orbits but for long periods of time are almost indistinguishable from regular orbits.

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plane up to thousands (and possibly more) of transits. Twoexamples of the surface of section plots are shown inFigures 2a and 2b. Note the high degree of symmetry acrossthe _X = 0 line exhibited by Figure 2a as compared toFigure 2b. The nested lines in the regions Ci correspond tothe asymptotic pitch angles, b, of the incoming particlesin 10° increments where the most field aligned (b = 180°)is the central dot and the outer most curves have a pitchangle of b = 90°. Whereas any value of the asymptoticphase angle � (for fixed b) is mapped to the appropriatecurve in the SoS, it is not a simple mapping. Rather onefinds that the full range of 0° ≤ � ≤ 360° can be greatlystretched and wrapped around the constant b curve in theSoS many times. The exact amount of stretching that occursis a function of the energy, the pitch angle, and the range ofphase angles. We have observed cases in which anasymptotic phase angle range of less than one degree hasbeen mapped onto the entire range of the constant b curvein the SoS.[8] If a SoS has a high degree of symmetry (e.g., Figure 2a),

the majority of particles will be transient, and there will besome maximum pitch angle beyond which particles may notenter into the chaotic region. If a surface of section is asym-metric (compare Figure 2b), even field aligned particles mayenter the chaotic region, and a large percentage of the parti-cles are chaotic. An early finding by Burkhart and Chen[1991] is that as the energy is increased, the surface of sec-tion alternates between symmetry and asymmetry. Numericalfits to the data show that the symmetric surface of sectionshave energies given by H1/4 = N + 0.6 where N is a integer. It

is this variation in the symmetry properties of the surface ofsection and the resulting partitioning of the particles intotransient and chaotic orbits that lead to the distributionfunction signatures that have been observed in satellite data[Chen et al., 1990a; Holland et al., 1999, 2006, 2008]. InFigure 3, we plot the value of bc as a function of H1/4. Notethat it also has minimum values at the resonant energies.

2. Fractal Boundaries

[9] Shortly after it was shown that the charged particledynamics in the magnetotail are chaotic, Chen et al. [1990b]examined the chaotic entry region at the midplane for thesingle case of H = 500 and bz = 0.1 and found that it didindeed have a fractal structure for the boundary betweenforward scattered and backscattered particles. Rather thanfocusing on the midplane, in this paper we look at the basinboundary in the asymptotic source region. We do this fortwo reasons. First off, in chaotic scattering systems such as thecurrent sheet, it is often the case that one wants to understandthe dynamics in terms of the initial conditions of the orbitrather than at some point in the middle of the system.Secondly when calculating the uncertainty exponent as insection 3), the variation of initial conditions corresponds tothe physically meaningful variations in pitch and phaseangles. Figures 4a and 4b show examples of the particlesource region boundaries for the entire range of incomingpitch and phase angles i.e. 90 ≤ b ≤ 180 and 0 ≤ � ≤ 360.Figure 4a is for H1/4 = 2.6 i.e. the second resonant energyand Figure 4b is for H1/4 = 2.1 the first off‐resonant energy.

Figure 2. Examples of surface of section plots for (a) the second resonance energy (H1/4 = 2.6) and (b) thefirst off‐resonance energy (H1/4 = 2.1). All particles approaching the asymptotic region initially enterthrough region C1 and then proceed through regions C2 and C3. Those particles that are in the escape region(the mirror image of C1 across the _Y = 0 line) escape and are transient orbits. The rest of the orbits are mir-rored back into the system and enter the chaotic region (B). Note that due to the symmetry (asymmetry) ofthe resonant (off‐resonant) surface the majority of the particles are transient (chaotic). Region A is occupiedby regular orbits. In the absence of collisions, particles in region A are trapped forever and will never enterregion B. The nested curves in the transient regions correspond to fixed asymptotic pitch angles with theboundary between regions B and C being particles with an asymptotic pitch angle of 90°.

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The light color represents forward scattered particles and thedark color represents backscattered. Even more detailedstructuring appears if rather than simply looking at forwardand backward scattering, we color code the data according toanother parameter, such as the number of midplane crossingsor the trapping time of the particle. Note that in Figure 4a,beyond bc ’ 120° there are no fractal boundaries becauseall of the particles fall in the transient region. For Figure 4bon the other hand, since even antifield aligned particlesmay enter the chaotic region, the fractal boundary exits forall possible values of b and �.[10] On first glance, it appears that the fractal boundaries

for the resonant and off‐resonant energies exhibit X and Olike structures at well defined locations in the entry regions.These structures are not real but are purely the effects ofusing uniformly spaced data points. This can be demon-strated either by using a different spacing of the data points,in which case the X and O structures change locations orby zooming in on regions that appears to have well definedO structures, in which case the apparent structuring dis-appears entirely. A similar artificial structuring was notedby Chen et al. [1990b] in the midplane entry region. InFigures 5a, 5b, 5c, and 5d, we zoom in on progressivelysmaller regions of the chaotic region for the case H1/4 =2.1 Similar effects can be seen for the case H1/4 = 2.6, orfor that matter any other energy level. More example canbe found on the Web site http://www.phy.ilstu.edu/ holland/publications.html.[11] In Figure 5a, we zoom in on the region 130 ≤ b ≤ 150

and 50 ≤ � ≤ 100, an area that apparently exhibits an O‐likestructure in Figure 4b. Note that at the higher resolution wecan clearly see that the actual structure has bands of forwardand back scattering interspersed with chaotic bands. InFigures 5b, 5c and 5d we show the chaotic structure of thebasin boundary at higher and higher levels of magnificationwith Figure 5d being magnified by a factor of 180000 in band 360000 in �. At each level of magnification we see thesame basic structuring of a well defined band of forwardscattered and backscattered particles interspersed with cha-otic regions. It is also interesting to note that in the chaoticregions, the structuring gets finer the closer one is to one of

Figure 3. Maximum allowable pitch angle bc for a particle to enter into the chaotic region of phasespace. (Note this assumes that particles are launched from the north. Similar arguments can be madefor particles launched from the south.) In comparing with Figure 2 we see that at resonance only parti-cles that have pitch angles near 90° can access the chaotic regions of phase space, whereas at off reso-nance even antifield aligned particles may enter the chaotic region.

Figure 4. Comparison of the basin boundary for two ener-gies. Black represents backscattered particles, and whiterepresents forward scattered particles. (a) The fractal forthe second resonant energy H1/4 = 2.6. (b) The fractalfor the first off‐resonant energy, H1/4 = 2.1. The patternsof ellipses in both images are not an image of the fractalstructure but merely a result of using a uniform grid atan angle with the lines in the fractal structure.

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the well defined forward scattered or backscattered bands.We also note in Figure 5d the seeming reappearance ofcurved bands in the chaotic regions. Once again these arenot real but are caused by the use of a regular grid.

3. Uncertainty Exponent and Fractal Dimension

[12] Having constructed the basic fractal in the previoussection, we now calculate the dimension of the fractal.Rather than directly calculating the fractal dimension of thebasin boundary, for chaotic scattering systems it is mucheasier to measure the uncertainty exponent. A given trajec-tory is said to be � uncertain if another trajectory with initialconditions that vary by a random amount � ends up in adifferent basin. In our case, the two basins are for forwardscattered and backscattered particles. Theoretically [Ott,2002], if we take a large number of initial trajectories, thenthe fraction of them that are � uncertain, f (�), is expected toscale as

f �ð Þ � �� ð9Þ

where a is the uncertainty exponent. Equation (9) indicatesthat if we wish to reduce f (�) by a factor of 10, we need toreduce � by a factor of 101/a. Once we have the uncertainty

exponent a, we may use it to determine the box countingdimension of the fractal, D0, from the relation

� ¼ N � D0 ð10Þwhere N is the imbedding dimension (in our case N = 2).[13] We begin our calculation by determining if a given

trajectory is chaotic or transient. (Note, for transient orbits,small changes in initial conditions do not significantly affectthe outcome unless they are right on the boundary betweenthe chaotic and transient regions of phase space.) If the orbitis found to be chaotic, we then select four nearby orbitswhose initial conditions vary in either pitch or phase angleby an amount � (Figure 6). If any one of the nearby tra-jectories ends up in a different basin, the initial orbit is takento be � uncertain. We then reduce the magnitude of � andrepeat the calculations for the range 10−9 ≤ � ≤ 10−3. Theprocess is repeated until we have 5000 initial conditionsfor a given energy level. In Figure 7 we show an example ofa log‐log plot of f (�) as a function of � for the first off‐resonant energy. For this case, the uncertainty exponent isa = 0.044, which yields a box counting dimension of 1.956.The fact that the uncertainty exponent is so small tells usthat in order to reduce f (�) by a factor of 10, we need toreduce � by a factor of almost 1011.

Figure 5. (a–d) Magnifications of the original fractal shown in Figure 4b. The circular structures dis-appear after the first magnification, providing evidence that they are not real but a result of using a regulargrid. Note in Figure 5c that as the lines get closer to a band of stability, they become narrower and morechaotic. Also note the recurrence of the circular structures in Figure 5d.

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[14] Due to the phase space partitioning detailed in theintroduction, many attributes of the particle dynamics havebeen found to exhibit different behaviors on and off theresonant energies defined by H1/4 = N + 0.6 [cf. Chen, 1992,and references therein]. The question naturally arises as towhat effect changing the particle energy may have on theuncertainty exponent/box counting dimension. In Figure 8(top, middle, and bottom) we have plotted a as a functionof H1/4 for bz = 0.05, 0.1, and 0.15. The first thing we noticeis that a has local minima at the resonant energies, and localmaxima at off‐resonant energies (which we take to be mid-

way between the resonant energies, i.e. H1/4 = N + 0.1).Secondly, we see that for smaller values of bz there are morewell defined resonances. This effect has been previouslynoted in other properties and is also discussed by Chen[1992]. Thirdly, in general, for any given energy, the valueof a, although it is still small, is larger for larger values of bz.This third effect is further investigated in Figure 9, where wehave plotted a as a function of bz for a resonant and an off‐resonant energy. In both cases we see that as bz → 0, a → 0.This is the expected result since, in the limit as bz → 0,equations (1) and (2) are completely integrable, and thereforewe will not have a chaotic region of phase space.

4. Discussion

[15] In their investigation of the fractal basin boundariesfor the single case of H = 500 and bz = 0.1 Chen et al.[1990b] also used the uncertainty exponent to determinethe dimension of the fractal. However, they did it in twodifferent ways. In the first, they threw out orbits that weretrapped for too long, and in the second they kept all orbits.The first method yielded an uncertainty exponent that issignificantly larger than what we measured; however, thereis no real reason to eliminate the long‐lived orbits. In thesecond case they kept all of the orbits regardless of thetrapping time and found a result that is identical to the onepresented in this paper for the identical case. This is not asurprising result since their second method only differs fromthe method used here in that the orbits are initialized fromthe midplane rather than the asymptotic region.[16] In this paper, we have significantly expanded on the

work of Chen et al. [1990b] by examining the fractal natureof the basin boundary between forward scattered andbackscattered particles in the modified Harris model of themagnetotail as a function of both the particle energy andthe magnetic field ratio bz. We have found that the value ofa tends to be lowest at resonant energies and for smallervalues of bz. In the case of bz going to zero, this has beeninterpreted as being the result of approaching a completelyintegrable system with no fractals and no chaos. For the caseof resonant energies, the interpretation is not as clear cutsince at all energies we do indeed have chaotic orbits. It isinteresting to note that the average exponential divergencerate for chaotic orbits displays similar behavior as a function

Figure 6. Method used to determine if a particle is � cer-tain. A particle with the coordinates �, b is shown with fourmore particles, each a distance � from the original particle.Two possible basin boundaries are represented by the solidand dashed lines. A particle is said to be � certain if all fiveparticles fall on the same side of the basin boundary. Thisparticle is � certain with respect to the solid line. A particleis � uncertain if one or more of the surrounding particles donot fall on the same side as the original particle. This particleis � uncertain with respect to the dashed line.

Figure 7. Log‐log plot of f (�) as a function of � for H1/4 = 2.1, the first off‐resonance energy, where f (�)is the fraction of particles that are f (�) uncertain.

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of energy and field ratio, i.e. it has a maximum value at off‐resonant energies and approaches zero as bz goes to zero[Chen, 1992; Chen and Holland, 1993]. One needs to becareful in relating the chaotic nature of the orbits (as mea-sured by the average exponential divergence rate forinstance) to the fractal nature of the source region. As is wellknown, they are both indicative of extreme sensitivity toinitial conditions but one is a temporal effect and the other aspatial effect. For chaotic orbits, no matter how well weknow the initial conditions, the particles will divergeexponentially in phase space. Thus, although two particleswith nearly identical conditions may (or may not) eventuallyland in the same basin, they will still diverge from eachother in phase space and may cross the midplane a vastlydifferent number of times. This effect was demonstrated by

Chen et al. [1990b] where they plotted the surface of sec-tion, color coding the initial conditions to the number oftimes the orbits actually cross the midplane. The same basicfractal structure is still present, but with additional bandsrepresenting the numbers of midplane crossing. As onewould expect, the fractal structuring based on the number ofmidplane crossing of the orbit (rather than the final state ofthe orbit) is also present in the asymptotic region. For par-ticles whose initial conditions are distributed throughout aregion with fractal basin boundaries, the actual particledynamics may or may not be chaotic, but there will still beextreme sensitivity on initial conditions. The magnetotailchaotic scattering system examined in this paper exhibitsboth the fractal basin boundaries and exponential divergenceof nearby orbits.

Figure 8. Shown is a as a function of H1/4 for bz = 0.15,0.1 and 0.05. Note that in all cases the a takeson local minima at the resonant energies.

Figure 9. Shown is a as a function of bz for the second resonant energy H1/4 = 2.6 (solid circles) andfirst off‐resonant energy H1/4 = 2.1 (solid squares). Note that in both cases, as bz → 0, a → 0. This is theexpected result since in this limit the equations of motion are completely integrable.

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[17] Acknowledgments. This work was supported by Illinois StateUniversity research grants.[18] Philippa Browning thanks Anastasios Anastasiadis and another

reviewer for their assistance in evaluating this paper.

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D. L. Holland, R. F. Martin, H. Matsuoka, and M. E. Presley, Departmentof Physics, Illinois State University, Normal, IL 61790‐4560, USA.(holland@phy.ilstu.edu)

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